Mas_thesis.

Mas_thesis.
[?]
Master thesis - Electrical Engineering
Modeling of wave propagation in open domains
A Krylov subspace approach
4047621
Delft University of Technology
CAS–MS–2014–04
Jörn Zimmerling
6 of June 2014
Specialization Mircoelectronics,
Faculty of Electrical Engineering,
Mathematics and Computer Science
Modeling of wave propagation in open domains (May 20, 2014)
Typeset by the author in Nimbus Roman 11 pt using LATEX.
c 2014 Jörn Zimmerling
Copyright All rights reserved.
Preface
In this Master thesis we investigate the use of several Krylov subspace methods for approximating timeand frequency-domain solutions of wave equations in open structures, including dispersive media. The
research presented here is in the process of being published at the time of the defense.
This thesis is submitted in partial fulfillment of the requirements for the degree of Master of Science in
Electrical Engineering at Delft University of Technology.
The research described in this thesis was a joint project between two research groups, namely, the Circuits and Systems Research Group and the Optics Research Group. The thesis committee consist in equal
parts of people form both groups. The Circuits and Systems Research Group is represented by its head
prof.dr.ir. A.-J. van der Veen and dr.ir. R.F. Remis, whereas the Optics Research Group is represented by
its head prof.dr. H.P. Urbach and ir. L. Wei.
Corrections
This is the second version of this thesis, where the feedback of the committee has been processed. In this
paragraph the changes with respect to the first version are listed. I would like to thank the committee for
the feedback on the thesis and pointing out mistakes.
• Figures 4.27 and 4.26 are updated as the medium parameters in the text did not correspond to the
ones used in the simulation.
• Minor language and notation corrections
Jörn Zimmerling
Delft, May 2014
Acknowledgments
First and foremost I would like to express my gratitude to my supervisor Rob Remis, for his guidance
throughout the past years. Further I am thankful to him for motivating me for science, and his patience
during reading and correcting this thesis.
I would like to thank my supervisor for optics related subjects Lei Wei, he offered great support especially in my first month at the institute. I learned a lot from Lei during our technical discussions.
I am sincerely grateful to Vladimir Druskin and Mikhail Zaslavsky from Schlumberger-Doll-Research
i
for their input and feedback on my work. Both are leading experts in their field and I could benefit greatly
from their advice and expertise.
I would like to acknowledge and express my gratitude to the Optics Research group for the valuable
discussions and motivational coffee breaks. Without the group the past year would have been less enjoyable.
Finally, I would like to thank my family and friends for their support during the past years.
ii
Modeling of wave propagation in open domains
A Krylov subspace approach
This thesis is submitted in partial fulfillment of the requirements for the degree of
M ASTER OF S CIENCE
in
E LECTRICAL E NGINEERING
by
Jörn Zimmerling B.Sc
born in Pinneberg, Germany
The work presented in this thesis was performed at:
The Circuits and Systems Group
Department Microelectronics & Computer Engineering
Faculty of Electrical Engineering, Mathematics and Computer Science
Delft University of Technology
and
The Optics Research Group
Department of Imaging Physics
Faculty of Applied Sciences
Delft University of Technology
iv
D ELFT U NIVERSITY OF T ECHNOLOGY
D EPARTMENT OF E LECTRICAL E NGINEERING
The undersigned hereby certify that they have read and recommend to the Faculty of Electrical
Engineering, Mathematics and Computer Science for acceptance a thesis entitled “ Modeling
of wave propagation in open domains ” by Jörn Zimmerling in partial fulfillment of the
requirements for the degree of Master of Science.
Dated: 6 of June 2014
Chairman:
prof.dr.ir. A.-J. van der Veen
Advisor:
dr.ir. R.F. Remis
Committee Members:
prof.dr. H.P. Urbach
ir. L. Wei
Contents
Preface
i
Acknowledgments
vii
List of Figures
ix
List of Tables
xi
1
Introduction
1
2
State-space representations for EM and acoustic wave fields
2.1 System formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Linear Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spatial discretization of the Maxwell equations . . . . . . . . . . . . . . . . .
2.2.1 Electromagnetic state-space representation for instantaneously reacting
media in three dimensions . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Electromagnetic state-space representation for media exhibiting relaxation in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Electromagnetic state-space representation for media exhibiting relaxation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Full order solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 PML, the complex-scaling method, and stability correction . . . . . . . . . . .
3
4
Krylov methods for MOR
3.1 Introduction . . . . . . . . . . . . . .
3.2 ROM via projection methods . . . . .
3.3 Projection subspace . . . . . . . . . .
3.3.1 Polynomial Krylov subspaces
3.3.2 Extended Krylov subspace . .
3.3.3 Rational Krylov subspace . .
3.3.4 Convergence Comparison . .
5
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23
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Results
4.1 Instantaneously reacting materials – 1D configurations
4.1.1 Dielectric slab . . . . . . . . . . . . . . . . .
4.1.2 Photonic crystal . . . . . . . . . . . . . . . . .
4.1.3 Conclusion . . . . . . . . . . . . . . . . . . .
4.2 Instantaneously reacting materials – 2D configurations
4.2.1 Dielectric box . . . . . . . . . . . . . . . . . .
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33
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42
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Conclusions
5.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
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4.3
4.4
4.5
4.6
5
4.2.2 Photonic crystal waveguide . . . . . . . . . . . . .
4.2.3 Layered Earth Example . . . . . . . . . . . . . . .
4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . .
Instantaneously reacting materials – A 3D configuration . .
4.3.1 Dielectric box . . . . . . . . . . . . . . . . . . . .
Media exhibiting relaxation – A 2D configuration . . . . .
4.4.1 Dispersive box with Drude relaxation . . . . . . .
Media exhibiting relaxation – A 3D configuration . . . . .
4.5.1 Spontaneous decay rate of a dipole near a nanorod
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Performance comparison . . . . . . . . . . . . . .
4.6.2 Summary of Results . . . . . . . . . . . . . . . .
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Bibliography
69
A List of abbreviations
73
B Nomenclature
75
C Electromagnetic system formulation
C.1 One-dimensional system formulation . . . . . . . .
C.1.1 Discretization . . . . . . . . . . . . . . . .
C.1.2 Symmetry properties . . . . . . . . . . . .
C.2 Two-dimensional system formulation . . . . . . .
C.2.1 System Formulation for H-polarized fields .
C.2.2 System Formulation for E-polarized fields .
C.3 Three-dimensional system formulation . . . . . . .
C.3.1 Discretization . . . . . . . . . . . . . . . .
C.3.2 Symmetry properties . . . . . . . . . . . .
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77
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90
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D Optimal PML formulation
D.1 Derivation of optimal step sizes .
D.2 Small angle approximation . . .
D.2.1 Determining D . . . . .
D.2.2 Calculation of step sizes
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List of Figures
2.1
3.1
3.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
Example of a support matrix given a one-dimensional domain. Grid points y2
and y3 are within the dispersive medium and thus y2 , y3 ∈ Ωsup . The rows of Isup
are the basis vectors of Ωsup expressed in the basis vectors of ΩDOI . . . . . . .
Illustration showing the influence of shifts on the spectrum of the operator A in
complex plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence of eigenvalues schematically shown for polynomial (PKS), extended (EKS) and rational (RKS) Krylov subspace projections. The stable poles
are shown in the fourth quadrant. . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated configuration: A one-dimensional slab with source and receiver on
one side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence of the ROM poles and time-domain solution with increasing ROM
dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Snapshot at t = 0 of some of the eigenfunctions that belong to the scatteringpoles.
Labeled scattering poles to show the eigenvalues of the resonances. . . . . . . .
Illustration of the simulated 1D periodic structure. The parameters are given by
ap = 3 µm, bp = 30 µm and ε1 = 3, ε2 = 1. . . . . . . . . . . . . . . . . . . .
Band structure of the photonic crystal, calculated via plane wave expansion, and
using the Krylov subspace approach. . . . . . . . . . . . . . . . . . . . . . . .
Two dimensional dielectric box with source and receiver inside the box. . . . .
Convergence of the ROM system response with increasing ROM dimension. . .
Expansion of the system response in the scattering poles of the system. Left:
the selected scattering poles to capture the field approximation. Right: resulting
field approximation. Early times are captured by selecting poles with a small
real part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Snapshots of two resonance eigenfunctions f res of the system in the time-domain.
The black box indicates the boundary of the box. . . . . . . . . . . . . . . . .
Rational Krylov subspace approximation of the box problem with equidistant
shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated photonic waveguide crystal. The source is marked by an X whereas
the receiver is marked as a triangle. . . . . . . . . . . . . . . . . . . . . . . . .
Convergence of the ROM system response with increasing ROM dimension. . .
Expansion of the system response in the scattering poles of the system. . . . . .
Rational Krylov subspace approximation of the spectrum of the crystal problem
with equidistant shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational Krylov subspace approximation of the crystal problem response with
equidistant shifts and single shifts. . . . . . . . . . . . . . . . . . . . . . . . .
Simulated configuration: Earth layer and drilling hole. Top receiver is labeled
as receiver 1 and the lowest as receiver 5. . . . . . . . . . . . . . . . . . . . .
ix
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4.18 Wavefield along a line from receiver 1 to 5 in the drilling hole. The yellow lines
mark the horizontal position of the Earth layer. . . . . . . . . . . . . . . . . .
4.19 Signal at the five receivers. Receiver 1 is the top receiver, receiver 5 the lower
one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.20 Scattering pole expansion of the layered Earth problem. . . . . . . . . . . . . .
4.21 Rational Krylov subspace approximation of the layered Earth response using
equidistance and spectrally adapted shifts. . . . . . . . . . . . . . . . . . . . .
4.22 Eigenfunctions of several poles of the layered Earth problem. . . . . . . . . . .
4.23 Three-dimensional dielectric box with dipole source next to the box. . . . . . .
4.24 Resonant field in a three-dimensional box. . . . . . . . . . . . . . . . . . . . .
4.25 Simulated configuration: Square nanorod excited by a line source with L=1 nm.
4.26 Scattering response using polynomial Krylov subspace reduction. . . . . . . .
4.27 Results of the RKS reduction for the two-dimensional dispersive box. . . . . .
4.28 Eigenvectors of the system matrix A. Absolute value of the magnetic field is
depicted on the left hand side, whereas Ex is depicted to the right. . . . . . . . .
4.29 Continued from Figure 4.28. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.30 Simulated configuration: cylindrical nanorod excited by a dipole molecule. . .
4.31 Spontaneous decay rate of the nanorod dipole configuration using the Lanczos
algorithm and the method described in [1]. The decay rate of the cylinder configuration was calculated using the Lanczos reduced-order model with m=3500.
5.1
49
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54
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57
57
58
59
61
62
Overview of the Krylov subspace approaches discussed in the literature and in
this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
C.1 Illustration of the grid used for the 1D simulation. Note that the step sizes
between points {ŷ−1 , ŷ1 } and {ŷN , ŷN+1 } are given by δ/2 plus the complex
distribution from the PML. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
x
List of Tables
2.1
Parameters to obtain common dispersion models with the general second-order
dispersion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Computation time comparison of FDTD, PKS, and RKS. . . . . . . . . . . . .
62
B.1 Nomenclature used in this document. . . . . . . . . . . . . . . . . . . . . . . .
75
4.1
xi
xii
List of Algorithms
1
2
3
4
5
6
Arnoldi MGS algorithm . . . . .
Modified Lanczos algorithm . .
Rational Arnoldi MGS algorithm
Power method . . . . . . . . . .
Inverse power method . . . . . .
Inverse shifted power method . .
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26
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xiv
Chapter 1
Introduction
Simulation and modeling are key driving forces in physics and engineering. Without powerful
simulation tools it is difficult if not impossible to design and optimize the complex systems
of today and tomorrow. The design of a ship, for example, or a bridge, or nanoantennas, or
electronic circuits, or sensors – all require accurate modeling and effective simulation tools.
In many of these complex systems, electromagnetic (or acoustic) wave fields play a crucial role.
The propagation of waves needs to be accurately modeled and simulated in a wide variety of
applications ranging from photonics to seismic exploration. To this end, numerous modeling
and simulation tools have been developed and many of these tools are based on the so-called
Finite-Difference Time-Domain method (FDTD method) proposed by Yee [2] already in 1966.
Around that time, a computer had no more than a hundred transistors and less than 200 kB of
memory. The number of applications that could be effectively solved by FDTD was therefore
quite limited. Times have changed, however, and presently essentially everyone owns and uses
a computer. The transistor count of a modern day processor exceeds a billion and FDTD implementations on (super)computers can now easily handle wave field problems with millions or
even billions of unknowns.
To push the limits of computable configurations beyond the line of hardware scaling, new computational algorithms are needed to reduce memory use and computation time. In this respect,
Krylov subspace methods have a proven track record in many different areas of scientific computing [3]. Their use in hyperbolic wave field problems has been limited, however. The main
reason is that most wave field problems encountered in practice take place on unbounded domains and up until recently it is was simply unclear how to simulate outward propagating waves
in a computationally tractable manner within a Krylov subspace setting. This problem was resolved in [4] by using a so-called global optimal complex-scaling method for domain truncation
combined with a stability correction procedure to obtain stable wave field approximations in the
time- and frequency-domain. Complex-scaling was already introduced in the 1970s to simulate
open quantum systems (see [5]) and can be seen as a special case of the well-known Perfectly
Matched Layer (PML) technique proposed by Berenger in 1994 [6]. The latter technique is
nowadays widely used in various FDTD implementations and commercial FDTD simulations
tools. It utilizes frequency-dependent sponge layers that completely surround the finite domain
of interest. The layers depend in a nonlinear manner on frequency and perfectly absorb outgoing
waves thereby simulating the extension to infinity. By simply fixing the frequency of the PML
the complex-scaling method can be obtained. In this way, a linearized, frequency-independent
PML is obtained that works well for frequencies of operation close to the fixed PML frequency.
The complex-scaling method presented in [4] is a generalization of the above approach in which
the frequency-independent PML optimally absorbs propagating waves for a whole range of frequencies not necessarily close to the fixed PML frequency. In [7], this approach was generalized
to optimally absorb evanescent and propagating fields simultaneously. The optimality in both
1
2
approaches leads to PMLs containing a small number of grid points. This is desirable, since a
PML is only used to simulate the extension to infinity and the field inside such a layer is of no
interest.
A consequence of implementing the optimal scaling-method for domain truncation is that the
resonances and anti-resonances of the open system are approximated simultaneously. Consequenty, the complex-scaled system cannot be solved in a similar manner as the unscaled system,
since otherwise unstable field approximations will result. Fortunately, in [4] it is shown that
stable field approximations can be obtained from the complex-scaled system by evaluating a
stability-corrected wave function. This function reduces to the unscaled wave field solution if
complex-scaling is switched off and produces accurate, symmetry-preserving field approximations.
With complex-scaling and stability-correction into place, it is now possible to apply Krylov
reduction to open wave field problems. Fast convergence can be expected, since infinity can
be seen as an absorber and Krylov techniques have been shown to exhibit fast convergence for
parabolic diffusion problems [8–11]. In [4] and [12] it was shown that so-called polynomial
and extended Krylov subspace techniques do indeed exhibit fast convergence and outperform
FDTD in the time-domain or conventional linear system solvers in the frequency-domain. Since
the Krylov field approximations are basically expansions in approximate open resonant modes,
convergence is particularly fast for configurations in which the fields are highly resonant. Examples of such configurations can be found in plasmonics, photonics, nano-antenna design,
etc.
In this thesis we apply and validate the above mentioned Krylov solution methods to a variety
of configurations and present a generalization of these techniques for media exhibiting general
second-order relaxation phenomena. Specifically,
• we show that the resonances found via Krylov subspace reduction coincide with the resonances of a simple one-dimensional open system for which the resonances can be determined analytically;
• we show that the bandstructure of periodic structures can be determined via Krylov reduction techniques;
• we validate the above mentioned Krylov methods by computing time-domain field responses for 1D, 2D, and 3D configurations;
• we determine and visualize the dominant scattering modes of various open structures via
Krylov reduction and show that significant order reduction can be achieved, especially for
highly resonating fields;
• we introduce resonance expansions in reduced order modeling to obtain small order approximations of frequency-domain transfer functions and time-domain Greens functions
after polynomial Krylov subspace reduction;
• we show the potential of rational Krylov subspace reduction for wave problems on open
domains;
and, finally,
• for media showing second-order relaxation effects, we present a new symmetry preserving
Krylov subspace formulation of Maxwell’s equations in 2D and 3D, and
• we use the newly developed Krylov subspace technique for 3D wave fields to efficiently
compute the spontaneous decay rate of a quantum emitter located near a golden nanorod.
Moreover, we also show that a single reduced-order model approximates the spontaneous
decay rate over a complete frequency band of interest.
This thesis is organized as follows: In Chapter 2 we discuss the basic equations that
govern the behavior of electromagnetic and acoustic wave fields. These equations are discretized on a so-called staggered Yee grid and the spatial derivatives are approximated by twoOutline
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 1. INTRODUCTION
3
point finite-difference formulas. The symmetry properties of the resulting state-space representation are discussed and media with and without dispersion are considered. Subsequently,
we briefly address our implementation of the complex-scaling method, the stability-correction
procedure, and a short discussion on resonant field expansions is included as well.
Chapter 3 introduces model order reduction via projection methods. Several Krylov subspace
methods are reviewed and compared. Algorithms to construct the subspaces are also presented.
To illustrate the reduction methods developed in Chapters 2 and 3, several electromagnetic
examples are presented in Chapter 4. Polynomial and rational Krylov subspace methods are
demonstrated and their performance is compared. Finally, Chapter 5 draws conclusions from
the research documented in this thesis and gives an outlook to future work.
All abbreviations used throughout the document are listed in Appendix A. The nomenclature
we tried to obey as consistently as possible is given in Appendix B.
A Krylov subspace approach
Modeling of wave propagation in open domains
4
Modeling of wave propagation in open domains
A Krylov subspace approach
Chapter 2
State-Space Representations for
Electromagnetic and Acoustic Wave Fields
In this chapter we discuss the basic equations that govern the behavior of electromagnetic
and acoustic wave fields. For electromagnetic fields, these equations are obviously given by
Maxwell’s equations, while for acoustic fields we consider the linearized acoustic wave equations. In Section 2.1 both sets of equations are written in a general matrix operator form that
serves as a starting point for our reduced-order modeling approach. Furthermore, in Section 2.2
we discuss the spatial discretization procedure for Maxwell’s equations. We do not discuss the
discretization of the acoustic wave equations, since this discretization procedure is essentially
the same as for Maxwell’s equations. The full-order solution of the resulting semi-discrete system is presented in Section 2.3. As suggested by the title of this thesis, we are interested in
wave equations on open domains. Therefore absorbing boundary conditions are introduced into
the formalism in Section 2.4. This will lead to instabilities, however, and a stability correction
is introduced in the same section to obtain a stable solution. Finally, resonance expansions
are introduced as an efficient tool to obtain small order representations of open wave problem
solutions.
2.1
2.1.1
System formulation
Maxwell Equations
The behavior of the electromagnetic field is governed by Maxwell’s equations
−∇ × H + ε0 ∂t E + Jind = −Jext
(2.1)
and
∇ × E + µ0 ∂t H + Kind = −Kext .
(2.2)
Equation (2.1) is also known as the Ampère-Maxwell law, while Eq. (2.2) is known as Faraday’s
law. In the above equations, E is the electric field strength (V/m) and H is the magnetic field
strength (A/m). Furthermore, ε0 (F/m) and µ0 (H/m) are the permittivity and permeability of
vacuum, respectively, while Jext (A/m2 ) and Kext (V/m2 ) are the external (impressed) electric
and magnetic current densities. Finally, Jind (A/m2 ) and Kind (V/m2 ) take the presence of matter
into account and represent electric and magnetic current densities that are induced inside a piece
of matter. To properly take into account the various physical processes that take place within a
material, it is customary to write the induced electric current density as
Jind = Jcond + ∂t P,
5
(2.3)
6
2.1. SYSTEM FORMULATION
where Jcond (A/m2 ) is the induced conduction current and P is the polarization vector (C/m2 ).
The induced magnetic current density is written as
Kind = µ0 ∂t M,
(2.4)
where M is the magnetization (A/m). For most materials, the induced conduction current and
the polarization vector depend only on the electric field strength, while the magnetization only
depends on the magnetic field strength. Explicitly, for a general class of materials we have
cond
J
Z t
(x,t) =
P(x,t) = ε0
Zt 0t=0
t 0 =0
κc (x,t 0 )E(x,t − t 0 ) dt 0 ,
(2.5)
κe (x,t 0 )E(x,t − t 0 ) dt 0 ,
(2.6)
κm (x,t 0 )H(x,t − t 0 ) dt 0 ,
(2.7)
and
Z t
M(x,t) =
t 0 =0
where
• κc is the conduction relaxation tensor (S/m · s),
• κe is the dielectric relaxation tensor (s−1 ), and
• κm is the magnetic relaxation tensor (s−1 ).
Notice that only past values of the electric and magnetic field strength can contribute to the
induced current densities at time t, since all materials are causal.
In this thesis, two types of materials will be considered, namely, instantaneously reacting
materials and materials with relaxation (noble metals, for example). We first discuss instantaneously reacting media and then consider media with relaxation.
Instantaneously reacting media
For instantaneously reacting materials, the above relaxation tensors become
κc = σ(x)δ(t),
κe = χe (x)δ(t),
and
κm = χm (x)δ(t),
(2.8)
where δ(t) is the Dirac distribution operative at t = 0, σ is the conductivity tensor (S/m), and χe
and χm are the electric and magnetic susceptibility tensors, respectively (both dimensionless).
Substitution of the above relaxation functions in the constitutive relations of Eqs. (2.5) – (2.7)
gives
Jcond (x,t) = σ(x)E(x,t),
P(x,t) = ε0 χe (x)E(x,t),
and M(x,t) = χm (x)H(x,t), (2.9)
and the induced currents for instantaneously reacting materials are given by
Jind = σ(x)E(x,t) + ε0 χe (x)∂t E(x,t) and Kind = µ0 χm (x)H(x,t).
(2.10)
Substituting these expressions in Maxwell’s equations (2.1) and (2.2) gives
−∇ × H + σE + ε∂t E = −Jext
(2.11)
and
∇ × E + µ∂t H = −Kext ,
Modeling of wave propagation in open domains
(2.12)
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS
where we have introduced the permittivity and permeability tensor as
h
h
i
i
ε(x) = ε0 I + χe (x)
and µ(x) = µ0 I + χm (x) ,
7
(2.13)
respectively, and I is the diagonal unit tensor.
Written out in full, the Maxwell system of Eqs. (2.11) and (2.12) can be written in matrix
form as (see [13])
(D + S + M ∂t )F = Q 0 ,
(2.14)
where the spatial derivatives are contained in the spatial differentiation matrix


