Enhancing the Efficiency of the Polynomial Chaos Expansion ...
Transcript of Enhancing the Efficiency of the Polynomial Chaos Expansion ...
Enhancing the Efficiency of the Polynomial Chaos ExpansionFinite-difference Time-domain Method
by
Zixi Gu
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
c© Copyright 2014 by Zixi Gu
Abstract
Enhancing the Efficiency of the Polynomial Chaos Expansion Finite-difference Time-domain Method
Zixi Gu
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2014
The polynomial chaos based finite-difference time-domain (PCE-FDTD) method is a promising tech-
nique for quantifying the impact of parameter variability on the performance metrics of electromagnetic
structures. With the aim to improve the versatility and computational efficiency of the PCE-FDTD
method, this thesis presents two novel formulations of the PCE-FDTD. First, a formulation and system-
atic study of the convolutional perfectly matched layer for terminating simulation domain constituting
random media is presented, and demonstrates excellent efficiency for the study of microwave structures
with substrate permittivity uncertainties. Second, a hybrid Monte Carlo / PCE-FDTD method based on
the control variate is developed to mitigate the large computation cost associated with multi-parametric
uncertainty analysis using PCE-FDTD. This method is applied to a Bragg reflector structure with un-
certain slab permittivities and leads to a considerable reduction in computation time over conventional
PCE-FDTD method.
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To my family.
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Acknowledgements
I would like to express my deepest and sincere gratitude towards my supervisor, Professor Costas Sarris,
for his dedicated support and diligent guidance throughout the duration of this work. In addition to the
immense technical expertise and rigorous insights he has provided me with, his admirable work ethics
and an unyielding drive for excellence have been a tremendous source of inspiration and motivation. I
am extremely fortunate and grateful to have had this invaluable opportunity to work under and to learn
from him.
I would also like to thank the members of my thesis committee, Professor Sean Hum, Professor Piero
Triverio, and Professor Shahrokh Valaee, for the valuable time which they have taken from their busy
schedule in order to evaluate and give feedback to this thesis.
I would like to thank Dr. Andrew Austin. Every one of the many conversations we had on the topic
of polynomial chaos had made me more interested and inspired by this subject. I have benefited greatly
from his extensive knowledge and experience on the subject of FDTD and numerical methods in general,
and I am grateful for all of the help I have received from him over the years.
I would like to acknowledge every member of the electromagnetic group here at the University of
Toronto for their camaraderie and friendship. It was an amazing experience to work with and be
surrounded by such an exemplary group of colleagues.
I would like to extend my gratitude to Hans-Dieter Lang, Colan Ryan, Neeraj Sood, Luyu Wang,
Muhammad Alam, Trevor Cameron, Mohammad Memarian, Tony Liang, Michael Chen, Xingqi Zhang
and Sameer Zaheer for their help on various aspects of this thesis which have led to a great amount of
improvements in this work.
And finally, nothing would have been possible nor mattered without my family, especially my parents,
Zhong Gu and Ying Zhou. This thesis is a testament of their endless and unconditional love and support.
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Contents
1 Introduction 1
1.1 Overview of Past Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Numerical Methods for Quantifying Uncertainty Propagation . . . . . . . . . . . . 6
1.1.2 Uncertainty Quantification in Electromagnetic Problems . . . . . . . . . . . . . . . 7
1.2 Thesis Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Background 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Finite-difference time-domain method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Central Finite-difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 The Yee’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.4 Numerical Dispersion and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.5 Perfectly Matched Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Quantifying Output Uncertainty by Uncertainty Propagation . . . . . . . . . . . . . . . . 22
2.3.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Generalized Polynomial Chaos Expansion . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Intrusive Polynomial Chaos Expansion-based Finite-difference Time-Domain Method . . . 30
2.4.1 PCE-FDTD Formulation for Modelling Material Uncertainties . . . . . . . . . . . 30
2.4.2 PCE-FDTD Update Equations for Geometric Uncertainties . . . . . . . . . . . . . 34
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 A PML Absorber for the Termination of Random Media 52
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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3.2 The PCE-FDTD PML absorber: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Random dielectric-filled two-dimensional domain . . . . . . . . . . . . . . . . . . . 55
3.3.2 Microstrip Low-pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Polynomial Chaos Expansion as Control Variate . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Conclusions 76
5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Bibliography 79
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List of Tables
2.1 Askey-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Uni-variate Hermite Polynomial Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Multi-variate Hermite Polynomial Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Comparison of Computation Time Between MCM, PCE, and CV-PCE . . . . . . . . . . . 74
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List of Figures
1.1 Transistor Threshold Voltage Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Transistor Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 FR-4 Substrate Dielectric Permittivity Variability . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Track Forecast Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Yee’s Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Leap Frog Time-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Monte Carlo algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Monte Carlo Standard Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Polynomial Chaos Expansion Surrogate Model . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Material Uncertainty Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Geometry of the Single-stub Microstrip Filter . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Rectilinear Mesh Cell Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Single-stub Filter |S11| Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 Single-stub Filter |S11| Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.12 Single-stub Filter |S21| Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.13 Single-stub Filter |S21| Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.14 Single-stub Filter Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.15 Geometry of the Two-stub Microstrip Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.16 Two-stub Filter |S11| Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.17 Two-stub Filter |S11| Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.18 Two-stub Filter |S21| Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.19 Two-stub Filter |S21| Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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3.1 Geometry of Two-dimensional Dielectric-filled Domain . . . . . . . . . . . . . . . . . . . . 55
3.2 Relative Error of Two-dimensional Domain CPML . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Relative Error of Two-dimensional Domain Reflection for Varying Permittivity Variance . 59
3.4 PCE Convergence of Relative Error of Two-dimensional Domain CPML . . . . . . . . . . 60
3.5 Monte Carlo Convergence of Relative Error of Two-dimensional Domain CPML . . . . . . 61
3.6 Geometry of Low-pass Microstrip Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Low-pass Microstrip Filter |S11| Statistical Moments . . . . . . . . . . . . . . . . . . . . . 62
3.8 Low-pass Microstrip Filter |S21| Statistical Moments . . . . . . . . . . . . . . . . . . . . . 63
4.1 Geometry of One-dimensional Bragg Reflector . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 PCE Convergence of the 8 Cell Bragg Reflector |S21| Statistical Moments . . . . . . . . . 69
4.3 8 Cell Bragg Reflector |S21| Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Correlation Coefficient of Control Variate . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Relative Error of 8 Cell Bragg Reflector |S21| Variance . . . . . . . . . . . . . . . . . . . . 72
4.6 Relative Error of 6 Cell Bragg Reflector |S21| Variance . . . . . . . . . . . . . . . . . . . . 73
4.7 Relative Error of 10 Cell Bragg Reflector |S21| Variance . . . . . . . . . . . . . . . . . . . 73
4.8 Comparison of Computation Time Between MCM, PCE, and CV-PCE . . . . . . . . . . . 74
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Chapter 1
Introduction
The seminal prediction made by Thomas Moore in 1965, known as Moore’s law, states that the transistor
density of integrated circuits doubles every year [1]. This is largely achieved by the dimensional scaling
of semiconductor transistors and drives the continual improvement of microchip performance, which
has increased five orders of magnitude over the last four decades [2]. It would have been impossible
otherwise to produce cheap and powerful electronics which have become a pervasive and integral aspect
of the modern society. However, the shrinking of transistors is not without its challenges. One of
which is the presence of variability in the physical properties of the transistors due to manufacturing
processes. These variabilities lead to inconsistent electrical properties of transistors produced on a single
chip. Especially as feature sizes of transistors approach the nanometer regime, issues with variability
are becoming extremely difficult, if not impossible, to solve. An example of this is given in [3]. To speed
up transistor switching speed, dopants are added to the silicon channels by bombarding wafers with
high-speed ions. While the exact numbers of dopants successfully implanted is difficult to control, large
transistors are able to accommodate tens of thousands of dopant atoms, and the impact of variations in
the amount of dopant is negligible. At present, transistors can only accommodate a few hundred atoms,
and the impact of deviations of a few atoms can lead to variations in the threshold voltage needed to turn
on the transistor [3]. As a result, transistors on a single chip can exhibit varying threshold voltages. The
impact of variability will become even more pronounced as transistor sizes continue to scale downwards.
Many other sources of variations, such as the roughness of the silicon gate used in transistors, and
the granularity of the metal electrode used to turn on and off a transistor, are becoming important
contributors to the variability of transistor performance[3]. Variability Expedition, an institution seek-
ing to reduce the problem of variability in microchip technology, identified three additional sources of
1
Chapter 1. Introduction 2
variability: fluctuations in environmental conditions, the wearing down of a device due to aging, and
differences in the devices from different vendors [4]. All of these sources of variability can greatly impact
the performance and reliability of computer hardware.
Threshold Voltage (volts)
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 1.1: Variability of threshold voltage as transistor feature sizes shrink from 28 nm to 14 nm [3].
Figure 1.2: Uneven rows of exposed photoresist which will become 30 nm long transistor gates [3].
In electromagnetic engineering, a prominent source of variability arises from the dielectric permittivity
of laminate material used as printed circuit board substrates. For example, FR-4 substrate is a relatively
inexpensive and commonly used material for PCB construction. Its dielectric permittivity can be affected
by a variety of factors, such as variations in moisture absorption, temperature, and substrate height [6].