0
0
0
0
∂z −∂y
0
0 −∂z 0
∂x 
 0


0
0
0
∂
−∂
0 
y
x
D =

.
0
0
0 
 0 −∂z ∂y
 ∂
0 −∂x 0
0
0 
z
−∂y ∂x
0
0
0
0
Furthermore, the medium parameters are contained in the medium matrices
ε 0
σ 0
and M =
.
S=
0 0
0 µ
(2.15)
(2.16)
From energy considerations it follows that the conductivity tensor σ is symmetric and semipositive definite, while the permittivity and permeability tensors ε and µ are symmetric and positive definite. Given the definitions of the medium matrices, it is easy to see that S is symmetric
and semi-positive definite as well, while M is symmetric and positive-definite. Furthermore,
for isotropic media we have
σ = σI,
ε = εI,
and µ = µI,
(2.17)
where σ, ε, and µ are the scalar conductivity, permittivity, and permeability medium parameters.
In this thesis, we consider isotropic media only.
Finally, the field and source vector are given by
F = [Ex , Ey , Ez , Hx , Hy , Hz ]T ,
(2.18)
Q 0 = −[Jxext , Jyext , Jzext , Kxext , Kyext , Kzext ]T ,
(2.19)
and
respectively. For many external sources as encountered in practice, the time dependence of the
source can be factored out and we have Q 0 = w(t)Q , where Q is time independent. The scalar
function w is called the source wavelet or source signature and vanishes for t < 0 if the external
source is switched on at t = 0.
Media exhibiting relaxation
We consider isotropic media with relaxation tensors given by
κc = 0,
κm (x,t) = χm (x)δ(t)I,
and κe = κe I.
(2.20)
We now have
cond
J
Z t
= 0,
M = χm H,
A Krylov subspace approach
and
P(t) = ε0
t 0 =0
κe (t 0 )E(t − t 0 ) dt 0 ,
(2.21)
Modeling of wave propagation in open domains
8
2.1. SYSTEM FORMULATION
where we have suppressed the spatial dependence for notational convenience. For most materials the reaction to the presence of an electromagnetic field is conveniently described by a
relaxation function of the form
κe (t) = (ε∞ − 1)δ(t) + χ̃e (t),
(2.22)
where ε∞ is the relative permittivity at high frequencies. The polarization vector now becomes
P = ε0 (ε∞ − 1)E + P̃ with P̃ = ε0
Z t
t 0 =0
χ̃e (t 0 )E(t − t 0 ) dt 0
(2.23)
and for the induced electric-current density we obtain
Jind = Jcond + ∂t P = ε0 (ε∞ − 1)∂t E + ∂t P̃.
(2.24)
Furthermore, the induced magnetic-current density is given by
Kind = µ0 ∂t M = µ0 χm ∂t H
(2.25)
and substituting these induced-current densities in Maxwell’s equations, we arrive at
−∇ × H + ∂t D = −Jext ,
∇ × E + µ∂t H = −Kext ,
(2.26)
(2.27)
where µ = µ0 (1 + χm ) and D = εE + P̃ with ε = ε0 ε∞ .
Different materials are described by different relaxation functions χ̃e . By specifying this function or by specifying the differential equation satisfied by P̃, we describe how a piece of matter
reacts to the presence of an electromagnetic field. Specifically, we consider the following different types of matter:
• Debye material: P̃ satisfies
τ∂t P̃ + P̃ = ε0 (εs − ε∞ )E.
(2.28)
Here, τ is the characteristic relaxation time of the material and εs is the static relative
permittivity. Debye materials describe relaxation of non-interacting dipoles. An example
of a Debye material is human tissue.
• Drude material: P̃ satisfies
∂t2 P̃ + γp ∂t P̃ = ε0 ω2p E.
(2.29)
Here, ωp is the volume plasma frequency and γp = 1/τ is the collision frequency with τ
the relaxation time of the material. Examples of Drude materials include noble metals in
the infrared region.
• Lorentz material: P̃ satisfies
∂t2 P̃ + 2δ∂t P̃ + ω20 P̃ = ε0 (εs − ε∞ )ω20 E.
(2.30)
Here, ω0 is the resonant plasma frequency of bound electrons and δ is the (mainly radiative) damping constant. Examples of Lorentz materials are noble metals in the visible
range.
All three cases can be covered using the generic form
β3 ∂t2 P̃ + β2 ∂t P̃ + β1 P̃ = β0 E,
(2.31)
where for each material the different medium parameters βi are listed in Table 2.1.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS
9
Table 2.1: Parameters to obtain common dispersion models with the general second-order dispersion model.
Medium
Normalization
Lorentz
Drude
Debye
Conductivity
β0
2
L /c20
ε0 (εs − ε∞ )ω20
ε0 ω2p
ε0 (εs − ε∞ )
σ
β1
2
L /c20
ω20
0
1
0
β2
β3
L/c0
2δ
γp
τ
1
1
1
1
0
0
Introducing the auxiliary variable
U = −∂t P̃
(2.32)
we can write the above second-order constitutive relation in first-order form as
β3 ∂t U + β2 U − β1 P̃ + β0 E = 0.
(2.33)
Putting everything together, we have arrived at the first-order representation
−∇ × H − U + ε∂t E = −Jext
U + ∂t P̃ = 0,
β2 U − β1 P̃ + β0 E + β3 ∂t U = 0,
∇ × E + µ∂t H = −Kext ,
(2.34)
(2.35)
(2.36)
(2.37)
and written out in full, the above Maxwell system can also be written in matrix-operator form
as
(D + S + M ∂t ) F = Q 0 ,
(2.38)
where the spatial differentiation matrix is given by









D =









0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 −∂z ∂y
∂z
0 −∂x
−∂y ∂x
0
A Krylov subspace approach
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0 0
∂z −∂y
0 −∂z 0
∂x 

0 ∂y −∂x 0 

0 0
0
0 

0 0
0
0 

0 0
0
0 

0 0
0
0 
0 0
0
0 

0 0
0
0 

0 0
0
0 

0 0
0
0 
0 0
0
0
(2.39)
Modeling of wave propagation in open domains
10
2.1. SYSTEM FORMULATION
and the medium matrices S and M are given by


0 0 0
0
0
0 −1 0
0 0 0 0
 0 0 0
0
0
0
0 −1 0 0 0 0 


 0 0 0
0
0
0
0
0 −1 0 0 0 


 0 0 0
0
0
0
1
0
0 0 0 0 


0
0
0
0
1
0 0 0 0 
 0 0 0


0 0 0
0
0
0
0
0
1 0 0 0 
S =


0
β2 0
0 0 0 0 
 β0 0 0 −β1 0
 0 β 0
0 −β1 0
0 β2 0 0 0 0 


0
 0 0 β
0
0 −β1 0
0 β2 0 0 0 


0

 0 0 0
0
0
0
0
0
0
0
0
0


 0 0 0
0
0
0
0
0
0 0 0 0 
0 0 0
0
0
0
0
0
0 0 0 0
and









M =









ε
0
0
0
0
0
0
0
0
0
0
0
0
ε
0
0
0
0
0
0
0
0
0
0
0
0
ε
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 β3 0 0
0 0 β3 0
0 0 0 β3
0 0 0 0
0 0 0 0
0 0 0 0
0
0
0
0
0
0
0
0
0
µ
0
0
0
0
0
0
0
0
0
0
0
0
µ
0
0
0
0
0
0
0
0
0
0
0
0
µ
(2.40)










,








(2.41)
respectively. Finally, the field and source vectors are given by
F = [Ex , Ey , Ez , P̃x , P̃y , P̃z ,Ux ,Uy ,Uz , Hx , Hy , Hz ]T
(2.42)
Q 0 = −[Jxext , Jyext , Jzext , 0, 0, 0, 0, 0, 0, Kxext , Kyext , Kzext ]T ,
(2.43)
and
respectively.
Media exhibiting relaxation – Two-dimensional configurations
We consider H-polarized fields in a two-dimensional configuration that is invariant in the zdirection. Consequently, we may set ∂z = 0 and for H-polarized fields the full set of Maxwell’s
equations simplifies to
∂x Ey − ∂y Ex + µ∂t Hz = −Kzext ,
−∂y Hz −Ux + ε∂t Ex = −Jxext ,
∂x Hz −Uy + ε∂t Ey = −Jyext ,
Ux + ∂t P̃x = 0,
Uy + ∂t P̃y = 0,
β3 ∂t Ux + β2Ux − β1 P̃x + β0 Ex = 0,
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
(2.49)
β3 ∂t Uy + β2Uy − β1 P̃y + β0 Ey = 0.
(2.50)
and
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 11
This system can also be written in the form
(D + S + M ∂t ) F = Q 0 ,
(2.51)
where the field and source vectors are now given by
F = [Hz , Ex , Ey , P̃x , P̃y ,Ux ,Uy ]T
(2.52)
Q 0 = −[Kzext , Jxext , Jyext , 0, 0, 0, 0]T ,
(2.53)
and
respectively, while the spatial differentiation matrix is given by

0 −∂y ∂x 0 0 0 0
 −∂y 0
0 0 0 0 0

0
0 0 0 0 0
 ∂x

0
0 0 0 0 0
D = 0

0
0
0 0 0 0 0

 0
0
0 0 0 0 0
0
0
0 0 0 0 0





.



Finally, the medium matrices are given by


0 0 0
0
0
0
0
 0 0 0
0
0 −1 0 


0
0
0 −1 
 0 0 0


0
0
1
0 
S = 0 0 0


0
0
0
1 
 0 0 0
 0 β 0 −β
0
β2 0 
0
1
0 0 β0 0 −β1 0 β2
(2.54)
(2.55)
and




M =




2.1.2
µ
0
0
0
0
0
0
0
ε
0
0
0
0
0
0
0
ε
0
0
0
0
0
0
0
1
0
0
0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 β3 0
0 0 β3





.



(2.56)
Linear Acoustics
To illustrate that other types of wave phenomena can be captured within the same framework, let
us consider acoustic wave motion in linear, time invariant, instantaneously reacting, anisotropic,
and inhomogeneous fluids. In the low-velocity approximation, the acoustic wave equations are
given by the equation of motion
∇p + ρ∂t v = f
(2.57)
∇ · v + κ∂t p = q.
(2.58)
and the deformation equation
A Krylov subspace approach
Modeling of wave propagation in open domains
12
2.2. SPATIAL DISCRETIZATION OF THE MAXWELL EQUATIONS
In these equations, p is the acoustic pressure (Pa) and v is the particle velocity (m/s). Furthermore, f is the volume source density of volume force (N/m3 ) and q is the volume density of
injection rate (s−1 ). Finally, ρ is the volume density of mass (kg/m3 ) and κ is the compressibility
tensor (Pa−1 ). Written out in full, we again arrive at a system of the form
(D + S + M ∂t ) F = Q 0
(2.59)
where this time the field and source vectors are given by
F = [p, vx , vy , vz ]T
(2.60)
Q 0 = [q, fx , fy , fz ]T ,
(2.61)
and
respectively, while the differentiation matrix is given by


0 ∂x ∂y ∂z
∂x 0 0 0 

D =
 ∂y 0 0 0  .
∂z 0 0 0
(2.62)
Finally, the medium matrices for linearized acoustics are


ρ 0
0
0
0 κxx κxy κxz 

S = 0 and M = 
0 κyx κyy κyz  .
0 κzx κzy κzz
(2.63)
From energy considerations it follows that the compressibility tensor κ is symmetric and positive definite. Since ρ > 0 it then follows that the medium matrix M is symmetric and positive
definite as well. Finally, we mention that for isotropic media, the compressibility tensor reduces
to κ = κI, where κ is the scalar compressibility.
2.2
Spatial discretization of the Maxwell equations
In this section we discuss the spatial finite-difference discretization of Maxwell’s equations for
isotropic media. Discretization of the linearized acoustic field equations is not discussed, since
these equations can be handled in essentially the same manner as Maxwell’s equations.
Spatial discretization is carried out using two-point finite-difference formulas for the spatial
derivatives that appear in Maxwell’s equations. To this end, we introduce primary and dual grids
in each Cartesian direction and subsequently define the finite-difference approximations of the
electric and magnetic field strength on certain Cartesian products of these grids. For instance,
the primary and dual grids in the y-direction are defined as
Ωpy = {yq ∈ R, q = 0, 1, . . . , Q + 1, yq > yq−1 },
(2.64)
Ωdy = {ŷq ∈ R, q = 1, . . . , Q + 1, ŷq+1 > ŷq },
(2.65)
and
respectively, with corresponding step sizes given by
δy,q = yq − yq−1
∀ q = 1, . . . , Q + 1
and δˆ y,q = ŷq+1 − ŷq
Modeling of wave propagation in open domains
∀ q = 1, . . . , Q.
(2.66)
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 13
Furthermore, in each Cartesian direction the first and last node of the primary grid are always
located on the boundary of the computational domain and only so-called staggered grids are
considered. These grids are characterized by the interlacing property
y0 < ŷ1 < y1 < ŷ2 < · · · < ŷQ+1 < yQ+1 ,
(2.67)
showing that primary and dual nodes always have neighbouring nodes of the opposite kind.
Primary and dual grids in the x- and z-directions are defined similarly with P + 2 primary nodes
in the x-direction and R + 2 nodes in the z-direction. As a result, we have a total of six onedimensional spatial grids given by
Ωpx , Ωdx , Ωpy , Ωdy , Ωpz ,
Ωdz .
and
Now to truncate our domain of interest, we impose Perfect Electrically Conducting (PEC)
boundary conditions at the boundary of our computational domain (absorbing boundary conditions are introduced later). Consequently, we have to set the tangential components of the
electric field strength to zero at this boundary. Denoting the finite-difference approximations
of the x-, y-, and z-components of the electric and magnetic field strength by {ex , ey , ez } and
{hx , hy , hz }, respectively, we therefore define
ex on Ωdx × Ωpy × Ωpz
ey on Ωpx × Ωdy × Ωpz
ez on Ωpx × Ωpy × Ωdz
and
hx on Ωpx × Ωdy × Ωdz
hy on Ωdx × Ωpy × Ωdz
hz on Ωdx × Ωdy × Ωpz
since the tangential components of the electric field strength are then located at the outer boundary of our computational domain. Consequently, we can properly take the PEC boundary conditions into account by simply setting these components to zero. Note that the normal components
of the magnetic field strength at the boundary then automatically vanish as well. The resulting
grid is also known as a Yee grid [2].
As mentioned above, we use two-point finite-difference formulas to approximate the partial
derivatives in Maxwell’s equations. These formulas are implemented using bidiagonal differentiation matrices. For example, y-differencing a field quantity defined on the primary grid in
the y-direction is carried out by acting with the finite-difference matrix
 −1

δy;1
0
··· ···
···
0
..


−1
 −δ−1

δy;2
0 ···
···
.
y;2


.


−1
..
−δ−1
δ
0
·
·
·
 0

y;3
y;3
 .

.
..
..

.
.
Y=
(2.68)
.
.
.
.



 .
.
.
.
..
..
..
 ..



 ..

−1
−1
 .

···
· · · · · · −δ
δ
y;Q
0
···
···
···
0
y;Q
−δ−1
y;Q+1
on these field quantities. Observe that this matrix takes the PEC material boundary conditions
into account and as a result matrix Y is a nonsquare (Q + 1)-by-Q matrix that maps from the
A Krylov subspace approach
Modeling of wave propagation in open domains
14
2.2. SPATIAL DISCRETIZATION OF THE MAXWELL EQUATIONS
primary to the dual grid in the y-direction. Introducing the (Q + 1)-by-(Q + 1) diagonal matrix
of primary step sizes as
Wy = diag(δy;1 , δy;2 , ..., δy;Q+1 )
(2.69)
and the Q-by-(Q + 1) bidiagonal matrix bidiagQ (−1, 1) with −1 on the diagonal and +1 on the
first upper diagonal, the above differentiation matrix can also be written as
Y = −Wy−1 bidiagQ (−1, 1)T .
(2.70)
The above difference matrix acts on field quantities defined on the primary grid in the ydirection. In a similar manner, we can define a difference matrix that acts on field quantities
defined on the dual grid. Specifically, if we introduce the Q-by-Q diagonal step size matrix
then the difference matrix
Ŵy = diag(δˆ y;1 , δˆ y;2 , ..., δˆ y;Q )
(2.71)
Ŷ = Ŵy−1 bidiagQ (−1, 1)
(2.72)
computes two-point finite-differences of field quantities defined on the dual grid in the ydirection. This matrix is clearly Q-by-(Q + 1) and maps from the dual grid to the primary
grid. Moreover, we also have the obvious symmetry relation
ŶT Ŵy = −Wy Y.
(2.73)
Difference matrices in the x- and z-direction are defined in an analogous manner. In particular, introducing the diagonal step size matrices
Wx = diag(δx;1 , δx;2 , ..., δx;P+1 ),
Wz = diag(δz;1 , δz;2 , ..., δz;R+1 ),
Ŵx = diag(δˆ x;1 , δˆ x;2 , ..., δˆ x;P ),
Ŵz = diag(δˆ z;1 , δˆ z;2 , ..., δˆ z;R ),
(2.74)
(2.75)
the difference matrices in the x- and z-direction are given by
X = −Wx−1 bidiagP (−1, 1)T ,
Z = −Wz−1 bidiagR (−1, 1)T ,
X̂ = Ŵx−1 bidiagP (−1, 1),
Ẑ = Ŵz−1 bidiagR (−1, 1),
(2.76)
(2.77)
and we have the symmetry relations
X̂T Ŵx = −Wx X
and
ẐT Ŵz = −Wz Z.
(2.78)
With the introduction of all these differentiation matrices the semi-discrete state-space representation of the Maxwell equations can be formulated. In this thesis we work out three cases,
namely, three-dimensional electromagnetic wave propagation in instantaneously reacting media
and in media exhibiting relaxation, and two-dimensional electromagnetic wave propagation in
media with relaxation. Other formulations can be found in Appendix C.
2.2.1
Electromagnetic state-space representation for instantaneously reacting media in
three dimensions
Now that we have described the spatial grid and the corresponding differentiation matrices, we
are in a position to discretize Maxwell’s equations for isotropic and instantaneously reacting
media. Using the Yee grid introduced above, approximating the partial derivatives by two-point
finite-difference formulas, and arranging the unknowns in lexicographical order, we arrive at
the state-space representation
(D + S + M∂t ) f = q0 .
(2.79)
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 15
The order of this system is denoted by n and it is typically very large for real-world 3D problems
(millions or even a billion of unknowns is not uncommon). Furthermore, the spatial differentiation matrix is given by
0 Dh
D=
,
(2.80)
De 0
with


Ẑ ⊗ IQ ⊗ IP+1 −IR ⊗ Ŷ ⊗ IP+1
0
IR ⊗ IQ+1 ⊗ X̂ 
−IR+1 ⊗ IQ ⊗ X̂
0
(2.81)

0
−Z ⊗ IQ+1 ⊗ IP IR+1 ⊗ Y ⊗ IP
0
−IR+1 ⊗ IQ ⊗ X  ,
De =  Z ⊗ IQ ⊗ IP+1
−IR ⊗ Y ⊗ IP+1 IR ⊗ IQ+1 ⊗ X
0
(2.82)
0
Dh =  −Ẑ ⊗ IQ+1 ⊗ IP
IR+1 ⊗ Ŷ ⊗ IP
and

and ⊗ is the Kronecker (tensor) product. Furthermore, matrix S is given by
Mσ 0
S=
,
0 0
(2.83)
where Mσ is a diagonal semi-positive definite matrix with (averaged) conductivity values on its
diagonal. The medium matrix M is given by
Mε 0
M=
0 Mµ
(2.84)
and both Mε and Mµ are diagonal and positive definite medium matrices with averaged permittivity and permeability values on their diagonal. The field vector is of the form
f = [eTx , eTy , eTz , hTx , hTy , hTz ]T ,
(2.85)
where all field quantities are stored in lexicographical order in the corresponding field vectors ei
and hi , i = x, y, z. Finally, the finite-difference approximations of the external sources are stored
in the source vector
, jext;T
, jext;T
, kext;T
, kext;T
, kext;T
]T .
q0 = −[jext;T
x
y
z
x
y
z
(2.86)
Before discussing the various symmetry properties of the state-space representation of Eq. (2.79),
we first introduce the system matrix, which characterizes the complete system. In particular, by
premultiplying Eq. (2.79) by the inverse of the medium matrix, we obtain
(A + I∂t ) f = M−1 q0 ,
(2.87)
where the system matrix is introduced as
A = M−1 (D + S).
(2.88)
The solution of the state-space representation can conveniently be written in terms of this matrix
as we will show in Section 2.3.
A Krylov subspace approach
Modeling of wave propagation in open domains
16
2.2. SPATIAL DISCRETIZATION OF THE MAXWELL EQUATIONS
Symmetry relations
To discuss the symmetry properties satisfied by the differentiation matrix, we first introduce the
diagonal and positive definite step size matrices