As a result the variation in its dielectric permittivity may be up to 10% or more of its average values.
The consequences of variability can be significant. For example, variability of electrical properties as a
result of the manufacturing process can lead to batches of chips where more than half will run 30 percent
Chapter 1. Introduction 3
Figure 1.3: The variability in the relative electric permittivity of FR-4 substrate materials [5].
slower than intended or consume 10 times more power on standby [3]. It was found that flash devices,
with nominally identical specifications, showed a 27 percent energy variation [4]. This represents a
considerable overhead cost for applications requiring computing hardware to meet precise specifications.
In some cases, failure to account for variability in the design process can lead to unforeseen and grave
consequences. A glaring example of this is the refurbishment project of a fleet of Nimrod MRA2 patrol
aircraft [7], which was contracted to BAE Systems by the Royal Air Force in 1996. This involved an
overhaul of the fuselage and the installation of new wings and engines to the aircraft. However, when
it was discovered that wide variations in the assembly of the fuselage had existed, the integration of
the newly designed wings posed numerous engineering difficulties. High cost overrun and long delays
ensued, and the project was ultimately canceled in 2010 at a cost of £3.6 Billion [8].
Therefore, a prudent design requires a critical assessment of the impact of variability that may
be present in any step of the engineering process. Since the precise value of a particular realization
of a system parameter is unknown a priori, variability is viewed as an “uncertainty”. Uncertainty is
defined by the AIAA as the deficiency in any phase or activity of the modelling process due to a lack of
knowledge [9]. This type of uncertainty is generally referred to as epistemic uncertainty, which is found
Chapter 1. Introduction 4
in situations where the parameter values of some material property are not known precisely but can be
found by repeating or refining the experiments to obtain more data or data precision [10]. Uncertainties
may also arise due to the inherent stochastic nature of the system. These uncertainties are referred to
as aleatory uncertainties and usually involve processes which we have very little control over [10]. The
previous example of variability in the transistor threshold voltage due to transistor dopant level is an
example of aleatory uncertainty.
Uncertainty quantification allows us to analyze the impact of the parameter variability, which may
involve the following [11]:
• Variance analysis: Quantify the variability of the output, such as by establishing a confidence
interval of the output quantity.
• Reliability analysis: Ensure the proper operations of the device by finding the likelihood the device
will perform outside of some critical threshold, probability of failure, or expected lifetime of the device.
• Sensitivity analysis: Evaluate the relative contribution of each parameter variability on the output
variability, in order to identify and minimize the parameter variability with the largest impact on the
output.
• Validation of Numerical Model: Validate a numerical model of output with uncertainty by com-
paring the measurement result of the physical processes.
To quantify uncertainties of some output of interest, the uncertain parameters, i.e., parameters with
random variations, are characterized within a probabilistic framework by representing them as random
variables or random fields. Then, the uncertainties in the model output or response are determined
by solving the numerical model with the uncertain parameters, where the parameter variations are
“propagated” to the model output. A visual example of uncertainty propagation is the “forecast cone”
used for hurricane predictions [12], shown in Fig. (1.4). Forecast models are used to determine the likely
path the hurricane will take based on the weather conditions of the region. However, the volatile nature
of a hurricane’s trajectory means any prediction about its future positions must be accompanied by a
measure of uncertainty. This is reflected by the area of the cone, which represents two-thirds of the
historical official forecast errors in the past five years. In the context of uncertainty propagation, the
uncertain parameter is the hurricane’s trajectory as predicted by the forecast model, the input of the
forecast model is the current position of the system, and the output is the position of the hurricane at
some point in the future. Attempts to estimate the trajectory further into the future are subjected to
larger forecast errors. As a result, the increasing uncertainty associated with the prediction is depicted
by a growing cone size in time.
Chapter 1. Introduction 5
Figure 1.4: A 5-day forecast cone representing the probable path the center of the hurricane will take. Aseries of circles are constructed by enclosing an area encompassing two-thirds of historical forecast errorover a 5-year period. The cone is then created by drawing a line that is tangent to each circle [12].
Numerical models used for uncertainty propagation generally employ numerical solvers for determin-
istic systems. Sometimes, we can simply use the deterministic model without modification to obtain
statistical information of the output uncertainty. This is referred to as a non-intrusive method. Intrusive
methods are also available where the numerical solver is reformulated to solve for a specific statistical
moment or function. In either case, for the purpose of uncertainty propagation, numerical solvers can
potentially involve a large parameter space, which require computation time or memory that are orders
of magnitude larger compared to simulations of deterministic models. For example, a simulation of a
physical system may involve uncertainties in geometric and material parameters, as well as boundary
and initial conditions, shown in Fig. (1.5). The output response must account for each of the random
variable present. This poses a significant barrier to the viability of uncertainty quantification methods,
particularly for large structures under the presence of numerous sources of uncertainties. As a result,
improving the efficiency of numerical methods for uncertainty quantification remain an important and
continuing endeavor.