Ŵz ⊗ Ŵy ⊗ Wx
0
0
,
We = 
(2.89)
0
Ŵz ⊗ Wy ⊗ Ŵx
0
0
0
Wz ⊗ Ŵy ⊗ Ŵx
and


Wz ⊗ Wy ⊗ Ŵx
0
0
.
Wh = 
0
Wz ⊗ Ŵy ⊗ Wx
0
0
0
Ŵz ⊗ Wy ⊗ Wx
Using the symmetry relations of Eqs. (2.73) and (2.78) it is now easily verified that
DTh We = −Wh De
and
DTe Wh = −We Dh .
(2.90)
(2.91)
Furthermore, with
We 0
W+ =
0 Wh
We
0
and W− =
0 −Wh
(2.92)
we also have
DT W+ = −W+ D and DT W− = W− D.
(2.93)
These symmetry relations can be translated into symmetry relations for the system matrix. In
particular, for lossless media the system matrix is given by A = M−1 D and satisfies the symmetry property
AT MW+ = −MW+ A.
(2.94)
In other words, for instantaneously reacting and lossless media, the system matrix is skewsymmetric with respect ot the energy inner product
hx, yien = yT MW+ x.
(2.95)
This inner product is positive definite and therefore induces a norm which is essentially equal
to the stored electromagnetic energy within the system. More precisely, with f any vector of the
form as given by Eq. (2.85), the norm 12 hf, fien approximates
1
1
E (D) =
ε|E|2 dV +
µ|H|2 dV,
(2.96)
2
2
x∈D
x∈D
which is the electromagnetic energy stored on the domain of interest D.
For lossy media, the system matrix is given by A = M−1 (D + S) and the skew-symmetry
property of Eq. (2.94) is lost. The system matrix does have another symmetry property, however.
To be specific, introducing the free-field Lagrangian bilinear form
ZZZ
ZZZ
hx, yiL = yT MW− x
(2.97)
and using the second symmetry relation of Eq. (2.93), we observe that the system matrix is
symmetric with respect to this form, that is,
hAx, yiL = hx, AyiL
for all x, y ∈ Rn .
(2.98)
1
2 hf, fiL
The reason for calling the above form a free-field Lagrangian bilinear form is that
approximates
ZZZ
ZZZ
1
1
2
Lfree (D) =
ε|E| dV −
µ|H|2 dV,
(2.99)
2
2
x∈D
x∈D
which is the free-field Lagrangian. We stress that the bilinear form hx, yiL does not induce a
norm, since it is not positive definite. Finally, we mention that from the symmetry relation of
Eq. (2.98) it follows that the discretized field quantities satisfy reciprocity (see [13]).
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 17
2.2.2
Electromagnetic state-space representation for media exhibiting relaxation in three
dimensions
Similar to the instantaneously reacting case, the discretized Maxwell’s equations for isotropic
media exhibiting relaxation can be obtained. The main difference is the presence of the polarization vectors P and U in the field equations. These vectors are only active at points where
the dispersive material is present. From a storage point of view, it is therefore advantageous to
only keep the finite-difference approximations of P and U at these points in memory. Since the
electric field strength approximations are defined over the total computational domain, we need
a support matrix to implement the local dispersion relations. To this end, we introduce selection
or logical projection matrices, which select the relevant electric field strength components from
sup
the total electric field vector. For example, if Iy is the support matrix of a dispersive material
in the y-direction and ey contains all finite-difference approximations of the y-component of the
sup
electric field strength, then the vector Iy ey contains only those y-components of Ey located
within the dispersive material. An illustration of how the support matrix is constructed is shown
in Figure 2.1.
Using this definition of the support matrices, the constitutive relation of Eq. (2.33) relating the electric and polarization fields to each other can be implemented in a straightforward
manner. For example, for the y-component of Eq. (2.33) we have
B3,y ∂t uy + B2,y uy − B1,y py + B0,y Isup
y ey = 0,
(2.100)
where the matrices B{0,1,2,3},y are diagonal matrices with (averaged) medium values on their
diagonal, according to Table 2.1.
y0
y1
y2
y3
y4
Isup =
Sup
DOI
Ω
Ω
0 0 1 0 0
0 0 0 1 0
(2.101)
Figure 2.1: Example of a support matrix given a one-dimensional domain. Grid points y2 and y3 are within the
dispersive medium and thus y2 , y3 ∈ Ωsup . The rows of Isup are the basis vectors of Ωsup expressed in the basis
vectors of ΩDOI .
Using the Yee grid introduced earlier, approximating the partial derivatives by two-point finitedifference formulas, and arranging the unknowns in lexicographical order, we now again arrive
at the state-space representation
(D + S + M∂t ) f = q0 .
(2.102)
In this equation, the spatial differentiation matrix is given by


0 0 0 Dh
0 0 0 0

D=
 0 0 0 0 ,
De 0 0 0
with Dh and De similar to the instantaneously reacting case.
Furthermore, matrix S is given by

0
0
−Isup;T
 0
0
I
S=
B0 Isup −B1
B2
0
0
0
A Krylov subspace approach

0
0
,
0
0
(2.103)
(2.104)
Modeling of wave propagation in open domains
18
2.2. SPATIAL DISCRETIZATION OF THE MAXWELL EQUATIONS
where B{0,1,2} are diagonal matrices only defined on the support of the dispersive media. In
addition, Isup is the total support matrix and the medium matrix M is given by


Mε 0 0 0
0 I 0 0

M=
(2.105)
 0 0 B3 0  ,
0 0 0 Mµ
where B3 is again a dispersion matrix and both Mε and Mµ are diagonal and positive definite
medium matrices with averaged permittivity and permeability values on their diagonal. The
field vector is now of the form
f = [eTx , eTy , eTz , pTx , pTy , pTz , uTx , uTy , uTz , hTx , hTy , hTz ]T ,
(2.106)
where all field quantities are stored in lexicographical order in the corresponding field vectors
ei , pi , ui , and hi , i = x, y, z. Finally, the finite-difference approximations of the external sources
are stored in the source vector
q0 = −[jext;T
, jext;T
, jext;T
, 0, 0, 0, 0, 0, 0, kext;T
, kext;T
, kext;T
]T .
x
y
z
x
y
z
(2.107)
Symmetry relations
The system matrix for media exhibiting relaxation given by
A = M−1 (D + S)
(2.108)
is symmetric with respect to the matrix WM with


We 0
0
0
 0 Wp
0
0 

W=
0
0 −Wu
0 
0
0
0
−Wh
(2.109)
and We and Wh as defined in Eqs. (2.89) and (2.90). Furthermore, Wu and Wp are given by
sup
sup;T
Wu = B−1
0 I We I
and
sup
sup;T
Wp = B1 Wu = B1 B−1
.
0 I We I
(2.110)
Introducing the bilinear form
hx, yidisp = yT WMx
(2.111)
this symmetry relation can also be expressed as
hAx, yidisp = hx, Ayidisp
for all x, y ∈ Rn .
(2.112)
Now if the dispersive material occupies the bounded domain Ddisp ⊂ D, we have that 21 hf, fidisp
approximates
1
Lfree (D) +
2
β1 2
1
|P| dV −
2
x∈Ddisp β0
ZZZ
ZZZ
β3
|∂t P|2 dV,
x∈Ddisp β0
(2.113)
where Lfree (D) is the free-field Lagrangian given in Eq. (2.99).
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 19
2.2.3
Electromagnetic state-space representation for media exhibiting relaxation in two
dimensions
For two-dimensional H-polarized problems the analysis is similar to the three-dimensional case.
We therefore only state the results. The state-space representation is again given by
(D + S + M∂t ) f = q0 .
(2.114)
In this equation, the spatial differentiation matrix is given by

0
−(Y ⊗ IP ) (IQ ⊗ X)
−(Ŷ ⊗ IP )
0
0

 (I ⊗ X̂)
0
0
 Q
D=
0
0
0


0
0
0


0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0

0

0

0
0
(2.115)
and matrix S is given by


0
0
0
0
0
0
0
sup;T
0
0
0
0
0
−Ix
0 


sup;T 
0
0
0
0
0
0
−Iy 

S=
0
0
0
0
I
0 
0
,
0
0
0
0
0
0
I 


0 B0,x Isup

0
−B
0
B
0
x
1,x
2,x
sup
0
0
B0,y Iy
0
−B1,y
0
B2,y
(2.116)
where B{0,1,2},{x,y} are diagonal matrices with (averaged) medium values on their diagonal.
Furthermore, Isup is the support matrix defining the support of the medium exhibiting relaxation
and the medium matrix M is given by

Mµ
0
0
 0 Mε;x
0

0
0
M

ε;y

0
0
M= 0
0
0
0

0
0
0
0
0
0
0
0
0
I
0
0
0

0 0
0
0 0
0 

0 0
0 

0 0
0 
I
0
0 

0 B3;x 0 
0 0 B3;y
(2.117)
with B3,x and B3,y dispersion matrices and Mε;x , Mε;y ,and Mµ diagonal and positive definite
medium matrices with averaged permittivity and permeability values on their diagonal. The
field vector is of the form
(2.118)
f = [hTz , eTx , eTy pTx , pTy , uTx , uTy ]T ,
where all field quantities are stored in lexicographical order in the corresponding field vectors
ei , pi , ui , and hi , i = x, y, z. Finally, the finite-difference approximations of the external sources
are stored in the source vector
q0 = −[kext;T
, jext;T
, jext;T
, 0, 0, 0, 0]T .
z
x
y
A Krylov subspace approach
(2.119)
Modeling of wave propagation in open domains
20
2.3. FULL ORDER SOLUTION
Symmetry relations
The system matrix for media exhibiting relaxation given by
A = M−1 (D + S)
(2.120)
is symmetric with respect to the matrix WM with


Wh 0
0
0
0
0
0
 0 We
0
0
0
0
0 
x


 0
0 Wey 0
0
0
0 


,
0
0
0
W
0
0
0
W=
p
x



 0
0
0
0
W
0
0
p
y


 0
0
0
0
0 Wux 0 
0
0
0
0
0
0 Wuy
(2.121)
where
Wh
Wex
Wey
Wpx
Wpy
Wux
= (Wy ⊗ Wx ),
= −(Ŵy ⊗ Wx ),
= −(Wy ⊗ Ŵx ),
sup
sup;T
= −B1,x B−1
,
0,x Ix (Ŵy ⊗ Wx )Ix
−1 sup
sup;T
= −B1,y B0,y Iy (Wy ⊗ Ŵx )Iy ,
sup
sup;T
= B−1
,
0,x Ix (Ŵy ⊗ Wx )Ix
(2.122)
(2.123)
(2.124)
(2.125)
(2.126)
(2.127)
and
−1 sup
Wuy = B0,y
Iy (Wy ⊗ Ŵx )Isup;T
.
y
2.3
(2.128)
Full order solution
As we have seen, the state-space representation of Maxwell’s equations can be written in terms
of the system matrix as
(A + I∂t ) f = M−1 q0 ,
(2.129)
where the system matrix is given by
A = M−1 (D + S).
(2.130)
From now on we consider source vectors for which the time dependence can be factored out,
that is, q0 = w(t)q, with w(t) the source wavelet that vanishes for t < 0 and q a time-independent
vector. With vanishing initial conditions, the field vector f is then given by
f(t) = w(t) ∗ η(t) exp(−At) M−1 q
for t > 0,
(2.131)
where the asterisk denotes convolution in time, η(t) is the Heaviside unit step function, and
exp(−At) is the matrix exponential or evolution operator. In the Laplace domain the solution is
given as
f̂ = ŵ(s)(A + sI)−1 M−1 q.
(2.132)
From the above two expressions it can be seen that computing field quantities in the timedomain amounts to evaluating the action of the matrix exponential function on the source vector,
while in the frequency-domain the action of the resolvent of A on the source vector is required.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 2. STATE-SPACE REPRESENTATIONS FOR EM AND ACOUSTIC WAVE FIELDS 21
Furthermore, if the source vector q is a finite-difference approximation of the Dirac distribution
and w(t) = δ(t), then
g(t) = η(t) exp(−At) M−1 q
(2.133)
is a numerical approximation of the continuous Green’s function.
For real-world applications, direct evaluation of the evolution operator using some decomposition of matrix A is not feasible, since the order of this matrix is simply too large (millions
or even a billion of unknowns is not uncommon). Fortunately, only the action of the evolution
operator on the source vector (or the action of the resolvent on the source vector in frequencydomain problems) is required to obtain the fields. This allows us to efficiently compute the
electromagnetic field quantities using Krylov subspace techniques. These techniques are discussed in the next chapter. Before doing so, however, we first briefly discuss our perfectly
matched layer implementation and some of its consequences. We are then in a position to address the problem of computing propagating wave fields in open domains via Krylov subspace
projection methods.
2.4
Perfectly matched layers, the complex-scaling method, and stability
correction
In a numerical approach, the computational domain is compact and has bounded support, since
the memory and computation power of all computational architectures are finite. Therefore a
boundary condition needs to be imposed on the boundary of the computational domain.
As already stated in the title of this thesis, one of the goals is modeling of resonances in open
systems. For this purpose, an optimal PML is inserted around the computational domain, which
itself has Dirichlet boundary conditions. This PML simulates the extension of the domain to
infinity.
Perfectly matched layers (PMLs) were introduced by Berenger in 1994 [6]. The problem
with this technique is that it leads to nonlinear eigenproblems for spatial dimensions larger
than one. The first step is to linearize the PML by fixing the PML frequency. The PML technique then essentially reduces to the so-called complex-scaling method, which was introduced
in the 1970s to study open quantum systems. However, this scaling method is only effective
for frequencies around the fixed PML frequency. This problem was resolved in [4], where an
optimal and global complex-scaling method was presented. The method is optimal for propagating waves and this optimality leads to PMLs with a small number of layers. Furthermore, a
global complex-scaling method means that the method is effective on an entire interval of frequencies not necessarily close to the fixed PML frequency. The technique was further refined
in [7] resulting in an optimal and global complex-scaling method for evanescent and propagating waves. In this thesis, we use these optimal scaling methods to simulate the extension to
infinity. In practice, this amounts to implementing complex step sizes within a layer (the PML
layer) that completely surrounds the computational domain. The particular values for the step
sizes follow from the optimization schemes presented in [4] and [7].
After incorporation of the optimal complex-scaling method, the system matrix is obviously
no longer real-valued, since the step sizes within the PML layer are complex. All symmetry
relations discussed in the previous sections still hold, however, provided we keep using the
transpose operator instead of the Hermitian transpose. In other words, Lagrangian or dispersion
symmetry on Rn turns into complex Lagrangian or dispersion symmetry on Cn .
Stability Correction
The problem with the (optimal) complex-scaling method is that both the open resonances and
anti-resonances of the system are approximated simultaneously. The anti-resonances lead to unA Krylov subspace approach
Modeling of wave propagation in open domains
22
2.4. PML, THE COMPLEX-SCALING METHOD, AND STABILITY CORRECTION
stable field representations in the time-domain and the question now arises how to obtain stable
time-domain solutions for the complex-scaled Maxwell system. This question was answered
by Druskin and Remis in [4], where it is shown that stable and causal time-domain wavefield
solutions can be computed via the stability-corrected wave function
f(t) = w(t) ∗ 2η(t)Re η(A) exp(−At) M−1 q
for t > 0,
(2.134)
where η(z) is the Heaviside unit step function defined as


1 if Re(z) > 0,
η(z) = 21 if Re(z) = 0,

0 if Re(z) < 0.
(2.135)
We note that without complex-scaling, the stability-corrected wave function of Eq. (2.134) reduces to Eq. (2.131). Finally, we remark that the real and imaginary parts of the complex
stability-corrected Green’s function
gc (t) = 2η(t)η(A) exp(−At) M−1 q
for t > 0
(2.136)
are related to each other via a generalized Hilbert transform.
Resonance expansions
To illustrate which scattering poles contribute to the solution for a given wavelet w(t), let us
assume for simplicity that matrix A can be diagonalized, that is, we can write
A = SΛST WM,
(2.137)
where the eigenvectors si of A form the columns of matrix S and the corresponding eigenvalues λi are stored on the diagonal of the diagonal matrix Λ. Substitution of the eigendecomposition of Eq. (2.137) in Eq. (2.134) gives
"
#
n
f(t) = w(t) ∗ 2η(t)Re
∑ hsi, M−1qidispη(λi) exp(−λit) si
for t > 0.
(2.138)
i=1
This result shows that only the resonant modes si that are excited by the source, that is, modes
for which
hsi , M−1 qidisp 6= 0
and with resonances in
Λres := {z ∈ C : 0 ≤ Re(z) < σmax , −ωmin < Im(z) < −ωmax }
(2.139)
essentially contribute to the solution on a given time interval [Tmin , ∞) with an error of the form
O [ exp(−σmax Tmin )] exponentially decaying in time [14]. Therefore, σmax determines the quality of the early-time approximation, while ωmin and ωmax are determined by the spectrum of the
source wavelet w(t).
For later reference we finally mention that the quality factor of a resonance λi is defined as
Q=
|Im(λi )|
.
2Re(λi )
Modeling of wave propagation in open domains
(2.140)
A Krylov subspace approach
Chapter 3
Krylov methods for model order reduction
3.1
Introduction
In this chapter model order reduction via projection methods is introduced in order to solve the
systems defined in the last chapter. Specifically, we focus on the computation of the action of the
evolution operator exp(−At) on the scaled source vector M−1 q. Since matrix A is large, sparse,
and symmetric with respect to the Lagrangian or dispersion bilinear form, we pay particular
attention to Krylov projection methods.
Section 3.2 introduces reduced-order modeling via general projection methods. It shows how a
projection approach leads to a reduced-order solution of the defined problem. After this section
the question of how to choose the projection subspace K is still open. As mentioned above,
in this thesis Krylov subspaces are chosen as projection subspaces. Polynomial, extended and
rational Krylov subspaces are introduced in Section 3.3. Their convergence behavior, performance, and construction is reviewed and discussed.
3.2
ROM via projection methods
In this section, reduced-order approximations via projection are introduced. Let us start with
the state-space representation
(A + I∂t )g = δ(t)M−1 q.
(3.1)
We are interested in the solution g(t) for t > 0 with vanishing initial conditions.
To construct an approximate solution to the above problem, we consider an m-dimensional
subspace K m (m < n) of Cn and seek an approximation gm to the above system that belongs to
this subspace.
Let the vectors v1 , v2 , ..., vm form a Ws -orthogonal set of basis vectors, where Ws = MW− for
instantaneously reacting media or Ws = MW for dispersive media. The approximate solution is
now expanded in terms of these basis vectors as
gm (t) = α1 (t)v1 + α2 (t)v2 + ... + αm (t)vm = Vm am (t),
(3.2)
where Vm has the column partitioning Vm = (v1 , v2 , ..., vm ) and am (t) is the m-by-1 vector of expansion coefficients. To find these coefficients, we first introduce the residual that corresponds
to the approximation gm as
rm = δ(t)M−1 q − Agm − ∂t gm
= δ(t)M−1 q − AVm am − Vm ∂t am .
23
(3.3)
24
3.2. ROM VIA PROJECTION METHODS
The coefficients now follow from the requirement that the residual is Ws -orthogonal to K m . In
other words, we require that the residual satisfies the pseudo-Galerkin condition
Ws
rm ⊥ K
m
T s
or Vm
W rm = 0
for t > 0.
(3.4)
Substitution of the residual in this condition gives
T s −1
T s
δ(t)Vm
W M q − Vm
W AVm am − Dm ∂t am = 0,
(3.5)
T Ws V = D = diag(δ , δ , ..., δ ) =: D , since the basis vectors are
where we have used Vm
m
m
m
m
1 2
s
W -orthogonal. Setting
T s
Tm = Vm
W AVm ,
(3.6)
the above equation can also be written as
−1 T s −1
(D−1
m Tm + Im ∂t )am = δ(t)Dm Vm W M q
for t > 0.
(3.7)
The solution of the above equation (with vanishing initial conditions) is easily found as
−1 T s −1
am (t) = η(t) exp −D−1
T
t
Dm Vm W M q
for t > 0
m
m
(3.8)
and the approximate projected solution follows as
−1 T s −1
gm (t) = η(t)Vm exp −D−1
m Tmt Dm Vm W M q
(3.9)
for t > 0.
To evaluate this approximation, the exponent of a small m-by-m matrix D−1
m Tm needs to be
evaluated instead of the exponent of the large matrix A. Furthermore, since we are interested in
modes that are excited by the source vector M−1 q, it makes sense to choose the first basis vector
proportional to the source vector. Specifically, if we take
−1
v1 = n−1
s M q
(3.10)
with ns = kv1 k2 so that v1 has a unit 2-norm, we have
M−1 q = ns v1 = ns Vm e1 ,
(3.11)
where e1 is first m-by-1 canonical basis vector. Substitution in our approximate solution then
gives
gm (t) = η(t)ns Vm exp −D−1
T
t
e1
for t > 0.
(3.12)
m
m
Finally, if we now consider a source with a source signature w(t) and incorporate stabilitycorrection, we arrive at the field approximation
−1
fm (t) = w(t) ∗ 2η(t)ns Re Vm η(D−1
for t > 0.
(3.13)
m Tm ) exp −Dm Tmt e1
Below we will show that by exploiting the symmetry property AT Ws = Ws A, we can construct
the basis vectors vi via a Lanczos algorithm using a three-term recurrence relation if K m is
a polynomial Krylov space generated by matrix A and the source vector M−1 q. Moreover,
matrix Tm is then given by
Tm = Dm Hm ,
(3.14)
where Hm is a tridiagonal m-by-m matrix that satisfies
HTm Dm = Dm Hm
which is precisely the reduced counterpart of AT Ws = Ws A.
(3.15)
The projected field approximation then becomes
fm (t) = w(t) ∗ 2η(t)ns Re [Vm η(Hm ) exp(−Hmt) e1 ]
Modeling of wave propagation in open domains
for t > 0.
(3.16)
A Krylov subspace approach
CHAPTER 3. KRYLOV METHODS FOR MOR
3.3
25
Projection subspace
As mentioned in the introduction, Krylov subspaces show good convergence while approximating matrix functions or eigenvalues of matrices. Iterative solvers based on Krylov subspace
approaches include GMRES, BiCGstab, and QMR [15]. They are widely used in commercial
software and industry. Three types of Krylov subspaces suggested in the literature are now introduced. Then algorithms to construct these subspaces in software are given, together with a
short literature review of the described methods.
In this thesis, polynomial, extended, and rational Krylov subspaces are considered, given in this
order by
2
m−1
K PKS
b},
m = span{b, Ab, A b, . . . , A
EKS
−n+1
−1
K n;m = span{A
b, . . . , A b, b, Ab, A2 b, . . . , Am−1 b},
(3.17)
(3.18)
m
K RKS
= span{(A + σ1 I)−1 b, (A + σ2 I)−1 (A + σ1 I)−1 b, . . . , ∏ (A + σi I)−1 b}.
m
(3.19)
i=1
Polynomial Krylov subspaces basically build polynomials in the system matrix A on the seed
vector b. The extended Krylov subspace builds polynomials in A and A−1 on the seed vector b and therefore extends polynomial Krylov subspaces with negative powers of the system
matrix A.
A rational Krylov subspace consists of products of rational matrix functions on the seed
vector b. The rational matrix functions thereby represent single (complex) frequency solutions
of the system itself. In the following subsections it is shown how a basis for these subspaces
can be constructed.
For extended and polynomial Krylov subspaces, recursion relations to construct the basis
can be found in the case the system matrix is symmetric with respect to a weighted pseudoinner product. This leads to Lanczos-type algorithms, whereas Arnoldi algorithms can be used
without the presence of such a symmetry. The Arnoldi and Lanczos algorithms for polynomial
Krylov subspaces as well as the Arnoldi algorithm for rational Krylov subspaces are presented
in the following. The Lanczos algorithm for extended Krylov subspaces can be found in [16,17].
3.3.1
Polynomial Krylov subspaces
Two algorithms are presented for constructing a basis for polynomial Krylov subspaces (PKS),
namely, the Arnoldi and Lanczos algorithms. The Arnoldi algorithm has no symmetry constraints on the iteration matrix and utilizes the standard inner product on Cn . All basis vectors
need to be kept in memory in the Arnoldi algorithm. By contrast, the Lanczos algorithm exploits the symmetry of the system matrix and uses the Lagrangian or dispersion bilinear form
to generate a basis. The basis vectors can be obtained via three-term recurrence relations and
consequently at each iteration only the last two Lanczos vectors need to be kept in memory.
Arnoldi decomposition
For Arnoldi type algorithms the system matrix has no symmetry constraint [15]. The Arnoldi
algorithm constructs an orthonormal Krylov basis Vm and produces the so-called Arnoldi decomposition
AVm = Vm Hm + pm eTm
(3.20)
of matrix A. In this expression, Hm is an m-by-m upper Hessenberg matrix whose eigenvalues
approximate the ones of A. The residual pm eTm is a rank one matrix and pm is orthogonal to
the Krylov subspace. Given a starting vector b, the Arnoldi decomposition can be computed
A Krylov subspace approach
Modeling of wave propagation in open domains
26
3.3. PROJECTION SUBSPACE
via a modified Gramm Schmidt (MGS) procedure as shown in Algorithm 1. In this algorithm,
the previous basis vector is multiplied by the system matrix. After that the projections on all
previous basis vectors are subtracted from the vector in a modified Gramm-Schmidt manner.
Algorithm 1 Arnoldi MGS algorithm
γ1 = ||b||2
v1 = b/γ1
for j = 1, 2, . . . , m − 1 do
pj
= Av j
for i = 1, . . . , j do
hi, j = hp j , vi i
p j = p j − hi, j vi
end for
h j+1, j = ||p j ||2
v j+1 = pi /h j+1, j
end for
Lanczos decomposition
In the case the system matrix is symmetric with respect to a weighted inner product, a three-term
recursion relation between subsequent vectors can be found. This allows fast decomposition
with an algorithm complexity linear proportional to the subspace size. The Arnoldi algorithm
needs projections on all earlier vectors, which makes its computational cost quadratically dependent on the subspace size.
Since a three-term recursion relation can be found, each new vector in the basis only needs
to be orthogonalized with respect to the two previous vectors. Therefore each iteration of the
Lanczos algorithm has the same computational costs, and only three Lanczos vectors need to fit
the memory of the computational architecture to achieve the decomposition.
As shown in the last section, the Arnoldi algorithm constructs the decomposition
AVm = Vm Hm + pm eTm
(3.21)
with the residual pm eTm a matrix of rank one and pm orthogonal to Vm .
To derive the Lanczos algorithm, we replace the standard inner product on Cn by the bilinear
form hx, yiWs = yT Ws x and we normalize the basis vectors such that each vector has unit 2norm. With δi = hvi , vi iWs , we then arrive at the decomposition
AVm = Vm Hm + pm eTm ,
(3.22)
T Ws V = diag(δ , δ , ..., δ ) = D and p is Ws -orthogonal to V . From the above
with Vm
m
m
m
m
m
1 2
decomposition, we obtain
T s
T s
W Vm Hm = Dm Hm
Vm
W AVm = Vm
(3.23)
and from this equation it follows that Dm Hm is symmetric, since A is symmetric with respect to
Ws . In other words, we have
HTm Dm = Dm Hm
which is the reduced counterpart of AT Ws = Ws A.
(3.24)
Furthermore, since Dm Hm is symmetric, the Hessenberg structure of Hm simplifies to a tridiagonal structure and consequently the basis vectors can be constructed via a three-term recurrence
relation. The following Lanczos algorithm implements the above steps by constructing a Ws orthogonal basis for the Krylov space K PKS
m .
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 3. KRYLOV METHODS FOR MOR
27
Algorithm 2 Modified Lanczos algorithm
ζ1 = ||b||2
v1 = b/ζ1
δ0 = 1
δ1 = hv1 , v1 iWs
for i = 1, 2, . . . , m − 1 do
αi = δi−1 hAvi , vi iWs
pi = Avi − αi vi − ζi δi δ−1
i−1 vi−1
ζi+1 = ||pi ||2
vi+1 = pi /ζi+1
δi+1 = hvi+1 , vi+1 iWs
end for
Observe from the algorithm that the basis vectors are generated via the three-term recurrence
relation
δi
ζi+1 vi+1 = Avi − αi vi − ζi
vi−1
(3.25)
δi−1
and consequently the ith basis vector (a.k.a. Lanczos vector) is given by
vi = pi−1 (A)b,
(3.26)
where pi−1 (·) is the so-called Lanczos polynomial. To show the relation between Lanczos
polynomials and orthogonal polynomials of one variable, Favard’s Theorem is introduced as
Theorem 1. Favard’s Theorem: Polynomials satisfying
pn+1 (x) = (an x + bn )pn − cn pn−1
x ∈ [−1, 1] an , cn > 0
(3.27)
are orthogonal with respect to a weighted inner product space.
Examples of such orthogonal polynomials are Legendre and Chebychev polynomials. The solution of the Laplacian operator on a spherical domain is given by Legendre polynomials given
in the notation used in Eq. (3.27) as
an =
n
2n + 1
, bn = 0, cn =
, p0 = 1, p1 = x
n+1
n+1
(3.28)
or Chebyshev polynomials of first kind widely used for interpolation are given by
an = 2, bn = 0, cn = 1, p0 = 1, p1 = x.
(3.29)
Comparing the recursion relation for the Lanczos vectors with Favard’s Theorem given in
Eq. (3.27), it can be seen that it fits the same formulation. In [18], an extensive treatment
on the relation between Lanczos vectors and orthogonal polynomials can found.
3.3.2
Extended Krylov subspace
For extended Krylov subspaces, five term recursion relations can be found as shown in [16, 17].
It was shown that extended Krylov subspaces can approximate matrix functions efficiently,
having smaller errors than polynomial subspaces for the same dimension. This paper further
discusses the relation between Laurent polynomials and the extended Krylov subspaces.
Approximation of general parameter-dependent matrix equations and of transfer function by
projection onto an extended Krylov subspace was investigated in [19]. This research showed the
A Krylov subspace approach
Modeling of wave propagation in open domains
28
3.3. PROJECTION SUBSPACE
effectiveness of this approach for realistic applications such as Sylvester equations and shifted
systems.
Within the extended Krylov subspace approach for electromagnetic systems, one basically
needs to solve Poisson systems while projecting the previous vector on A−1 . For wave equations in open domains it was recently shown that extended Krylov subspace approximations can
outperform conventional methods especially when low-frequency solutions are required [12].
3.3.3
Rational Krylov subspace
In the rational subspace approach, the system is projected on a sequence of single frequency
solutions of the system with b as source. A general rational Krylov subspace is a subspace of
the form
m
K RKS
= span{(A + σ1 I)−1 b, (A + σ2 I)−1 (A + σ1 I)−1 b, . . . , ∏ (A + σi I)−1 b}.
m
(3.30)
i=1
With properly selected shifts these subspaces can approximate a matrix function within a region
of interest with only a small fraction of the dimension needed in polynomial Krylov subspaces.
The systems that need to be solved are Helmholtz like systems for different complex frequencies
σi . Because of a lack of efficient Helmholtz preconditioners for iterative solvers, these systems
are computationally expensive, which for certain problems however is outweighted by the fast
convergence.
The convergence is strongly dependent on the choice of the initial shifts. Several shift selection techniques have been developed in literature. The work done in [20] presents an algorithm
which adapts its shifts and tunes its spectrum towards the spectrum of operator A. It was shown
that this approach was superior to equidistant shifts in the case A has a non-uniform spectrum.
The problem of selecting optimal shifts for rational Krylov subspaces for the Maxwell diffusion
equation was reduced to a third kind Zolotarev problem in [11, 21] and showed improved convergence compared with equidistance shifts. Finally, in [22] it was shown that rational Krylov
subspace approaches are attractive for solving the inverse problem. The paper [23] presents an
iterative H2 optimal RKS algorithm for approximating dynamic systems.
Using modified Gramm-Schmidt projections for orthogonalization of the basis and given certain
shifts denoted by σ = {σ1 , . . . , σm }, a rational Arnoldi MGS algorithm can be defined as shown
in Algorithm 3.
Algorithm 3 Rational Arnoldi MGS algorithm
γ1 = ||b||2
v0 = b/γ1
for j = 1, 2, . . . , m do
pj
= (A + σi )−1 v j−1
for i = 1, . . . , j − 1 do
p j = p j − hp j , vi ivi
end for
vj
= p j /||p j ||2
end for
Within this algorithm the system p j = (A + σi )−1 v j−1 can be solved iteratively or directly using
sparse solvers. If only a single shift is used so σi = σfix ∀i = 1, . . . , m, an LU (or other) decomposition of the shifted system (A + σfix ) is computationally attractive as the same LU decomposition can be used in every iteration. In this thesis the use of iterative solvers was not explored,
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 3. KRYLOV METHODS FOR MOR
29
only direct sparse solvers were used.
3.3.4
Convergence Comparison
To obtain insight into how PKS, EKS, and RKS approximate the spectrum of a matrix, we compared these methods from a viewpoint of spectral approximation via power methods. The power
method, inverse power method, and shifted inverse power method are reviewed and linked to
PKS, EKS, and RKS subspace approximations. All power methods are well described by Saad
in [24].
In these power methods, the matrix A, A−1 , or (A − σI)−1 is raised to a large power and
multiplied by a vector in an iterative manner. In the case the matrix has one dominant eigenvalue, and the starting vector has a component in the direction of the associated eigenvector, the
method converges towards the dominant eigenpair.
In Algorithm 4 the value α j converges to the eigenvalue with largest modulus λmax , and v j
to its associated eigenvector under the condition that there is only one eigenvalue of largest
modulus λmax . Furthermore, v0 needs to have a component in the invariant subspace of λmax for
convergence.
Using the inverse of a matrix in the power method, the eigenvalue with the smallest modulus
can be found. In the inverse power method as presented in Algorithm 5, the value of α j converges to 1/λmin with λmin the eigenvalues of smallest modulus. Furthermore, v j converges to
the associated eigenvector. For convergence similar constrains as for the power method holds.
The whole spectrum of matrix A can be shifted by a constant value which leads to shifted
power methods. By subtracting a value σ from the diagonal of A the whole spectrum shifts by
−σ. The eigenvalue λσ of A which is closest to σ becomes the smallest eigenvalue of the shifted
system as illustrated in Figure 3.1. Therefore, given specific convergence constraints similar to
the power method, the shifted inverse power method as shown in Algorithm 6 converges. The
value α1 converges to λσ − σ and v j to the eigenvector of λσ .
Algorithm 4 Power method
Chose nonzero start vector v0
Chose tolerance tol
j=0
while ||Av j − α j v j || > tol do
j = j+1
α j = Element of Av j−1 with max. mod.
v j = α1j Av j−1
end while
Algorithm 5 Inverse power method
Chose nonzero start vector v0
Chose tolerance tol
j=0
while ||Av j − α1j v j || > tol do
j = j+1
α j = Element of A−1 v j−1 with max. mod.
v j = α1j A−1 v j−1
end while
Algorithm 6 Inverse shifted power method
Chose nonzero start vector v0
Chose tolerance tol
j=0
while ||Av j − ( α1j + σ)v j || > tol do
j = j+1
α j = Element of (A − σI)−1 v j−1 with max. modulus
v j = α1j (A − σI)−1 v j−1
end while
A Krylov subspace approach
Modeling of wave propagation in open domains
30
3.3. PROJECTION SUBSPACE
Krylov subspace methods can be seen as a variation of power methods. Where power methods
target to approximate one specific eigenpair, Krylov subspace methods target to approximate
the whole or at least a substantial part of the spectrum of a matrix. At each iteration the new
vector is orthogonalized with respect the whole basis, such that once an eigenvector is within
the subspace, the following vectors have no component in the invariant subspace of the associated eigenvalue. The latter is a requirement for convergence towards that specific eigenvalue.
Therefore, the method approximates new eigenvectors with every iteration.
By comparing the basic iteration of the rational, extended, and polynomial Krylov subspace
with the described power methods, similarities can be found. The ordinary power method has
the same basic iteration as the PKS method, while the EKS combines inverse and ordinary
power method. Finally, the rational Krylov subspace method has the same core iteration as the
inverse shifted power method.
Although this comparison is a large simplification of the convergence of Krylov subspace
methods, especially when multiple shifts are used, it leads to basic and rough insight into how
the subspace methods converge.
The PKS approach approximates large eigenvalues first. With increasing dimension of the
subspace the poles of the reduced-order system converge towards smaller eigenvalues. The
polynomial Krylov subspace approach therefore converges from the outside to the inside of the
spectrum of matrix A.
For a RKS approach the shifts σ can be selected in such a manner that the projected system
converges form a point in the complex plane outwards, or on a certain domain of interest in
the region of the selected shifts. This domain of interest can be defined as a rectangle in the
complex plane via the bandwidth of the input signal and the time of interest. The former puts
a constraint on the imaginary part and the latter on the real part of the poles of relevance. The
illustration in Figure 3.1 shows which choice of shifts leads to well- and ill-conditioned systems
and also shows the effect of shifting a system. The shifts in the RKS approach can be used to
adapt the spectrum of the approximation subspace to the spectrum of the operator. According to
the power method analogy, the shifts should be chosen as mirror images of the poles of interest.
(a) Poles of the original system.
(b) Poles of the shifted system.
Figure 3.1: Illustration showing the influence of shifts on the spectrum of the operator A in complex plane.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 3. KRYLOV METHODS FOR MOR
31
The EKS approach can be viewed as a rational Krylov subspace approach with one interpolation
point at the origin of the extended complex plane and the other at infinity. It therefore converges
from the origin outwards and from infinity inwards. Figure 3.2 shows a schematic comparison
of the convergence behavior of the three Krylov methods.
In the numerical experiments with RKS presented in the next chapter, direct solvers are to
solve the Helmholtz-type systems. However, for larger systems iterative solvers might need
to be incorporated into the RKS algorithm, such that a shift selection in the well-conditioned
quadrant might be beneficial. Using the conjugated transpose instead of the mirror image of the
poles of interest for instance leads to well-posed systems. In fact for poles belonging to a high
quality factor resonance, both choices are very similar and both show good convergence.
Im(s)
Re(s)
Figure 3.2: Convergence of eigenvalues schematically shown for polynomial (PKS), extended (EKS) and rational
(RKS) Krylov subspace projections. The stable poles are shown in the fourth quadrant.
A Krylov subspace approach
Modeling of wave propagation in open domains
32
Modeling of wave propagation in open domains
3.3. PROJECTION SUBSPACE
A Krylov subspace approach
Chapter 4
Results
The numerical examples illustrating the developed methods are presented in this chapter. First
model order reduction of instantaneously reacting media is presented in one, two, and three spatial dimensions, given in Section 4.1, Section 4.2 and Section 4.3, respectively. The examples
are chosen to illustrate the convergence properties of the various Krylov methods in the timeand frequency-domain, as well as the capability of the methods to deal with very large systems.
Two numerical examples are presented for media exhibiting relaxation, namely, a twodimensional and a three-dimensional one using a Drude dispersion model. The two-dimensional
example shown in Section 4.4 is used to validate the method and shows convergence of the timedomain result to an ADE-FDTD comparison simulation.
The three-dimensional example given in Section 4.5 is used two compare the frequencydomain result of the reduced-order model with the frequency-domain result of an aperiodic
Fourier modal method used in [1] to calculate the spontaneous decay rate of a quantum emitter
in close proximity to a golden nanorod.
Polynomial Krylov subspaces are used in all numerical experiments. In addition, model
order reduction using rational Krylov subspaces is presented for all two-dimensional examples.
Finally, the performance of the different algorithms for the chosen examples are compared in
Section 4.6.
4.1
Instantaneously reacting materials – One-dimensional configurations
In this section two one-dimensional examples of instantaneously reacting media are given to
show that the numerical scattering poles coincide with the analytical poles. Therefore it can be
concluded that the numerical method describes the physics correctly. It validates the approach
of truncating a domain with an optimal PML to simulate its extension to infinity and find its
resonances.
The Lanczos algorithm as presented in Algorithm 2 is used for all examples shown in this
section. All domains are truncated with an optimal PML for propagating and evanescent waves.
The optimization interval is tuned to the frequency of operation.
4.1.1
Dielectric slab
The first investigated configuration is a one-dimensional electromagnetic wave problem. A
dielectric slab with εslab
= 4 and width of λpeak is simulated. The configuration is shown in
r
Figure 4.1. As a source wavelet, the time derivative of a Gaussian pulse is used, with a maximum
of the spectrum at λpeak .
The acoustical equivalent of the described configuration is studied in [25]. With this first
example it will be shown that the results obtained in [25] can be reproduced with a model order
33
34
4.1. INSTANTANEOUSLY REACTING MATERIALS – 1D CONFIGURATIONS
reduction approach. In the cited work the full equations are solved in the frequency-domain.
Therefore the approach presented in this thesis is advantageous in two ways. First, the model
order reduction approach allows for smaller models and faster computation. Second, the cited
approach can only calculate Fourier domain solutions and no direct solutions in the time-domain
can be obtained.
In order to check the convergence of the results calculated with the reduced-order model,
an FDTD algorithm with a leapfrog time stepping scheme is used for comparison as described
in [26, 27]. The domain is truncated using the optimal PML and the problem size is n = 1019.
(500 secondary grid points, 499 primary grid points, 4×5 PML layers). The convergence of the
Lanczos reduced-order model towards the FDTD comparison response is show in Figure 4.2.
Simulated configuration
2.5
Contrast
Source
Receiver
PML
Contrast (ε µ)0.5
2
1.5
1
0.5
0
0
1
2
3
Length in λpeak
4
5
Figure 4.1: Simulated configuration: A one-dimensional slab with source and receiver on one side.
The analytical poles of the system are calculated in [28] and follow from the condition
1=
Y vac −Y slab
Y vac +Y slab
2
q
slab
slab
slab
slab
exp −2 sµ (σ + sε )d
with
r
Y=
σ + sε
.
sµ
For the given configuration and with normalized parameters for numerical stability, the scattering poles are given by
1
iπ
ssc = slab ln(9) − slab n ∀n ∈ N+ .
4d
2d
Figure 4.2 shows the location of the analytical poles, the poles of the operator A, and the poles
of the Lanczos reduced-order model as they converge. It can be seen that the late scattering
events that happen inside the slab are modeled correctly, as soon as the Lanczos poles converge
to the analytical scattering poles.
Furthermore, it can be seen that the Lanczos poles approximate the poles of A. Furthermore,
for high imaginary values of s, the poles of A do not match the analytical poles, since the
spatial discretization and PML are not tuned for those frequencies. The figure also shows that
the scattering poles are relatively small in absolute value compared with the other poles of A.
Therefore the convergence of the Lanczos algorithm is not optimal, as it usually finds the large
eigenvalues first.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
35
Scattering Poles
Lanczos m=800
FDTD
1
−0.05
m=800
−0.1
0.8
Poles of the operator A
Analytical poles
Exitation frequency
Lanczos Poles
−0.15
−0.2
−0.25
−0.3
0.6
Electric field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
0
−0.35
0.4
0.2
0
−0.2
−0.4
−0.6
−0.4
−0.8
−0.45
−0.5
−1
0
0.005
0.01
0.015
0.02
0.025
Real axis, (normalized frequency Re(s))
0
0.03
0.5
1
1.5
2
Time [ps]
2.5
3
3.5
4
(b) Time-domain result for m = 800 .
(a) Poles of the system in comparison with the Lanczos
approximation for m = 800 .
Scattering Poles
Lanczos m=900
FDTD
1
−0.05
m=900
−0.1
0.8
Poles of the operator A
Analytical poles
Exitation frequency
Lanczos Poles
−0.15
−0.2
−0.25
−0.3
0.6
Electric field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
0
−0.35
0.4
0.2
0
−0.2
−0.4
−0.6
−0.4
−0.8
−0.45
−0.5
−1
0
0.005
0.01
0.015
0.02
0.025
Real axis, (normalized frequency Re(s))
0
0.03
0.5
1
1.5
2
Time [ps]
2.5
3
3.5
4
(d) Time-domain result for m = 900 .
(c) Poles of the system in comparison with the Lanczos
approximation for m = 900 .
Scattering Poles
Lanczos m=1000
FDTD
1
−0.05
m=1000
−0.1
0.8
Poles of the operator A
Analytical poles
Exitation frequency
Lanczos Poles
−0.15
−0.2
−0.25
−0.3
−0.35
0.6
Electric field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
0
0.4
0.2
0
−0.2
−0.4
−0.6
−0.4
−0.8
−0.45
−0.5
−1
0
0.005
0.01
0.015
0.02
0.025
Real axis, (normalized frequency Re(s))
0.03
(e) Poles of the system in comparison with the Lanczos
approximation for m = 1000 .
0
0.5
1
1.5
2
Time [ps]
2.5
3
3.5
4
(f) Time-domain result for m = 1000 .
Figure 4.2: Convergence of the ROM poles and time-domain solution with increasing ROM dimension.
The eigenfunctions shown in Figure 4.3 show good agreement with the eigenfunctions given
in [25], where the acoustic equivalent case was studied. In the results shown, even and uneven
eigenfunctions can be found, equivalent to the cited work. The amplitude of the waveform
increases with increasing distance from the origin. In addition, the eigenfunctions decay with
increasing time or once they have reached the PML. When viewed in time it can be seen that
standing waves can be found inside the slab and traveling waves outside the slab. The eigenA Krylov subspace approach
Modeling of wave propagation in open domains
36
4.1. INSTANTANEOUSLY REACTING MATERIALS – 1D CONFIGURATIONS
functions modeled correctly correspond to 2π − 12π resonances. The corresponding poles have
been labeled in Figure 4.4.
basis vector of pole s=0.0058807−0.18707i at t=0. Pole nr 1
basis vector of pole s=0.0054472−0.12533i at t=0. Pole nr 5
basis vector of pole s=0.0054827−0.047114i at t=0. Pole nr 10
1
2
1
0.8
0.8
1.5
0.6
0.6
1
0.4
0.2
0
−0.2
0.5
Electrical field [a.u.]
Electrical field [a.u.]
Electrical field [a.u.]
0.4
0
−0.5
−0.4
0.2
0
−0.2
−0.4
−1
−0.6
−0.6
−1.5
−0.8
−0.8
−1
−2
−1
0
0.5
1
1.5
2
2.5
3
Domain length in λ
3.5
4
4.5
5
0
(a) 12π resonane of pole #1.
0.5
1
1.5
2
2.5
3
Domain length in λ
3.5
4
4.5
5
(b) 8π resonance of pole #5.
0
0.5
1
1.5
2
2.5
3
Domain length in λ
3.5
4
4.5
5
(c) 3π resonance of pole #10.
Figure 4.3: Snapshot at t = 0 of some of the eigenfunctions that belong to the scatteringpoles.
Scattering Poles
Sc.Pole # 11
Sc.Pole # 10
Sc.Pole #9
Imaginary Axis, (normalized frequency Im(s))
−0.05
Sc.Pole # 8
m=1250
Poles of the operator A
Analytical poles
Exitation frequency
Lanczos Poles
Sc.Pole # 7
Sc.Pole # 6
−0.1
Sc.Pole # 5
X: 0.01483
Y: −0.1186
Sc.Pole # 4
Sc.Pole # 3
−0.15
Sc.Pole # 2
Sc.Pole # 1
−0.2
−0.25
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Real axis, (normalized frequency Re(s))
0.016
0.018
0.02
Figure 4.4: Labeled scattering poles to show the eigenvalues of the resonances.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
4.1.2
37
Photonic crystal
The second one-dimensional example is a periodic structure. Structures like this can for instance
be found in lasers or chips of integrated circuits. The simulated configuration is illustrated in
Figure 4.5. The source is placed in the middle of the configuration. The thickness of the layers
is ap = 3 µm and bp = 30 µm and the total finite lattice consists of 41 primitive cells. Finally,
we use a step size of δ = 1 µm and nine PML layers resulting in a total number of n = 2761
unknowns.
Figure 4.5: Illustration of the simulated 1D periodic structure. The parameters are given by ap = 3 µm, bp = 30 µm
and ε1 = 3, ε2 = 1.
An infinite periodic structure can also be analyzed using the plane wave expansion method
described in Chapter 4 of [29]. With this method the band structure of an infinite periodic
structure can be calculated. Now the Lanczos poles, which match the poles of the original
operator A, of the finite periodic structure can be compared to the band structure obtained by
the plane wave expansion method.
The comparison is shown in Figure 4.6, where the normalized imaginary Laplace frequencies
are depicted on the y-axis. On the x-axis the PWE method depicts the wave vector in the
fundamental interval (Brillouin zone) [−π/(ap + bp ), π/(ap + bp )]whereas the Krylov subspace
approximation plots the normalized real Laplace frequencies.
The figure shows that the Krylov subspace approach is able to capture the band structure of a
periodic structure. Moreover, the frequencies associated with the middle of a band have higher
radiation losses that the frequencies at the edges of a band. In other words, the quality factor of
the resonances in the structure of the frequencies at the sides of a band is higher.
PWE method
Krylov subspace approximation
0.7
Imaginary Axis, (normalized frequency Im(s))
0.7
Relative frequency ωr=ω L/c
0.6
0.5
0.4
0.3
0.2
0.1
0
−5
0
5
Wave vector k, [10−4 m−1]
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
Real axis, (norm. freq. Re(s)[10−3] )
Figure 4.6: Band structure of the photonic crystal, calculated via plane wave expansion, and using the Krylov
subspace approach.
A Krylov subspace approach
Modeling of wave propagation in open domains
38
4.1.3
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Conclusion
In this section it was shown that truncating the domain with an optimal PML is a valid approach to simulate an open system. The numerically calculated resonances match the analytical
resonances.
The one-dimensional slab experiment showed that the analytical poles match the poles of
the truncated operator A. The one-dimensional periodic structure showed that even multiple
scattering events in a periodic structure are modelled correctly by the chosen approach. The
resulting scattering poles showed a clear band structure. It can be concluded that the numerical
method describes the physics correctly.
4.2
Instantaneously reacting materials – Two-dimensional configurations
In this section two-dimensional systems are analyzed in order to show that the order of a system
can be reduced and the correct time-domain behavior can be obtained without loss of accuracy.
Furthermore, it will be shown that the field can be expanded in a small number of scattering
poles using the scattering expansions of Eq. (2.138). The analyzed configurations are a dielectric box, a photonic crystal waveguide, and a layered Earth model.
4.2.1
Dielectric box
Polynomial Krylov MOR
The first investigated configuration is a two-dimensional electromagnetic wave problem. A
dielectric box with a relative permittivity εr = 4 and width of 50 µm is simulated. The source
and receiver are placed inside the box. The configuration is shown in Figure 4.7. The time
derivative of a Gaussian pulse, with a maximum of the spectrum at λpeak = 94 µm, is used as
source wavelet for an H-polarized electromagnetic field.
The domain is truncated using an optimal PML and the problem size is n = 37000. The convergence of the Lanczos reduced-order model towards the FDTD comparison response takes
1500 iterations and is shown in Figure 4.8.
Simulated configuration
0
10
20
x−length in L
30
Object ε=4
Background ε=1
Source
Receiver
PML
40
50
60
70
80
90
100
0
20
40
60
y−Length in L
80
100
Figure 4.7: Two dimensional dielectric box with source and receiver inside the box.
We can now try to expand the field in a reduced set of resonances that are located in a
rectangular subdomain of the complex plane determined by the frequency band of the pulse
and the length of the time interval of observation. To select the dominant scattering poles, two
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
39
1.2
1.2
Lanczos m=1000
FDTD
1
0.8
0.8
Magnetic field [a.u.]
Magnetic field [a.u.]
0.6
0.4
0.2
0
−0.2
0.6
0.4
0.2
0
−0.2
−0.4
−0.4
−0.6
−0.8
Lanczos m=1100
FDTD
1
0
0.5
1
1.5
2
Time in [ps]
2.5
3
−0.6
3.5
0
(a) Time-domain result for m = 1000 .
1.5
2
Time in [ps]
2.5
3
3.5
1.2
Lanczos m=1200
FDTD
1
0.8
0.8
0.6
0.6
0.4
0.2
0
0.4
0.2
0
−0.2
−0.2
−0.4
−0.4
0
0.5
1
1.5
2
Time in [ps]
2.5
3
(c) Time-domain result for m = 1200 .
Lanczos m=1500
FDTD
1
Magnetic field [a.u.]
Magnetic field [a.u.]
1
(b) Time-domain result for m = 1100 .
1.2
−0.6
0.5
3.5
−0.6
0
0.5
1
1.5
2
Time in [ps]
2.5
3
3.5
(d) Time-domain result for m = 1500 .
Figure 4.8: Convergence of the ROM system response with increasing ROM dimension.
aspects are of importance. First, the time window of interest puts a constraint on the real part of
the relevant poles and second, the imaginary part of the poles can be bounded by the frequency
spectrum of the input signal.
For the studied problem, 26 scattering poles lie in the rectangular subdomain of interest. The
selected poles and corresponding time-domain signal are shown in Figure 4.9. It can be seen
that the 26 poles describe the system response well except for times earlier than 0.4 ps. Here
the impact of the expected error dropping exponentially in time can be seen, as all late times are
modelled correctly.
By selecting the thirteen poles with an 1/e decay faster and slower than 1 ps, we can choose
whether early or late time events are modeled correctly. It can be seen that changing the constraint on the real part of the poles of interest changes the time interval on which the expansion
converges.
The experiment shows that the scattering response can be expanded in a small number of scattering poles. The Lanczos reduced-order model is already a factor 20 smaller than the full order
model, but the result shows that dependent on the time interval of observation a model order
reduction of up to 1000 is possible.
Snapshots at t = 0 of two of the resonances f res that belong to the scattering poles are shown
in Figure 4.10 (calculated via Eq. (2.138)). Only the fundamental magnetic field is shown.
When viewed in time, the eigenfunctions show a standing wave pattern inside the box and
traveling waves are present outside the box.
A Krylov subspace approach
Modeling of wave propagation in open domains
40
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Scattering Poles
m=1500
1.2
Lanczos Poles
Scattering Poles
Exitation frequency
−0.1
0.8
−0.2
−0.3
−0.4
0.6
0.4
0.2
0
−0.2
−0.5
−0.4
0
0.01
0.02
0.03
0.04
Real axis, (normalized frequency Re(s))
Scattering Poles
0
−0.6
0.05
(a) The 26 scattering poles needed to model the whole
time-domain solution.
m=1500
0.5
1
1.5
2
Time in [ps]
2.5
3
3.5
1.2
Lanczos Poles
Scattering Poles
Exitation frequency
−0.1
0
(b) Time solution expanded in 26 scattering poles.
Lanczos only scatteringpoles m=1500; #scp13
FDTD
1
0.8
−0.2
Magnetic field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
Lanczos only scatteringpoles m=1500; #scp26
FDTD
1
Magnetic field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
0
−0.3
−0.4
0.6
0.4
0.2
0
−0.2
−0.5
−0.4
0
0.01
0.02
0.03
0.04
Real axis, (normalized frequency Re(s))
−0.6
0.05
0
0.5
1
1.5
2
Time in [ps]
2.5
3
3.5
(c) The 13 scattering poles needed to model the late time- (d) Time solution expanded in 13 scattering poles with the
domain solution.
smallest real part.
Scattering Poles
m=1500
1.2
Lanczos Poles
Scattering Poles
Exitation frequency
−0.1
Lanczos only scatteringpoles m=1500; #scp13
FDTD
1
0.8
0.6
−0.2
Magnetic field [a.u.]
Imaginary Axis, (normalized frequency Im(s))
0
−0.3
−0.4
0.4
0.2
0
−0.2
−0.4
−0.5
−0.6
0
0.01
0.02
0.03
0.04
Real axis, (normalized frequency Re(s))
0.05
−0.8
0
0.5
1
1.5
2
Time in [ps]
2.5
3
3.5
(e) The 13 scattering poles needed to model the early (f) Time solution expanded in 13 scattering poles with the
time-domain solution.
largest real part.
Figure 4.9: Expansion of the system response in the scattering poles of the system. Left: the selected scattering
poles to capture the field approximation. Right: resulting field approximation. Early times are captured by selecting
poles with a small real part.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
41
(a) 2π− resonance of the box.
(b) 5π− resonance of the box.
Figure 4.10: Snapshots of two resonance eigenfunctions f res of the system in the time-domain. The black box
indicates the boundary of the box.
A Krylov subspace approach
Modeling of wave propagation in open domains
42
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Rational Krylov MOR
Magnetic field [a.u.]
1
RKS m=48
FDTD
0.5
0
−0.5
2
4
6
8
10
Time in [ps]
12
14
16
(a) System response of the reduced model.
Imaginary Axis, (normalized frequency Im(s))
The dielectric box example together with the RKS approach is used to show that the method
converges using equidistant shifts. Initial pole selection is done in a manner similar to defining
a selection criterion for resonance expansions. The imaginary part of interest is defined by the
bandwidth of the used pulse, whereas the real part is set by the time interval of interest.
Subsequently, equidistant shifts are used to fill the defined square. Using these shifts to
construct the rational Krylov subspace, an approximate solution can be obtained by projection
of the system matrix. In Figure 4.11 the response approximated by the RKS and FDTD are
compared. It can be seen that the whole system response is approximated well after m = 48
iterations.
Furthermore, the interpolation points of the RKS together with the final system poles and
the eigenvalues of the non-reduced system are shown in the right part of the figure. It can be
seen from the figure that most poles of the projected system approximate an eigenvalue of the
system matrix and the operator is well approximated inside the rectangle defined by the initial
shifts, whereas only individual poles are approximated outside the rectangle.
0
−0.1
−0.2
m=48
−0.3
Poles of projected system
Mirrored interpolation poles
eigs − systempoles
Exitation frequency
−0.4
−0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
Real axis, (normalized frequency Re(s))
(b) Poles of the reduced and original system.
Figure 4.11: Rational Krylov subspace approximation of the box problem with equidistant shifts.
4.2.2
Photonic crystal waveguide
In the second two-dimensional example, a photonic crystal waveguide is simulated as depicted
in Figure 4.12. The configuration is simulated using E-polarized fields and has n ≈ 0.7 million
unknowns. The depicted cylinders have a relative permittivity of εr = 11.56, whereas vacuum is
assumed in between the cylinders. As a current source, a Gaussian pulse shifted to a frequency
of fs = 192.21 THz was used. This frequency lies in the band gap of the simulated photonic
crystal, such that the wave can only propagate in the path of crystal.
Polynomial Krylov MOR
The convergence of the electrical field at the receiver towards an FDTD comparison result is
shown in Figure 4.13. After roughly 3000 Lanczos iterations the signal converged to the FDTD
result. Again the signal at the receiver can be expanded in resonance eigenfunctions of the
system. In the rectangle of interest, only a single pole (!) can be found as shown in Figure 4.14(a). This single pole can describe the time-domain solution at the receiver as shown in
Figure 4.14(b). Essentially this means that a single mode is present in the photonic waveguide.
This example clearly shows the impact of resonance expansions for certain type of structures.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
43
Figure 4.12: Simulated photonic waveguide crystal. The source is marked by an X whereas the receiver is marked
as a triangle.
A Krylov subspace approach
Modeling of wave propagation in open domains
44
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Lanczos m=1000
FDTD
1
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
Time in [ps]
0.4
0.5
0.6
(a) Time-domain result for m = 1000.
1
Lanczos m=2000
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
Time in [ps]
0.4
0.5
0.6
(b) Time-domain result for m = 2000.
1
Lanczos m=3000
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.1
0.2
0.3
Time in [ps]
0.4
0.5
0.6
(c) Time-domain result for m = 3000.
Figure 4.13: Convergence of the ROM system response with increasing ROM dimension.
Modeling of wave propagation in open domains
A Krylov subspace approach
45
Imaginary Axis, (normalized frequency Im(s))
CHAPTER 4. RESULTS
Scattering Poles
−30
m=3000
Lanczos Poles
Scattering Poles
Exitation frequency
−35
−40
−45
−50
0
1
2
3
4
Real axis, (normalized frequency Re(s))
5
(a) Used scattering pole in the expansion.
1
Lanczos only scatteringpoles m=3000; #scp1
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.1
0.2
0.3
Time in [ps]
0.4
0.5
0.6
(b) Time-domain result for m = 3000 with one scattering pole.
Figure 4.14: Expansion of the system response in the scattering poles of the system.
A Krylov subspace approach
Modeling of wave propagation in open domains
46
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Rational Krylov MOR
m=10
10
Poles of projected system
Mirrored interpolation poles
Exitation frequency
0
−10
−20
−30
−40
−50
−60
−70
0
1
2
3
4
5
Real axis, (normalized frequency Re(s))
6
Imaginary Axis, (normalized frequency Im(s))
Imaginary Axis, (normalized frequency Im(s))
In the last section it was shown that the response of the photonic crystal waveguide example
could be expanded in a single pole. Therefore, next to the standard method of using equidistant
shifts, also single shifts are used in this example. As a single shift, the pole found in the
scattering expansions of the last chapter is used. Obviously, this is a priori information which is
normally not available. However it shows that convergence is influenced by the choice of shifts.
The initial shifts and final system poles of both approaches are shown in Figure 4.15, where
equidistant shifts are used in Figure 4.15 (a,c), while a single shift is used in Figure 4.15 (b).
The resulting time-domain responses are shown in Figure 4.16.
After ten iterations, the early-time response is well approximated independent of the choice
of shifts. The RKS using equidistant shifts converged after 20 iterations, where only 15 iterations are needed in the case of using a single shift. In Figure 4.15(b) it can be seen that the
interpolation pole (mirror image of σ) coincides with a system pole. This system pole is the
same pole which lead to a resonance expansion with only one mode in the last chapter.
m=15
10
Poles of projected system
Mirrored interpolation pole
Exitation frequency
0
−10
−20
−30
−40
−50
−60
−70
0
1
2
3
4
5
Real axis, (normalized frequency Re(s))
6
Imaginary Axis, (normalized frequency Im(s))
(a) Poles of the reduced system and interpolation shifts (b) Poles of the reduced system using a single shift and
for m=10.
m=15
m=20
10
Poles of projected system
Mirrored interpolation poles
Exitation frequency
0
−10
−20
−30
−40
−50
−60
−70
0
1
2
3
4
5
Real axis, (normalized frequency Re(s))
6
(c) Poles of the reduced system and interpolation shifts
for m=20.
Figure 4.15: Rational Krylov subspace approximation of the spectrum of the crystal problem with equidistant
shifts.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
47
1
RKS m=10
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
50
100
150
200
250
Time in [ps]
300
350
400
450
500
(a) System response for a RKS dimension of m=10.
1
RKS m=15
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
50
100
150
200
250
Time in [ps]
300
350
400
450
500
(b) Converged system response for a RKS dimension of m=15 using a single shift.
1
RKS m=20
FDTD
0.8
0.6
Electric field [a.u.]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
50
100
150
200
250
Time in [ps]
300
350
400
450
500
(c) Converged system response for a RKS dimension of m=20.
Figure 4.16: Rational Krylov subspace approximation of the crystal problem response with equidistant shifts and
single shifts.
A Krylov subspace approach
Modeling of wave propagation in open domains
48
4.2.3
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Layered Earth Example
In this example, a layered Earth structure is analyzed. The simulated configuration is shown in
Figure 4.17 and includes a horizontal Earth layer and a vertical drilling hole. For this problem
the problem size is n ≈ 545000. In industry these kind of structures are simulated for drilling
hole characterization. Within inversion schemes they are simulated using FDTD codes followed
by Prony-type methods to obtain the dominant poles of the transfer function responses between
a source and a receiver. Using a Krylov model order reduction approach has the advantage that
the transfer function follows from the configuration and algorithm itself.
10
0
Background ε
Layer ε
Drilling Hole ε
Source
Receiver
PML
0.2
0.4
x−Length in L
0.6
9
8
7
6
0.8
air
λ air =0.375L
λ air =1L
NF:
LF:
ELF: λ
1
1.2
5
=3L
4
1.4
3
1.6
2
1.8
1
2
0
0.5
1
y−length in L
1.5
2
Figure 4.17: Simulated configuration: Earth layer and drilling hole. Top receiver is labeled as receiver 1 and the
lowest as receiver 5.
Polynomial Krylov MOR
As a source wavelet, again the time derivative of a Gaussian pulse is used (Ricker wavelet). The
wave field along a line from receiver 1, the top receiver, to receiver 5, the lowest receiver, is
shown in Figure 4.18. The wave field at every receiver separately is shown in Figure 4.19. The
response converged after roughly 3500 iterations of the polynomial Krylov subspace method.
It can be seen that the pulse at receiver 4 resonates much stronger than for the other receivers.
The poles in the region of interest are shown in Figure 4.20. The poles close to the imaginary axis are the scattering poles of interest. They can be related to resonances in the two slabs,
namely, the Earth layer and the drilling hole.
Now we can expand the solution in the scattering poles labeled by the circles in Figure 4.20.
There are 76 poles in the bandwidth of the pulse and have a reasonable small real part. The
result for receiver 3 is shown in Figure 4.20, which shows that a good approximation can be
found with 76 poles for the chosen configuration. It can be seen that only a very small error
arises for early times. For this example this is no problem, since the error occurs for time
instances smaller than the physical arrival time, which makes filtering easy.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
49
Seismic traces along the receiver line
0
Time [a.u.]
0.02
0.04
0.06
0.08
1
2
3
4
5
Location on receiver line (yellow=earth layer)
Figure 4.18: Wavefield along a line from receiver 1 to 5 in the drilling hole. The yellow lines mark the horizontal
position of the Earth layer.
Rational Krylov MOR
In this example, two different choices for the shifts of the rational Krylov subspace were made.
The shifts of the rational subspace where chosen either as 120 poles along a line parallel to the
imaginary axis, or based on the 76 eigenvalues of the polynomial subspace, which were used
to expand the field in Figure 4.20. The RKS method converges for both chosen sets of poles,
but fails to converge in the case less poles are used, as shown in Figure 4.21. This shows that
adaption of the interpolation points of the RKS to the spectrum of the approximated operator
improves convergence.
After decomposition, the weight of the poles can be sorted, in order to identify the most
contributing modes. These modes are illustrated in Figure 4.22. It was shown that the pole
with an eigenfunction as shown in Figure 4.22(a) is dominant at all receivers except for the one
next to the Earth layer. The field resonates in the drilling hole. The pole with an eigenfunction
shown in Figure 4.22(b) is dominant at the receiver next to the Earth layers. This eigenfunction
shows a resonating field at the cross section of the Earth layer and the drilling hole, with outgoing waves to the surrounding medium. In Figure 4.22(c-d) other resonance modes are shown.
Typical quality factors are between one and 30.
A Krylov subspace approach
Modeling of wave propagation in open domains
50
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Field at Receiver :1
Field at Receiver :2
1
1
Lanczos m=3500
FDTD
Response [a.u.]
Response [a.u.]
Lanczos m=3500
FDTD
0.5
0
−0.5
0
0.5
0
−0.5
0
0.05
0.1
Time [a.u.]
(a) Response at receiver 1 with m=3500.
(b) Response at receiver 2 with m=3500.
Field at Receiver :3
Field at Receiver :4
1
1
Lanczos m=3500
FDTD
Response [a.u.]
Response [a.u.]
Lanczos m=3500
FDTD
0.5
0
−0.5
0
0.05
0.1
Time [a.u.]
0.5
0
−0.5
0
0.05
0.1
Time [a.u.]
(c) Response at receiver 3 with m=3500.
0.05
0.1
Time [a.u.]
(d) Response at receiver 4 with m=3500.
Field at Receiver :5
1
Response [a.u.]
Lanczos m=3500
FDTD
0.5
0
−0.5
0
0.05
0.1
Time [a.u.]
(e) Response at receiver 5 with m=3500.
Figure 4.19: Signal at the five receivers. Receiver 1 is the top receiver, receiver 5 the lower one.
Modeling of wave propagation in open domains
A Krylov subspace approach
51
Scattering Poles
1
0
Lanczos m_scp=76
FDTD
Response [a.u.]
Imaginary Axis, (normalized frequency Im(s))
CHAPTER 4. RESULTS
−50
−100
m=3500
Lanczos Poles
Scattering Poles
Exitation frequency
−150
0
5
10
15
20
25
30
Real axis, (normalized frequency Re(s))
(a) Reduced system and expansion poles.
0.5
0
−0.5
0
35
0.05
0.1
Time [a.u.]
(b) Field at receiver 3 expanded in the scattering poles.
Magnetic field [a.u.]
1
RKS m=76
FDTD
0.5
0
−0.5
0
0.05
0.1
Time in [a.u.]
Imaginary Axis, (normalized frequency Im(s))
Figure 4.20: Scattering pole expansion of the layered Earth problem.
150
100
m=76
Poles of projected system
interpolation Poles
Exitation frequency
50
0
−50
−100
−150
0
5
10
Real axis, (normalized frequency Re(s))
15
Magnetic field [a.u.]
1
RKS m=120
FDTD
0.5
0
−0.5
0
0.05
0.1
Time in [a.u.]
Imaginary Axis, (normalized frequency Im(s))
(a) System response for a RKS dimension of m=76 with spec- (b) Interpolation poles and system poles for m=76 with spectrally adapted interpolation points.
trally adapted interpolation points.
150
100
m=120
Poles of projected system
interpolation Poles
Exitation frequency
50
0
−50
−100
−150
0
5
10
Real axis, (normalized frequency Re(s))
15
(c) System response for a RKS dimension of m=120 with (d) Interpolation poles and system poles for m=120 with
equidistant interpolation points.
equidistant interpolation points.
Figure 4.21: Rational Krylov subspace approximation of the layered Earth response using equidistance and spectrally adapted shifts.
A Krylov subspace approach
Modeling of wave propagation in open domains
52
4.2. INSTANTANEOUSLY REACTING MATERIALS – 2D CONFIGURATIONS
Real part eigenfunction of pole 2.67369−45.9461i
Abs part eigenfunction of pole 2.82652−38.9101i
50
50
100
100
150
150
200
200
250
250
50
100
150
200
250
50
100
150
200
250
(a) Dominant mode at the receivers 1,2,3,5. Position in (b) Dominant mode at the receiver 4. Position in top 10
top 10 poles:(1,1,5,X,3)
poles:(X,X,X,1,9)
real part eigenfunction of pole 1.962707−107.9345i
Abs part eigenfunction of pole 1.962707−107.9345i
50
50
100
100
150
150
200
200
250
250
50
100
150
200
250
(c) Real part of resonance in drilling hole.
50
100
150
200
250
(d) Absolute part of resonance in drilling hole.
Figure 4.22: Eigenfunctions of several poles of the layered Earth problem.
4.2.4
Conclusion
The two-dimensional experiments show that the transient system response converges towards
the corresponding FDTD results. The resonance expansions showed that the solution to a scattering problem can be expanded in a small number of scattering poles. In the example of the
box with H-polarized fields, 26 poles were sufficient to approximate the time-domain solution.
Using a rational Krylov subspace approach, it was shown that convergence can be reached
within very small subspaces compared with the problem size. The small subspace dimension
makes it feasible to store all basis vectors in memory, which allows for the computation of
modes even for large problems. The results showed that the convergence is strongly dependent on the shift selection, as spectral adaption of the shifts towards the problem enhanced the
convergence for the photonic crystal waveguide and layered Earth example.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
4.3
4.3.1
53
Instantaneously reacting materials – A three-dimensional configuration
Dielectric box
In this section a three-dimensional example is analyzed. This case is studied in order to show
that the method can handle large systems and significantly reduce their order.
The investigated configuration is a three-dimensional electromagnetic wave problem. A
dielectric box with a relative permittivity εr = 4 and width and height of 50 µm is simulated. As
a source, an x-directed dielectric dipole is used. The time derivative of a Gaussian pulse, with a
maximum of the spectrum at λpeak = 94 µm was used as excitation wavelet. The configuration
is shown in Figure 4.23.
The domain is truncated using the optimal PML and the problem size is n ≈ 8.4 · 106 . In
this section only the approximated resonance fields are presented, as it has already been shown
several times throughout the document that the method converges.
Figure 4.23: Three-dimensional dielectric box with dipole source next to the box.
Two of the approximated resonance modes are given in Figure 4.24 computed using m=10000
Lanczos iterations. The eigenmodes found by the Lanczos approximation are dependent on
source location, as it is used as a seed vector in the Lanczos algorithm. Only resonances which
are excited and thus contribute to the wavefield are calculated.
The resonance mode given in Figure 4.24 has a resonance wavelength of 2πc/ωres
1 = 32.6 +
0.27i µm. This resonance field is the Ex field which is the dominant field quantity inside the
box. The surface plots correspond to the 60% maximum of the resonance field. The resonance
shows a 4π resonance field in the y- and z-direction, but only a fundamental resonance in the
x-direction, which is the direction of the excitation dipole. Finally, the Ex field of the second
pole 2πc/ωres
2 = 38.3 + 0.47i µm shows a 3π resonance in all spatial directions.
A Krylov subspace approach
Modeling of wave propagation in open domains
54
4.3. INSTANTANEOUSLY REACTING MATERIALS – A 3D CONFIGURATION
(a) Imaginary part of the scattering resonance for Ex corresponding to the pole 2πc/ω = 38.3 + 0.47i µm
(b) Imaginary part of the scattering resonance for Ex corresponding to the pole
2πc/ωres = 32.6 + 0.27i µm
Figure 4.24: Resonant field in a three-dimensional box.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
4.4
4.4.1
55
Media exhibiting relaxation – a two-dimensional configuration
Dispersive box with Drude relaxation
The simulated configuration is shown in Figure 4.25. A two-dimensional H-polarized configuration was chosen. The dielectric function of the box is given by the Drude model using the model
parameters of gold as given in [30] (ωp = 13.6 PHz, γp = 0.1 PHz, ε∞ = 1). As a source wavelet
a Ricker Wavelet with a maximum in the spectrum at λpeak = 350 nm is used. At this frequency
the relative permitivity is εr (λpeak ) ≈ −40. The configuration measures 100 nm×100 nm and
the box has dimensions 50 nm×50 nm. A uniform grid with a step size of δ = 1 nm is used. The
problem size is n = 47000. If we compare this with the corresponding nondispersive case, we
observe that the total number of unknowns increased by approximately 10000. As a reference
simulation, an ADE-FDTD scheme as described in [31] is used.
Simulated configuration
0
x−length in L
20
Dispersive Object
Background ε=1
Source
Receiver
PML
40
60
80
100
0
20
40
60
y−Length in L
80
100
Figure 4.25: Simulated configuration: Square nanorod excited by a line source with L=1 nm.
Polynomial Krylov MOR
The convergence of the Lanczos reduced-order model towards the ADE-FDTD result is given
in Figure 4.26(a). It can be seen that the model converged after 6500 Lanczos iterations on the
interval of interest. Furthermore, it is shown that the late-time behavior can be expanded in 16
scattering poles with high quality factor.
For the PKS approach to converge it needs a subspace dimension which is more than four
times bigger than the non-dispersive one. The small increase in the problem size from
n = 37k to n = 47k cannot explain this fourfold increase in dimensionality. The location of
the main contributing scattering poles in Figure 4.26(c) shows, however, that they are small in
modulus compared with the wide range of the other poles. Most other modes have a relatively
large real part, such that they describe damped modes. All contributing poles have a high quality factor and are clustered at the imaginary axis. In the case of the dielectric box, the main
contributing poles did not cluster at one point in the spectrum, but were spread out in a region.
Therefore, the PKS can approximate distinct poles in the dielectric case with low dimensional
subspaces, but it needs a large dimension to obtain distinct modes for the dispersive case. PKS
methods converge from the pole of largest modulus towards the pole of smallest modulus. Thus
the increase in subspace dimension needed for convergence can be linked to the highly resonating structure, with modes of small modulus clustered at the imaginary axis. In the last sections
A Krylov subspace approach
Modeling of wave propagation in open domains
56
4.4. MEDIA EXHIBITING RELAXATION – A 2D CONFIGURATION
it was shown that the convergence of RKS is superior to PKS for systems described by a few
resonances. Therefore an RKS approach is used to obtain a faster and low-dimensional field
approximation.
Rational Krylov MOR
The whole system can also be projected onto a rational Krylov subspace. The poles describing
resonances with high quality factors are of interest, such that purely imaginary shifts are used.
As interpolation points, chosen with no a priori information about the system, equidistant pure
imaginary shifts between σmin = 5i and σmax = 70i were used. Using a subspace size of m=50,
convergence of the time-domain result is obtained. The converged response and poles of the
system are given in Figure 4.27. In Figure 4.28 the resonance fields of some of the poles are
shown and it can be seen that the electric field in the x-direction indeed satisfies the boundary
conditions which have to hold at the surface of the box.
The wave field of the resonances is localized at the surface of the dispersive object, whereas
standing waves inside the object were found in the non-dispersive case. For negative index
materials, surface plasmons can be excited forming an oscillation between the free-electron
gas of the metal and the electric field of the wave. The resonance field shows typical plasmon
behavior, as it is localized at the surface and decays exponentially away from the surface of the
object. Typically, the quality factors of the resonances range between Q = 100 and Q = 300.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
57
1
1
Lanczos only scatteringpoles m=6500; #scp16
FDTD
0.5
Magnetic field [a.u.]
Magnetic field [a.u.]
Lanczos m=6500
FDTD
0
−0.5
−1
0
5
10
Time in [fs]
15
0.5
0
−0.5
−1
0
20
5
10
Time in [fs]
15
20
m=6500
0
Lanczos Poles
Scattering Poles
Exitation frequency
−100
−200
−300
−400
−500
−600
−700
0
20
40
60
80
Real axis, (normalized frequency Re(s))
100
(c) Poles of the reduced-order model.
Imaginary Axis, (normalized frequency Im(s))
Imaginary Axis, (normalized frequency Im(s))
(a) Time-domain response at the receiver of the Lanc- (b) Expension of the scattering response in the scattering
zos reduced-order model of dimension m = 6500 and
poles of interest as shown in (d).
ADE-FDTD comparison simulation.
0
−10
−20
−30
m=6500
Lanczos Poles
Scattering Poles
Exitation frequency
−40
0
2
4
6
8
10
Real axis, (normalized frequency Re(s))
(d) Poles of the reduced-order model with high quality
factor in the bandwidth of the pulse.
1
Magnetic field [a.u.]
RKS m=50
FDTD
0.5
0
−0.5
−1
0
5
10
Time in [ps]
15
Imaginary Axis, (normalized frequency Im(s))
Figure 4.26: Scattering response using polynomial Krylov subspace reduction.
m=50
60
40
Poles of Projected System
Exitation frequency
Shifts in RKS
20
0
−20
−40
−60
0
2
4
6
8
Real axis, (normalized frequency Re(s))
10
(a) Time-domain response at the receiver of the RKS (b) Approximated poles of the RKS reduction and intermodel of dimension m = 50 and ADE-FDTD compolation poles.
parison simulation.
Figure 4.27: Results of the RKS reduction for the two-dimensional dispersive box.
A Krylov subspace approach
Modeling of wave propagation in open domains
58
4.4. MEDIA EXHIBITING RELAXATION – A 2D CONFIGURATION
(a) Absolute H-field of pole λres = 166 + 0.23i nm.
(b) Real Ex -field of pole λres = 166 + 0.23i nm.
(c) Absolute H-field of pole λres = 166 + 0.73i nm.
(d) Real Ex -field of pole λres = 166 + 0.73i nm.
(e) Absolute H-field of pole λres = 168 + 0.75i nm.
(f) Real Ex -field of pole λres = 168 + 0.75i nm.
(g) Absolute H-field of pole λres = 163 + 0.70i nm.
(h) Real Ex -field of pole λres = 163 + 0.70i nm.
Figure 4.28: Eigenvectors of the system matrix A. Absolute value of the magnetic field is depicted on the left
hand side, whereas Ex is depicted to the right.
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
59
(a) Absolute H-field of pole λres = 174 + 1.14i nm.
(b) Real Ex -field of pole λres = 174 + 1.14i nm.
(c) Absolute H-field of pole λres = 162 + 0.5i nm.
(d) Real Ex -field of pole λres = 162 + 0.5i nm.
Figure 4.29: Continued from Figure 4.28.
A Krylov subspace approach
Modeling of wave propagation in open domains
60
4.5
4.5.1
4.5. MEDIA EXHIBITING RELAXATION – A 3D CONFIGURATION
Media exhibiting relaxation – A three-dimensional configuration
Spontaneous decay rate of a dipole near a nanorod
The last example targets to calculate the electromagnetic field of a quantum emitter in close
proximity to a plasmonic resonator. This electromagnetic field is needed to calculate the enhancement of the spontaneous decay rate of a quantum emitter by the resonator. This enhancement is known as the Purcell effect.
The spontaneous decay rate of a two-level quantum system located at x = x0 can be computed
classically according to the formula [30]
2 Γ = Im p · Ê(x0 ) ,
~
(4.1)
where Ê is the electric field at the location of the electrical dipole Ĵext = −iωpδ(x − x0 ) with
dipole moment p. Using our Krylov reduction approach, the spontaneous decay rate can be
computed using the formula
(4.2)
p · Ê(x0 ) ≈ iωγ eT1 r(Hm ) + r(H̄m ) e1 ,
where the constant γ and the function r(z) are given by
γ = hM−1 q, M−1 qidisp
and
r(z) =
η(z)
.
z − iω
(4.3)
In other words, a single model provides decay rate approximations for all frequencies (wavelengths). To illustrate the effectiveness and impact of the approach, we consider a configuration
close to the one simulated in [1]. In this paper the authors simulate the spontaneous decay rate
of a cylindrical golden nanorod. Equivalent to the cited work, a Drude medium is chosen with
medium parameters ωp = 12.6 PHz, γp = 0.141 PHz and ε∞ = 1 embedded in a background
medium with a relative permittivity εr = 1.52 . After discretization the problem size is about
n = 17 · 106 .
In [1], the cylindrical golden nanorod is simulated using an aperiodic Fourier modal based
method (a.k.a. rigorous coupled-wave analysis (RCWA)). A version developed to evaluate axial
symmetric resonators is used, as described in [32]. By using the axial symmetry of the structure,
analytic expressions for the modes in the radial direction can be found, such that the modes need
to be calculated on an effectively two-dimensional grid. According to the paper, a single mode
calculation takes about 15 minutes on a modern personal computer, with no indication of the
problem size.
In the method used in this thesis, all excited modes are approximated at the same time.
Moreover, for this example a full three-dimensional simulation was used, whereas the cited
work effectively solves two-dimensional systems. Obviously, the axial symmetry constraint of
the method in [1] limits the class of objects that can be simulated. Finally, our method is able to
transform the solution directly to the time-domain, whereas the aperiodic Fourier model method
is a pure frequency-domain method.
Building a Krylov subspace of m =3500 via the developed Lanczos algorithm takes about
1.3 hours. However, once the reduced system is obtained, the Fourier domain solution at the
source for 1000 different wavelengths between 0.7 µm and 1.2 µm is obtained in 0.7 seconds
by a direct sparse solver. The cited work in [1] needs 15 minutes for a single frequency point.
Therefore, our method is superior after evaluating only five frequency points even though our
approach solves a full three-dimensional problem as oppose to the two-dimensional ones solved
in [1].
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
61
Figure 4.30: Simulated configuration: cylindrical nanorod excited by a dipole molecule.
Within 48 hours, the full order system could neither be solved directly or by the iterative solver
GMRES on the used personal computer, because of memory limitations. Five cores of the
Server HPC16 could not solve a single frequency point using restarted GMRES. It did not
converge to the set tolerance of 10−6 after 80 outer and 20 inner iterations, which took 4.3
hours. At that point the residual had a value of 2 · 10−4 . This directly shows the impact of the
developed PKS Lanczos algorithm for large system matrices.
In Figure 4.31 the spontaneous decay rate of the simulated cylindrical dipole configuration
and the cylinder configuration of [1] are compared. The peak shapes look very similar, and the
maximum in the decay rate is reached at the same wavelength.
4.6
4.6.1
Conclusion
Performance comparison
In this section the computation times of the numerical examples and different approaches are
compared. The computation times1 for all two-dimensional examples are provided in Table 4.1.
The compared methods are FDTD, RKS and PKS. FDTD is only included in the comparison,
to show that speed-ups can be achieved even for small time intervals. Furthermore, it shows
that the Lanczos decomposition of a large matrix is faster than FDTD, for the same number of
iterations, which is shown by the computation time of the photonic crystal. No such speed-up
can be seen for the two-dimensional dielectric box example as the system matrix is too small.
In this case the computational cost of the Lanczos algorithm is not dominated by taking matrix
vector product computations.
The computation time of the PKS approach is split up into Lanczos decomposition and
total computation time. This choice is made as a large fraction of the computation time is
spend on solving the eigenvalue problem of the reduced-order system, without making use of
1 Matlab
implementation on an Intel i5-3470 CPU @ 3.2 GHz 8GB 64-bit Windows 7 architecture.
A Krylov subspace approach
Modeling of wave propagation in open domains
62
4.6. CONCLUSION
Normalized decay rate
1
0.9
SDR Lanczos ROM cylinder
0.8
Total SDR cylinder
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.7
0.8
0.9
1
λ [µm]
1.1
1.2
Figure 4.31: Spontaneous decay rate of the nanorod dipole configuration using the Lanczos algorithm and the
method described in [1]. The decay rate of the cylinder configuration was calculated using the Lanczos reducedorder model with m=3500.
Table 4.1: Computation time comparison of FDTD, PKS, and RKS.
Box n = 37k
Iterations Time [s]
FDTD
PKS Lanczos
PKS total
RSK
10000
1500
1500
48
16
6.2
17.8
7.7
Crystal n = 695k
Iterations Time [s]
16000
3500
3500
15
3228
99
217
68
layered Earth n = 545k
Iterations
Time [s]
3000
3500
3500
76
Drude box n = 47.4k
Iterations Time [s]
245
77.5
195
275
8200†
6500
6500
50
36†
28
754
12
† ADE-FDTD schemes were used to compute the result in a medium exhibiting relaxation
its symmetry and tridiagonal structure. The eigenvalues of a symmetric tridiagonal structure
can be efficiently calculated via recursion relations, which can speed up the PKS algorithm.
Therefore significant speed up can be achieved by developing a dedicated eigenpair solver for
this problem.
The computation times in Table 4.1 show that the RKS approach outperforms FDTD and
PKS for the dielectric box and photonic crystal example. No speed-up can be seen for the layered Earth example. However, for all structures the model order is significantly reduced using
the RKS approach with respect to the PKS approach. This makes the RKS attractive for inversion schemes, as the transfer function between two points can be obtained in terms of a small
number of modes. The Earth layer and drilling hole structure extends into the PML, such that
the resonances found in the structure are rather lossy and of low quality factor. A structure with
a response dominated by a few scattering poles can be approximated by a lower order model
and thus gain from an RKS approach.
The system response of the example exhibiting relaxation is dominated by a few surface modes
with high quality factor. All contributing modes can be approximated by an RKS of dimension
m = 50, which shows superior convergence over PKS. Given the fact that the system response
is described by a few resonances with high quality factor, the RKS approach outperforms PKS.
The used example has a relatively small problem size, such that the systems inside the RKS
can be solved fast using sparse solvers. For this example, the comparison with ADE-FDTD
is only included to give a rough indication how the speed of the Lanczos algorithm compares
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 4. RESULTS
63
to the speed of ADE-FDTD. The quality factor of the plasmonic resonance is high, and even
8200 ADE-FDTD steps are not enough to cover the complete response of the pulse as shown in
Figure 4.26. The computation time of the reduced-order model could be reduced significantly
by using an RKS approach. It outperformed the ADE-FDTD method after 2800 time steps.
The impact of the standard eigenvalue solver which was used in all experiments can clearly
be seen in this example. As the PKS approach needs a high dimensional subspace the eigenvalue
problem takes about 96% of the computation time. A dedicated eigenvalue solver can yield
significant speed up especially for high dimensional subspaces. For large matrices one could
also use a PKS approach to reduce the size of the system matrix. After that an RKS approach
can be used on the reduced system matrix, such that neither large eigenvalue problems nor very
large sparse systems need to be solved.
The two-dimensional dispersive box problem leads to a system matrix of small dimension
such that the PKS runtime is not dominated by matrix vector products, which normally makes it
superior for large systems. The impact of the developed Lanczos algorithm for dispersive media
is of increasing impact for larger system matrices. The three-dimensional dispersive example
for instance could only be solved with the PKS approach, solving single frequency systems with
a direct sparse solver appeared to be infeasible. The developed Lanczos algorithm allows decomposition of a system matrix with a computation time proportional to m · n, rather than m2 · n
in an Arnoldi algorithm. Furthermore, the memory used by the Lanczos algorithm is reduced
by a factor 3/m compared with the Arnoldi algorithm.
4.6.2
Summary of Results
In this chapter, a novel model order reduction approach for dispersive media was presented.
Polynomial and rational Krylov subspace approaches together with resonance expansions were
used to model wave propagation. Convergence in the time- and frequency-domain was demonstrated. The time-domain results of the reduced-order models match the ones produced by
FDTD. Moreover, the spontaneous decay rate of a quantum emitter close to a three-dimensional
object has been calculated, showing convergence in the frequency-domain.
In conclusion, the RKS approach shows a better performance on highly resonating structures with only a few excited modes. PKS shows superior performance, when a large number
of modes contribute or the system matrix is too large to efficiently obtain single frequency
solutions for RKS projections.
The advantage of Krylov subspace methods over FDTD increases with increasing size of the
time interval of interest, as the Krylov method converges on the complete causal time-domain
and FDTD only on a bounded time interval. Especially resonating structures with high quality
factors need long simulation times, which motivates the choice of Krylov methods for micro
cavities, photonic crystals, and for plasmonic configurations.
The RKS convergence is heavily dependent on shift selection. In this chapter, domains of
interest were used to adapt the spectrum of the reduced-order model to an interval of interest. However, no optimal shift selection was used. To obtain optimal rational Krylov subspace
reduced-order models the process of shift selection should be based on a mathematical formulation of optimality.
For use in inversion schemes, H2 optimal models have desirable properties. The error of the
reduced-order transfer function lies in the null space of the approximate Jacobians, which makes
H2-optimal models insensitive to approximation errors. Finding constraints for H2 optimal pole
selection is therefore a desirable goal for future work.
A Krylov subspace approach
Modeling of wave propagation in open domains
64
Modeling of wave propagation in open domains
4.6. CONCLUSION
A Krylov subspace approach
Chapter 5
Conclusions
5.1
Summary of the Results
In this Master thesis we investigated the use of several Krylov subspace methods for the modeling of time- and frequency-domain wave propagation on open domains in instantaneously
reacting and dispersive media. The basic wave equations were discretized on a staggered
finite-difference grid and the extension of the domain to infinity was simulated via an optimized complex-scaling method (PML). The resulting state-space representation has a large
order which can easily run into the millions. To reduce this order, we exploited the sparsity
and symmetry properties of the system matrix in Lanczos-type and Arnoldi algorithms and
projected the state-space representation onto several different Krylov subspaces. Approximate
field solutions were obtained by exactly solving the projected Krylov system and incorporating
stability correction.
In Chapter 2, the discrete symmetry preserving operator formulation of the Maxwell equations
was introduced. A novel symmetry preserving formulation for media exhibiting general secondorder relaxation was developed. Furthermore, resonance expansions were introduced as an
efficient tool to obtain low-order models and transfer functions.
In Chapter 3, model-order reduction via projection onto subspaces was discussed. To obtain
the field approximations in the time- and frequency-domain, polynomial and rational Krylov
subspaces were used and the efficient construction of their bases was also described briefly. We
also presented a short discussion on how the different algorithms approximate the spectrum of
the system matrix by making a comparison with the power method and its variants.
In Chapter 4, we illustrated the theory developed in the previous chapters by several numerical experiments. Polynomial and rational Krylov subspace approaches were applied to wave
propagation problems involving instantaneously reacting and dispersive media in one-, twoand three-dimensions. For simple one-dimensional systems, it was shown that the resonances
obtained via model-order reduction match the analytical resonances of the configuration. Our
method was further validated in 1D, 2D, and 3D by comparing the Krylov reduction responses
with field responses computed by FDTD. In all cases, our reduced-order models showed good
agreement with FDTD results. We also showed that with our reduced-order modeling approach
it is possible to determine the bandgap structures of (quasi-)periodic arrays. Furthermore, we
showed that it is possible to determine the dominant resonant modes of an open system and we
illustrated these modes for a wide range of applications again in 1D, 2D, and 3D. In addition,
it was demonstrated that the full time-domain response of a photonic crystal can be determined
using a single resonance only. Finally, the newly developed symmetry preserving formulation
of the general second-order dispersive Maxwell system allowed for model-order reduction of
dispersive media via the Lanczos algorithm. Convergence in the frequency- and time-domain
was shown by comparing computed field responses with ADE-FDTD field responses. The im65
66
5.2. FUTURE WORK
pact of this new formulation was illustrated by a full vectorial three-dimensional example with
approximately 17 million unknowns.
Our experiments clearly revealed some of the advantages and disadvantages of PKS and RKS.
It was shown that RKS outperforms PKS for structures dominated by a few open resonance
modes. Next we found that the convergence of RKS strongly depends on the choice of interpolation points. Spectral adaption of the interpolation points to the spectrum of the system matrix
was shown to be desirable, which leads to the need of a mathematical formulation of optimal
shift selection. Furthermore, it was shown that the semi-discrete nature of our approach makes
it superior to conventional methods like the FDTD method, since a Krylov method constructs
an interpolatory polynomial or rational function that provides us with field approximations for
all causal times, whereas FDTD is a time stepping method that needs to compute the field response step by step, starting at the time instant when the source was switched on. Moreover, the
transfer function between a given source and a given receiver is explicitly calculated in a Krylov
method and in the frequency-domain the Krylov subspace approach outperforms conventional
methods as soon as the solution needs to be evaluated at more than one frequency.
At this point we want to put the presented work into perspective. Figure 5.1 gives an
overview of Krylov subspace approaches developed over the years for the full-wave Maxwell
system and for diffusive electromagnetic fields on unbounded domains. In this thesis we used
the PKS approach for wave equations with no relaxation and extended it to electromagnetic
wave problems involving media exhibiting relaxation. We further started to explore the use
of RKS approaches previously used for diffusion equations in case of instantaneously reacting
materials and media exhibiting relaxation.
Figure 5.1: Overview of the Krylov subspace approaches discussed in the literature and in this thesis.
5.2
Future work
More research is needed in the field of rational Krylov subspace model-order reduction. In this
work we only showed the potential of this approach. A significant reduction in computation
times was achieved, especially when the interpolation points (shifts) are adapted to the spectrum
Modeling of wave propagation in open domains
A Krylov subspace approach
CHAPTER 5. CONCLUSIONS
67
of the system matrix. However, a robust model-order reduction framework requires a mathematical basis for optimal shift selection. For example, H2 optimality constraints lead to rational
reduced-order models satisfying the necessary conditions for the time-integrated squared-error
to be minimum. Further speed-ups of the RKS method can be achieved by implementing effective preconditioners for Helmholtz-type systems. Another reasonable approach for Maxwell
equations involving media exhibiting relaxation is the EKS approach. Recently the impact of
the EKS approach was shown in [12]. For plasmonic structures this approach shows potential,
as plasmonic structures are usually dominated by a few modes of small modulus clustered close
to the imaginary axis.
In the field of optics, the easily measurable far-field is of main importance. Thus near-to-far
field transformations can be implemented as documented in Chapter 8.2 of [27] in order to
obtain the diffraction orders of a system as output. Moreover, future work should include the
incorporation of Floquet quasi-periodic boundary conditions in order to make the method suited
for structures like gratings and antenna arrays.
Finally, to illustrate the impact of the model-order reduction framework on full wavefield inversion, the developed framework should be incorporated into Gauss-Newton inversion schemes.
A Krylov subspace approach
Modeling of wave propagation in open domains
68
Modeling of wave propagation in open domains
5.2. FUTURE WORK
A Krylov subspace approach
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A Krylov subspace approach
Modeling of wave propagation in open domains
72
Modeling of wave propagation in open domains
BIBLIOGRAPHY
A Krylov subspace approach
Appendix A
List of abbreviations
List of abbreviations
et al.
etc.
a.k.a
PWE
PKS
EKS
RKS
FDTD
FVM
FEM
ROM
PML
DE
NtD
ADE
PEC
ABC
EM
MOR
et alii (lat. and others)
et cetera (lat. and so forth)
also known as
Plane Wave Expansion
Polynomial Krylov Subspace
Extended Krylov Subspace
Rational Krylov Subspace
Finite-Difference Time-Domain
Finite Volume Method
Finite Element Method
Reduced Order Model
Perfectly Matched Layers
Differential Equation
Neuman to Dirichlet
Auxiliary differential equations
Perfectly Electrically Conducting
Absorbing Boundary Conditions
Electromagnetic
Model Order Reduction
73
74
Modeling of wave propagation in open domains
A Krylov subspace approach
Appendix B
Nomenclature
The Nomenclature used throughout the thesis is shown in table B.1.
Table B.1: Nomenclature used in this document.
Quantity
Notation
Scalar
Continuous Operator
Continuous Field Quantity
Kronecker Product
Vector
Matrix
Lexicographical Ordering
Identity Matrix (P by P)
Matrix Transpose
Hermitan Transpose
Truncated/Stabilized Matrix
Laplace Transform
Fourier Transform
Hilbert Transform
heaviside step function
delta function
Euclidean norm
Complex conjugate
bidiagonal Matrix of size N×N+1
tridiagonal Matrix
Upper Hessenberg matrix with elements hi j
polinomial in · of order i
Rank two tensor
75
a
D
E
⊗
e
M
e = vec(E)
IP
AT
A†
Ã
F̂(s) = L { f (t)}
F̆(s) = F { f (t)}
g(t) = H { f (t)}
η(t)
δ(t)
|| · ||
¯·
bidiagN (a, b)/bidiagN (ai , bi )
tridiag(a, b, c)/tridiag(ai , bi , ci )
upperhess(hi j )
pi (·)
ε
76
Modeling of wave propagation in open domains
A Krylov subspace approach
Appendix C
Electromagnetic system formulation
In this appendix we present the discrete system formulation for one-dimensional problems
(Section C.1), two-dimensional problems (Section C.2) and finally three-dimensional problems
(Section C.3).
C.1
One-dimensional system formulation
In this section the algorithms used to simulate 1D EM-wave propagation are documented. In the
Maxwell equation we choose to set all fields to zero except Ez and Hx . Therefore the EM-wave
propagates along the y-direction. The Maxwell equations in the Laplace domain now read
or in operator form
∂y Êz + µr sĤx = −K̂ ext ,
∂y Ĥx + εr sÊz + σÊz = −Ĵ ext ,
(C.1)
(C.2)
(C.3)
(D + S + M s)F̂ = Q̂ 0 .
(C.4)
With D the spatial differentiation operator, M the medium matrices, F̂ containing the field
quantities and Q̂ 0 the vector containing the electromagnetic sources.
C.1.1
Discretization
In order to simulate the system we introduce a finite difference grid. At both ends of the domain
an optimal PML is used to simulate the extension of the system to infinity. Therefore we first
define two regions in our domain being Ωy,PML and Ωy,DOI . The latter denotes the domain of
interest, whereas the first denotes the region that truncates the DOI with a PML. Let k denote
the width of the PML and N the size of the domain of interest. We now define the primary grid
as
p
Ωy,DOI = {y p , q = 0, 1, . . . , N, yq > yq−1 },
(C.5)
and the dual grid as
Ωdy,DOI = {ŷ p , q = 1, . . . , N, ŷq > ŷq−1 }.
(C.6)
The grid in the PML is given by
1,p
Ωy,PML = {y p , q = −k, . . . , −1, yq > yq−1 },
and
2,p
Ωy,PML = {y p , q = N + 1, . . . , N + k, yq > yq−1 }.
77
(C.7)
(C.8)
78
C.1. ONE-DIMENSIONAL SYSTEM FORMULATION
The dual grid inside the PML is given by
Ω1,d
y,PML = {ŷ p , q = −k, . . . , N, ŷq > ŷq−1 },
(C.9)
Ω2,d
y,PML = {ŷ p , q = N + 1, . . . , N + k, ŷq > ŷq−1 }.
(C.10)
and
The step sizes in the DOI are given by
δy,p = y p − y p−1
and
δˆ y,p = ŷ p − ŷ p−1 .
(C.11)
We choose a uniform grid with all step sizes to be equal to δ. The optimal step sizes inside the
PML are calculated using the approach described in [7]. The step size ĥ1 is given by 2δ plus the
complex contribution given by the algorithm presented in [7], in order for the primary and the
dual PML to coincide at both ends where the DOI is truncated by the PML.
The step sizes inside the PML are denoted
∀ q = 1, 2, . . . , k
hq and ĥq
(C.12)
and are chosen symmetric on both ends of the domain. The grid is illustrated in Figure C.1.
A homogeneous Dirichlet boundary condition is imposed at the electric field approximation at
the outer most grid point. This boundary condition is known as PEC boundary condition in EM
theory, as a perfect electrically conducting contact does not allow any tangential electric field.
The full discrete system can now be written as
(D + S + Ms)f̂ = q̂0 ,
(C.13)
with f = [ez (y−(k−1) ), . . . , ez (yN+k−1 ), hx (ŷ−k ), . . . , hx (ŷN+k )], where the gridpoint labeled 0 is
only included for the electric fieldstrength approximation. The gridpoints ez (y−k ) and ez (yN+k )
are not included in the field vector as they are boundary points. The discrete spatial differentiay-1PML y0
h2 h1
PML
y-k
hk
hk
y-kPML
y1
δ
y;1
δy;1
h2 h1
y-1 δ y1
PML
Ω
δ
y2
2
δy;2
δy;3
δy;3
δy;2
y2
y3
δy;4
yN-2
yN-1
δy;N
δy;N-1
δy;N-1
δy;N-2
y3
yN-1
yN
DOI
DOI
Ω
Ω
yN
yN+kPML
hk
yN+1PML
h1 h2
h1 h2 PML
yN+1
hk
yN+kPML
PML
Ω
Figure C.1: Illustration of the grid used for the 1D simulation. Note that the step sizes between points {ŷ−1 , ŷ1 }
and {ŷN , ŷN+1 } are given by δ/2 plus the complex distribution from the PML.
tion operator matrix D is given by
0 Ŷ
D=
,
Y 0
Modeling of wave propagation in open domains
(C.14)
A Krylov subspace approach
APPENDIX C. ELECTROMAGNETIC SYSTEM FORMULATION
with Ŷ and Y the dual and primary differentiation matrix given by
 −1
hk
 −h−1 h−1
k−1
k−1


...
...


−1

−δ−1
y;1 δy;1


..
..

.
.
Y=
−1

−δy;N δ−1
y;N

−1
−1

−h

k−1 hk−1

...
...



−h−1
k−1
79

h−1
k−1
−h−1
k


















(C.15)
Ŷ is the equivalent upper bidiagonal matrix containing the dual step sizes.
Ŷ = diag(ĥk , . . . , ĥ1 , δˆ 1 , . . . , δˆ N−1 , ĥ1 , . . . , ĥk )−1 · bidiagN+2k−1 (−1, 1)
(C.16)
We now further introduce the step size matrixes Wy and Ŵy as
Wy = diag(hk , . . . , h1 , δ1 , . . . , δN , h1 , . . . , hk )
(C.17)
and the dual step size matrix as
Ŵy = diag(ĥk , . . . , ĥ1 , δˆ 1 , . . . , δˆ N−1 , ĥ1 , . . . , ĥk ).
(C.18)
The differentiation matrices can now comfortably be written as
Ŷ = Ŵy−1 · bidiagN+2k−1 (−1, 1) and
Y = −Wy−1 · bidiagN+2k−1 (−1, 1)T ,
(C.19)
where the function bidiagK (a, b) makes a upper bidiagonal matrix of size K × K + 1 with diagonal elements a and upper diagonal elements b. Further we define the full step size matrix
as
Ŵy 0
W=
.
(C.20)
0 Wy
The medium matrix M is defined as
M = diag(ε(y−(k−1) ), . . . , ε(yN+k−1 ), µ(ŷ−k ), . . . , µ(ŷN+k )),
(C.21)
S = diag(σ(y−(k−1) ), . . . , σ(yN+k−1 ), 0, . . . , 0).
(C.22)
and
There are N + 2k + 1 primary grid points and N + 2k dual grid points. The outer most primary
grid points have PEC boundary conditions. Therefore the matrix M is square and has a length
of 2N + 4k − 1. At both interfaces of two media the average of the neighbouring medium
parameters are chosen.
C.1.2
Symmetry properties
We note that the operator matrix M−1 D is in general neither symmetric nor anti-symmetric.
However we note that
ŶT Ŵy = −Wy Y = bidiag(−1, 1)T ,
(C.23)
A Krylov subspace approach
Modeling of wave propagation in open domains
80
C.2. TWO-DIMENSIONAL SYSTEM FORMULATION
which leads to the anti-commutativity relation
DT W = −WD.
(C.24)
Using this it can be shown that the opterator A is anti-symmetric with respect to the MW
weighted inner product.
h x | Ay iMW = yT AT MWx
= yT DT Wx
= −yT WDx
= −yT WMM−1 Dx
= −yT MWM−1 Dx
= − h Ax | y iMW
(C.25)
(C.26)
(C.27)
(C.28)
(C.29)
(C.30)
Therefore we can use a modified Lanczos algorithm using this anti-symmetry. Since A is antisymmetric in the weighted inner product we know that H is anti-symmetric in the classical sense
(HT = −H). Therefore we use the three term recurrence relation Avi = βi+1 vi+1 − βi vi−1 . The
term containing vi drops out for anti-symmetric matrices, as αi = −αi is only true for αi = 0.
In the lossy case we have a nonzero matrix S; The solution to Eq. (C.13) in the Laplace domain
is now given by
f̂ = (M−1 D + M−1 S + sI)−1 M−1 q̂0 .
(C.31)
Therefore we see that the system matrix is now given by
A = M−1 D + M−1 S
(C.32)
where the first part is skew-symmetric with respect to the WM weighted inner product and
the second part is a diagonal matrix which is obviously symmetric. Therefore the latter part
commutes with every diagonal matrix. Now we use the fact that the skew-symmetry of A can
be transformed into a symmetry by adding the weight factor δ− to the inner product with
δ− = diag(1, 1, . . . , 1, −1, −1, . . . , −1)
| {z } |
{z
}
# N+2·k-1
(C.33)
# N+2·k
Therefore the commutation relation
AT δ− WM = δ− WMA,
(C.34)
holds which leads to a symmetry in the δ− WM weighted pseudo inner product.
C.2
Two-dimensional system formulation
In this section the formulation used to simulate 2D EM-wave propagation for H- and E-polarized
fields is presented. The detailed derivation of the H-polarized system matrix can be found
in [31]. For the E-polarized waves it can be found in [26]. A short derivation of the H-polarized
algorithm will be presented, whereas the results for E-polarization are simply given. The first
part deals with H-polarized fields.
At last the symmetry matrices used in the Lanczos algorithm are explained.
Modeling of wave propagation in open domains
A Krylov subspace approach
APPENDIX C. ELECTROMAGNETIC SYSTEM FORMULATION
C.2.1
81
System Formulation for H-polarized fields
In the Maxwell equation we choose to set all fields to zero except Êx , Êy and Ĥz . Therefore the
EM-wave propagates in the xy-plane. The Maxwell equations in the Laplace domain equations
now read
−∂y Ĥz + sεÊx + σÊx = −Ĵxext ,
∂x Ĥz + sεÊy + σÊy = −Ĵyext ,
(C.35)
(C.36)
∂x Êy − ∂y Êx + sµĤz = −K̂zext ,
(C.37)
(D + S + M s)F̂ = Q̂ 0
(C.38)
or in operator form
With D the spatial differentiation operator, M , S the medium matrices, F̂ containing the field
quantities and Q̂ 0 the vector containing the electromagnetic sources.
Discretization
We introduce primary and secondary grids in x- and y-direction. There are P secondary grid
points in the x-direction and Q secondary points in the y-direction. Further there are k PML
layers at all sides. Therefore we define Qk = Q + 2 · k and Pk = P + 2 · k
Following this the discrete field approximations, excluding the Dirichlet boundary grid points
are given by
Ĥz
Êx
Êy
(Pk) × (Qk)
(Pk) × (Qk − 1)
(Pk − 1) × (Qk)
size
size
size
(C.39)
(C.40)
(C.41)
Seen the fact that our discretization has PEC at the outer most grid points Ĥz (x̂, ŷ) is defined on
the dual grid points in the x- and y-direction. Further Êy (x, ŷ) is defined on the primary x- and
dual y-grid. Finally Êx (x̂, y) is defined on the dual x- and primary y-grid.
We introduce the discretized spatial differentiation matrices as
 −1

hk
 −h−1 h−1

k−1
k−1




...
...




−1
−1


δ
−δ
y;1
y;1




..
..


.
.
Y=
(C.42)

−1
−1


−δ
δ
y;Q
y;Q


−1


−h−1
h


k−1
k−1


.
.
..
..




−1
−1


−h
h
k−1
k−1
−h−1
k
and the dual spatial discretization matrix Ŷ is the equivalent upper bidiagonal matrix containing
the dual step sizes.
Wy = diag(ĥk , . . . , ĥ1 , δˆ y;1 , . . . , δˆ y;Q−1 , ĥ1 , . . . , ĥk )
A Krylov subspace approach
and
Ŷ = Wy−1 · bidiagQ+2k−1 (−1, 1).
(C.43)
Modeling of wave propagation in open domains
82
C.2. TWO-DIMENSIONAL SYSTEM FORMULATION
Equivalently we can define the matrixes
Wx = diag(ĥk , . . . , ĥ1 , δx;1 , . . . , δx;P , ĥ1 , . . . , ĥk )
and
X = −Wx−1 · bidiagP+2k−1 (−1, 1)T .
(C.44)
and
X̂ = Ŵx−1 · bidiagP+2k−1 (−1, 1),
(C.45)
in the x-direction. Using these differentiation matrixes the Maxwell equations can be written in
their discrete form as
Ŵx = diag(ĥk , . . . , ĥ1 , δˆ x;1 , . . . , δˆ x;P−1 , ĥ1 , . . . , ĥk )
and
−Ĥz ŶT + εsÊx + σÊx = −Jx ,
X̂Ĥz + εsÊy + σÊy = −Jy ,
XÊy − Êx YT + µsĤz = −Kz .
(C.46)
(C.47)
(C.48)
In these equations Ê is a matrix containing the electric field approximation. In order to be able
to define a overall system matrix A in the form
(A + sI)f = M−1 q0 ,
(C.49)
all matrixes containing field quantities need to be transformed into vectors. We use lexicographical ordering of the 2D quantities. Let the operator vec() be the lexicographical mapping
operator with
êy = vec(Êy ).
(C.50)
In Matlab this is obtained by using h=reshape(H,Pk*Qk,1). Further let the Kronecker product
of two matrixes A and B be given by A ⊗ B. The identity connecting lexicographical ordering
and the Kroneker product used to derive the final system matrix is given by
vec(BXAT ) = (A ⊗ B)vec(X).
(C.51)
−(Ŷ ⊗ IPk )ĥz + sMε;x êx + Mσ;x êx = −jext
x ,
(IQk ⊗ X̂)ĥz + sMε;y êy + Mσ;y êy = −jext
y ,
(IQk ⊗ X)êy − (Y ⊗ IPk )êx + sMµ ĥz = −kext
z ,
(C.52)
(C.53)
(C.54)
with M the medium matrices containing the medium parameters in lexicographical ordering.
We now introduce D, the differentiation matrix given by,


0
−(Ŷ ⊗ IPk )
0
0
(IQk ⊗ X)  .
D =  −(Y ⊗ IPk )
(C.55)
0
(IQk ⊗ X̂)
0
The medium matrices are given by,


Mε;x 0
0
0 
M =  0 Mµ
0
0 Mε;y
and

Mσ;x 0
0
0
0 .
S= 0
0
0 Mσ;y
(C.56)

Modeling of wave propagation in open domains
(C.57)
A Krylov subspace approach
APPENDIX C. ELECTROMAGNETIC SYSTEM FORMULATION
83
Finally the field and source vectors are given by
T
ext T
ext T T
f̂ = [êTx , ĥTz , êTy ]T and q̂0 = [(−jext
x ) , (−k̂z ) , (−jy ) ] ,
(C.58)
respectively. Using these definitions the system can be written as
or as
(D + Ms + S)f̂ = q̂0 ,
(C.59)
(A + sI)f̂ = M−1 q̂0 ,
(C.60)
where the system matrix has been defined as
A = M−1 D + M−1 S.
(C.61)
Symmetry properties
For the lossless Maxwell equations we define the following anti-commutation relationship,
with
DT W = −WT D,
(C.62)

(Ŵy ⊗ Wx )
0
0
.
0
(Wy ⊗ Wx )
0
W=
0
0
(Wy ⊗ Ŵx )
(C.63)

For the lossy Maxwell equations we define the following commutation relationship,
with
C.2.2
T
DT W− = W−
D,
(C.64)

−(Ŵy ⊗ Wx )
0
0
.
0
(Wy ⊗ Wx )
0
W− = 
0
0
−(Wy ⊗ Ŵx )
(C.65)

System Formulation for E-polarized fields
We write the system of Maxwell equations as
(D + M1 + M2 s)F̂ = Q̂ 0 ,
(C.66)
F̂ = [Ĥx , Êz , Ĥy ]T
(C.67)
with
and
ext
ext
Q̂ 0 = [−K̂ x , −Ĵzext , −K̂ y ]T
(C.68)
Discretization
Seen the fact that our discretization has PEC at the outer most grid points Êz (x, y) is defined on
the primary grid points in the x- and y-direction. Further Ĥx (x, ŷ) is defined on the primary xand dual y-grid. Last Ĥy (x̂, y) on the dual x- and primary y-grid. We define the field quantities
not coinciding with the PEC as
Êz
Ĥx
size
size
A Krylov subspace approach
(Pk) × (Qk)
(Pk) × (Qk + 1)
(C.69)
(C.70)
Modeling of wave propagation in open domains
84
C.2. TWO-DIMENSIONAL SYSTEM FORMULATION
Ĥy
(Pk + 1) × (Qk)
size
Using this discretization we can define the following differentation matrix


0
(Y ⊗ IPk )
0
D =  (Ŷ ⊗ IPk )
0
−(IQk ⊗ X̂)  .
0
−(IQk ⊗ X)
0
(C.71)
(C.72)
The medium matrices are given by,

Mµ;x 0
0
0 ,
M =  0 Mε
0
0 Mµ;y
(C.73)

0 0 0
S =  0 Mσ 0  ,
0 0 0
(C.74)

and

where Mε , Mσ etc. contain the (averaged) medium parameters on the grid points. The exact
definitions can be found in [26]. Finally the field and source vectors are given by
T
ext T
ext T T
f̂ = [ĥTx , êTz , ĥTy ]T and q̂0 = [(−kext
x ) , (−ĵz ) , (−ky ) ] ,
(C.75)
respectively.
Using these definitions the system can be written as
or as
(D + Ms + S)f̂ = q̂0 ,
(C.76)
(A + sI)f̂ = M−1 q̂0 ,
(C.77)
where the system matrix has been defined as
A = M−1 D + M−1 S.
(C.78)
Symmetry properties
For the lossless Maxwell equations we define the following anti-commutation relationship,
with
DT W = −WT D,
(C.79)

(Wy ⊗ Ŵx )
0
0
.
W=
0
(Ŵy ⊗ Ŵx )
0
0
0
(Ŵy ⊗ Wx )
(C.80)

For the lossy Maxwell equations we define the following commutation relationship
with
T
DT W− = W−
D,
(C.81)

−(Wy ⊗ Ŵx )
0
0
.
W− = 
0
(Ŵy ⊗ Ŵx )
0
0
0
−(Ŵy ⊗ Wx )
(C.82)

Modeling of wave propagation in open domains
A Krylov subspace approach
APPENDIX C. ELECTROMAGNETIC SYSTEM FORMULATION
C.3
85
Three-dimensional system formulation
For three-dimensional systems our standard formulation as
(D + S + M ∂t )F = Q 0 = w(t)Q ,
(C.83)
is used again with
F = [Ex , Ey , Ez , Hx , Hy , Hz ]T ,
and Q 0 = [−Jxext , −Jyext , −Jzext , −Kxext , −Kyext , −Kzext ]T .
(C.84)
and



D =




0
0
0
0
∂z −∂y
0
0
0 −∂z 0
∂x 

0
0
0
∂y −∂x 0 
,
0 −∂z ∂y
0
0
0 
∂z
0 −∂x 0
0
0 
−∂y ∂x
0
0
0
0
ε 0
M=
,
0 µ
σ 0
,
S=
0 0
(C.85)
with ε, µ and σ the permittivity, permeability and conductivity tensor.
C.3.1
Discretization
We use the Yee discretization of the three dimensional Maxwell equation as given in [2]. Therefore we define primary and dual grids in all spatial coordinates. The letters P, Q, R denote the
number of dual step sizes in the x, y, z- direction, respictively. Using this form of discretization
we can rewrite Eq. (C.83) in its discrete form as
(D + S + M∂t )f = q0
(C.86)
with
and q0 = [−jext;T
, −jext;T
, −jext;T
, −kext;T
, −kext;T
, −kext;T
]T .
x
y
z
x
y
z
(C.87)
We assume Yee discretization and introduce the step size matrices as
f = [eTx , eTy , eTz , hTx , hTy , hTz ]T ,
Wk = diag(δk;1 , δk;2 , . . . )
and
Ŵk = diag(δˆ k;1 , δˆ k;2 , . . . ) ∀k = x, y, z.
(C.88)
and the second order differentiation matrices, with Dirichlet boundaries as
K = −Wk−1 ·bidiag(−1, 1)T
K̂ = Ŵk−1 ·bidiag(−1, 1) ∀{K, k} = {X, x}, {Y, y}, {Z, z}.
(C.89)
Now we can define the discrete spatial differentiation operator as
0 Dh
D=
,
(C.90)
De 0
and
with


(Ẑ ⊗ IQk ⊗ IPk+1 ) −(IRk ⊗ Ŷ ⊗ IPk+1 )
Dh =  (−Ẑ ⊗ IQk+1 ⊗ IPk )
0
(IRk ⊗ IQk+1 ⊗ X̂)  ,
(IRk+1 ⊗ Ŷ ⊗ IPk ) −(IRk+1 ⊗ IQk ⊗ X̂)
0
0
A Krylov subspace approach
(C.91)
Modeling of wave propagation in open domains
86
C.3. THREE-DIMENSIONAL SYSTEM FORMULATION
and

0
−(Z ⊗ IQk+1 ⊗ IPk ) (IRk+1 ⊗ Y ⊗ IPk )
0
−(IRk+1 ⊗ IQk ⊗ X)  .
De =  (Z ⊗ IQk ⊗ IPk+1 )
−(IRk ⊗ Y ⊗ IPk+1 ) (IRk ⊗ IQk+1 ⊗ X)
0

(C.92)
Furthermore we define


Ŵz ⊗ Ŵy ⊗ Wx
0
0
.
We = 
0
Ŵz ⊗ Wy ⊗ Ŵx
0
0
0
Wz ⊗ Ŵy ⊗ Ŵx
and
C.3.2
(C.93)


Wz ⊗ Wy ⊗ Ŵx
0
0
.
Wh = 
0
Wz ⊗ Ŵy ⊗ Wx
0
0
0
Ŵz ⊗ Wy ⊗ Wx
(C.94)
Symmetry properties
We note the commutation relations
− Wk K = K̂T Ŵk
∀{K, k} = {X, x}, {Y, y}, {Z, z},
(C.95)
such that we can define the symmetry matrix that satisfies the commutation relation with the
operator matrix A as
AT Ws = Ws A,
(C.96)
where the symmetry matrix is given by
We
0
W =M
.
0 −Wh
s
Modeling of wave propagation in open domains
(C.97)
A Krylov subspace approach
Appendix D
Optimal PML formulation
D.1
Derivation of optimal step sizes
To enhance the understanding in what way the PML is considered as optimal, the derivation for
propagating waves for a 2D Helmholtz operator will be followed as documented in [33]. The
derivations for more general optimal PMLs follow, besides being more complicated, the same
principle.
Considering a Helmholtz equation on a two-dimensional half space Ω ⊂ R2 with Ω := {y ∈
R, x ∈ R|x < 0}
∆u + ω2 u = 0 on Ω.
(D.1)
The half space x > 0 is used to insert a PML in order to absorb waves traveling in positive
x-direction, under various angles. After a Fourier transform with respect to y-coordinates the
Helmholtz systems is rewritten as
uxx − λu = 0
on Ω.
(D.2)
with λ = κ2y − ω2 . The solution to the upper equation is
 q
 κ2y − ω2
√ √
q
u = a exp − λx with λ =
 i ω2 − κ2
y
κ
if | ωy | ≥ 1 describing evanescent waves
κ
if | ωy | < 1 describing propagating waves
(D.3)
The solution given in Eq. (D.3) at x = 0 directly defines the impedance condition
u 1
(D.4)
= −√ ,
ux x=0
λ
mathematically also known as Neuman to Dirichlet (NtD) map, as it converts Neuman to Dirichlet data.
Within this thesis coupled first order differential equations are of interest such that optimal step
sizes on a primary and dual grid need to be obtained. Therefore the second order formulation is
changed to first order formulation as
√
√
ux = λv and vx = λu,
(D.5)
such that the NtD seen at the interface map of Eq. (D.4) becomes uv x=0 = −1. The step sizes
of the PML now need to approximate this NtD over a predefined range of angles. Therefore an
expression for the impedance inside the PML, inserted in the half space x > 0, is derived. The
87
88
D.1. DERIVATION OF OPTIMAL STEP SIZES
complex stretching function χ is introduced such that the first order differential equation inside
the PML becomes
√
√
χ−1 ux = λv and χ−1 vx = λu
(D.6)
with the complex stretching function
χ(x, iω) = α(x) +
β(x)
.
iω
(D.7)
Introducing k primary and dual step sizes inside the PML denoted by h j and ĥ j for j = 1, 2, . . . , k,
the stretched differential equation can be written in finite difference form as
u(x j+1 ) − u(x j ) √
= λv(x̂ j ),
χ(xˆj )h j
v(x̂ j+1 ) − v(x̂ j ) √
= λu(x j ),
χ(x j )ĥ j
∀ j = 1, . . . , k.
(D.8)
Therefore the stretched step sizes inside the PML are given by
hcj = χ(x̂ j )h j and ĥcj = χ(x j )ĥ j
∀ j = 1, . . . , k
(D.9)
The finite difference NtD is now given by
u(x1 )
(α,β)
= − fk
v(x̂1 )
(D.10)
(α,β)
where fk
can be found as continued fraction by subsequent substitution of Equations (D.8)
into each other, assuming a Dirichlet boundary condition at the fictive node xk+1 , as
(α,β)
fk
1
=
ĥc1
√
λ+
(D.11)
1
√
hc1 λ +
1
√
ĥc2 λ + · · · +
1
√
ĥck λ +
1
√
hck λ
Now a minimization problem arises as the difference between the upper NtD map and the NtD
of the half space x < 0 needs
√ to be minimalized. For this example propagating waves are
considered only. Therefore λ is imaginary and is redefined as
√
κy √
√
λ = iω γ, with γ = 1 − ( )2 γ ∈]0, 1]
(D.12)
ω
√
Therefore with γ the angle of the incoming wave is defined. The PML is only optimized over
a range incoming wave vectors, such that the optimization interval is defined as γ ∈ [γmin , 1].
Using this redefinition the stretched step sizes are defined as
√
hcj = [β(x̂ j ) + iωα(x̂ j )]h j γ and ĥcj = [β(x j ) + iωα(x j )]ĥ j ∀ j = 1, . . . , k
(D.13)
(α,β)
For fk
to only depend implicitly via γ on frequency, α is set to zero. This is unique for the
derivation of PMLs for propagating waves, as pure imaginary step sizes only add damping, but
no propagation delay, which damps evanescent waves. Further it can be seen that changing the
step sizes and keeping β = 1 has the same impact as fixing the step sizes and changing β.
The problem is thus reformulated from finding the optimal stretching function β to finding the
optimal step sizes that minimize the impedance mismatch at the interface x = 0. The NtD map
Modeling of wave propagation in open domains
A Krylov subspace approach
APPENDIX D. OPTIMAL PML FORMULATION
89
(0,1)
is no longer a function of α and β such that the notation of fk
is changed to fk .
Now the PML error is given by the difference of the NtD maps at the interface x = 0, such that
the minimization problem reads
δk = min max |1 − fk (γ)|
h j ,ĥ j γmin ≤γ≤1
(D.14)
This problem can be rewritten as a third type Zolotarev problem of rational approximation in
γ
the complex plane. Therefore a change of variable is used introducing z = γmin
and the rational
function m(z) as
√
zm(z) = fk (γmin z).
(D.15)
√
fk (γmin z) is a continued fraction in γ and can therefore be rewritten as rational function of
√
order [(2k − 1)/2k] in γ. Therefore m(z) can be written as a rational function of order [(k −
1)/k] in γ. Finding the optimal rational function that minimizes δk as
√
δk = min max 1 − zm(z) ,
(D.16)
h j ,ĥ j 1≥z≥ γ 1
min
was solved by the Russian mathematician Y.I.Zolotarev in the 19th century in the scope of
rational approximation theory. Zolotarevs solution denoted as mZ (z) is given by
(z + c2n )
∏k−1
mz (z) = D k n=1
∏n=1 (z + c2n−1 )
with
√
nK( 1−γmin ) √
; 1 − γmin
2k
√
nK( 1−γmin ) √
2
; 1 − γmin
cn
2k
sn2
cn =
(D.17)
n = 1, 2, . . . , 2k − 1
(D.18)
where sn and cn are Jacobi elliptic functions and K(x) denotes the complete elliptic integral.
The scalar D is defined via the symmetry condition
√
√
max [1 − zmZ (z)] = − max [1 − zmZ (z)].
(D.19)
[1,1/γmin ]
[1,1/γmin ]
Rewriting Zolotarevs function in pole-residue form allows to uniquely retrieve the imaginary
step sizes.
In order to obtain PML step sizes that are matched to the grid the discretized NtD map should
be used in Eq. (D.4) rather than
√ the continuous NtD map. To optimize the PML for evanescent
wave and propagating waves λ is not rewritten in the form of Eq. (D.12).
For one-dimensional problems the optimal PML can be simplified as shown in Appendix D.
D.2
Small angle approximation
In this section the expressions obtained for a 2D optimal PML are simplified for small angles
of incidence and for 1D problems. Therefore the limit of
κy → 0
(D.20)
is of interest, such the wave has no spatial frequency in the direction perpendicular to the PML.
This approach can also be used when designing 1D PMLs.
A Krylov subspace approach
Modeling of wave propagation in open domains
90
D.2. SMALL ANGLE APPROXIMATION
We start with our derivation at Eq. (D.12) by noting that γ = 1, and further that in the limit of
κy → 0 we also obtain γmin = 1. Further the eliptic modulus in Eq. (D.18) is given by
p
ϑ = 1 − γmin = 0.
(D.21)
The solution to Zolotarevs problem as shown in Eq. (D.17) is given by
(z + c2n )
∏k−1
mZ (z) = D k n=1
,
∏n=1 (z + c2n−1 )
with
cn =
sn2
nK(ϑ)
2K ; ϑ
cn2
nK(ϑ)
2K ; ϑ
(D.22)
,
(D.23)
with sn and cn the Jacobi elliptical functions, K(ϑ) the complete elliptic integral and k the
number of PML layers. Since ϑ = 0 in the small angle approximation we note that
π
K(0) = ,
2
sn(x; 0) = sin(x) ,
and
cn(x; 0) = cos(x) .
(D.24)
Therefore Zolotarevs rational function is now given by
z + tan2 nπ
∏k−1
n=1
2k
.
mZ (z) = D
(2n−1)π
k
2
∏n=1 z + tan
4k
D.2.1
(D.25)
Determining D
The constant D is uniquely determined by Eq. (D.19). Since the interval z = C[1, 1/γmin ] becomes a single point we can analytically determine D via
√
√
1 − √zmZ (z) = −1 + zmZ (z)
(D.26)
zmZ (z) = 1
(D.27)
(2n−1)π
∏kn=1 1 + tan2
4k
D=
(D.28)
k−1
nπ
2
∏n=1 1 + tan 2k
We now write 1 + tan2 (x) =
1
cos2(x)
and obtain
2 nπ
∏k−1
n=1 cos
2k
.
D=
(2n−1)π
k
2
∏n=1 cos
4k
We now use the identity
k
(2n − 1)π
∏ cos
4k
n=1
√
2
= k,
2
(D.29)
(D.30)
which can be obtained from setting a k-th order Butterworth low pass filter transfer functions
equal to the cut of frequency. Squaring the identity we can write
D=2
2k−1
k−1
nπ · ∏ cos
.
2k
n=1
Modeling of wave propagation in open domains
2
(D.31)
A Krylov subspace approach
APPENDIX D. OPTIMAL PML FORMULATION
91
To replace the second identity we first rewrite the cosine product into a sine product. For odd k
we can write
k−1
k−1
2
2 nπ
2 nπ
2 (k − n)π
(D.32)
∏ cos 2k = ∏ cos 2k cos
2k
n=1
n=1
k−1
2
2 (k − n)π
2 nπ
sin
(D.33)
= ∏ sin
2k
2k
n=1
nπ k−1
(D.34)
= ∏ sin2
2k
n=1
(D.35)
and for even k we can write
k−1
2
∏ cos
n=1
π k−2
2
(k − n)π
= cos
∏ cos 2k cos
2k
4 n=1
2k
k−2
2
2 nπ
2 (k − n)π
2 π
= sin
∏ sin 2k sin
4 n=1
2k
nπ k−1
.
= ∏ sin2
2k
n=1
nπ 2
2
nπ 2
(D.36)
(D.37)
(D.38)
(D.39)
Now we use the identity
k−1
∏ sin
n=1
nπ k
=
k
2k−1
,
and rewrite it by noting that the sin function is symmetric with respect to the point
√
nπ k−1
k
∏ sin 2k = 2k−1 .
n=1
Therefore we can square this identity and obtain our final expression for D as
k−1
k
2 nπ
cos
= 2k−2
∏
2k
2
n=1
to obtain
D=
D.2.2
k · 22k−1
= 2k.
22·k−2
(D.40)
π
2
as
(D.41)
(D.42)
(D.43)
Calculation of step sizes
To obtain the optimal step sizes the rational function mZ (z) if written in its pole-residue form
k
z + tan2 nπ
wi
∏k−1
n=1
2k
= D ∑
mZ (z) = D
.
(D.44)
(2n−1)π
k
z
−
θ
2
i
i=1
z
+
tan
∏n=1
4k
Obviously the poles of the functions coincide such that θi = −c2n−1 = − tan2 (2n−1)π
. The
4k
numbers wi needed to convert the Zolotarev function to its pole residue form is now given by
wi =
(z − θi )
mZ (z)
D
A Krylov subspace approach
(D.45)
Modeling of wave propagation in open domains
92
D.2. SMALL ANGLE APPROXIMATION
2 (2i−1)π + tan2 nπ
−
tan
∏k−1
n=1
4k
2k
,
=
(2i−1)π
(2n−1)π
(2i−1)π
(2n−1)π
i−1
k
2
2
2
2
−
tan
+
tan
−
tan
+
tan
∏n=1
∏n=i+1
4k
4k
4k
4k
(D.46)
where it appears that
k
∑ wi = 1.
(D.47)
i=1
Following the derivation in the previous section it can be seen that the optimal DtN map is given
by
k √γ
√
min wi
.
(D.48)
fk (γ) = γD ∑
i=1 γ − γmin θi
This DtN map must be the same as the numerical one of the PML. Therefore the DtN map needs
to be computed and brought into pole-residue form. This creates an inverse-eigenvalue problem
which can be solved uniquely by a Lanczos algorithm. From the derivation it follows that
1
1
2 (2i − 1)π
ĥ1 = = , µi = − tan
,
(D.49)
D 2k
4k
and
si =
√
wi .
(D.50)
Further by following the first step of the Laczos algorithm it can be found that
k
k
1
2 (2i − 1)π
a1 = −
= ∑ wi µi = − ∑ wi tan
,
4k
ĥ1 h1 i=1
i=1
(D.51)
with a the diagonal and b the off diagonal coefficients in the Lanczos coefficient matrix. Following the Lanczos algorithm the step sizes are found as
h1 =
1
1
,
ĥ1 ∑k wi tan2 (2i−1)π
i=1
4k
(D.52)
which after substitution of wi can be written as
h1 =
2k
4 2
1
3k − 3
=
6k
.
4k2 − 1
(D.53)
Following the Lanczos algorithm we now find
ĥ2 =
or
ĥ2 =
9 · 2k
(4k2 − 1)2
1
ĥ1 · (b2 h1 )2
(D.54)
1
∑ki=1
4 2
1
3k − 3
Modeling of wave propagation in open domains
−
tan2 (2i−1)π
4k
2
.
(D.55)
wi
A Krylov subspace approach
APPENDIX D. OPTIMAL PML FORMULATION
A Krylov subspace approach
93
Modeling of wave propagation in open domains
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