Chapter 1. Introduction 6
“Uncertainty Propagation”
within
Numerical Model
Output Response
Y(p1(ξ1), p2(ξ2), p3(ξ3), p4(ξ4))
Geometry
p2(ξ2)
Material
Properties
p3(ξ3)
Boundary
Conditions
p4(ξ4)
Initial
Conditions
p1(ξ1)
Figure 1.5: Uncertainty propagation of model parameters, e.g., geometric and material properties, asfunctions of random variables ξ. The output response becomes a function of the multi-parameteric spaceconsists of all uncertain parameters.
1.1 Overview of Past Works
1.1.1 Numerical Methods for Quantifying Uncertainty Propagation
In electromagnetic engineering, popular numerical methods for quantifying uncertainty propagation
are based on the Monte Carlo method (MCM) [13, 14, 15], the perturbation method [16, 17], and
the polynomial chaos expansion (PCE) method [18, 19, 20, 21, 22]. The MCM is perhaps the most
popular method in use today [23, 11]. In this method, samples of the random parameters are generated
according to their probability distributions and a separate deterministic numerical model is solved for
each parameter sample. The model output statistical moments are then estimated from the ensemble
of output samples. Due to its ease of implementation and the capability to increase the accuracy of
the estimate by simply increasing the number of deterministic simulations, the MCM has been widely
adopted for uncertainty quantification and is often employed as the standard for benchmarking other
uncertainty quantification methods. However, the MCM converges slowly with the mean converging
at a rate proportional to 1/√N for N samples, and is therefore computationally expensive for large
simulations.
Chapter 1. Introduction 7
In the perturbation method, the output random variable is expressed as a truncated Taylor expan-
sion in terms of the input variables about their mean [23]. The moments of the output are directly
approximated from the moments of the truncated expansion. The expansion is usually truncated up
to second-order, as higher order expansions leads to a more complicated implementation and a larger
computational cost. Therefore the perturbation method is only valid in circumstances where the input
and output variations are small.
Norbert Wiener introduced the concept of homogeneous chaos in his work on the study of Brownian
motion, where a Gaussian process is represented by an expansion of random Hermite polynomials [24].
Cameron and Martin demonstrated that the convergence of the homogeneous chaos for any Gaussian
process with a finite second-order moment is optimal [25]. By utilizing the relationship between the
orthogonal polynomial weight functions and the probability distribution functions of random processes,
Xiu and Karniadakis extended the homogeneous chaos to a class of common random processes using
orthogonal polynomials in addition to Hermite polynomials as basis functions [26]. This approach is
referred to as general polynomial chaos or the polynomial chaos expansion method.
The numerical implementation has been previously outlined in [26]. The PCE seeks a representation
of the random output by a truncated linear expansion of orthogonal polynomials. Each expansion
coefficient is determined by a Galerkin-based projection of the numerical model on the polynomial
basis of corresponding order. The expansion coefficients are used to reconstruct the polynomial chaos
expansion (PCE), which gives the output as a direct function of the input random variables, from which
the statistics of interest can be extracted. The polynomial chaos method has demonstrated excellent
computation efficiency compared to the Monte Carlo approach. However, the PCE method requires a
reformulation of the deterministic numerical solver for handling inner product integrals, which may be
difficult to implement. For these reasons, the development of the polynomial chaos method has become
an area of active research within the electromagnetic community.
1.1.2 Uncertainty Quantification in Electromagnetic Problems
The development of uncertainty quantification and propagation methods for electromagnetic structures
have received special attentions in recent years. We briefly review the development of a few finite-
difference time-domain (FDTD) and PCE-based methods for quantifying uncertainties.
A perturbation-based stochastic FDTD method was presented in [16], where the electric and magnetic
fields are expanded in terms of a second-order Taylor expansion and the FDTD update equations are
reformulated to time-step the mean and variance of the fields. This was applied to a layered tissue model
Chapter 1. Introduction 8
exhibiting variations in the electric permittivity and conductivity. This method is computationally
inexpensive compared to MCM, however, as with most perturbation methods, the accuracy of this
method relies on a low input parameter variance. In addition, computing higher order moments may
further complicate the implementation of the stochastic FDTD.
One of the earlier application of polynomial chaos expansion to electromagnetic problems was given
in [27] using a high-order discontinuous Galerkin method as the numerical solver. The polynomial chaos
expansion was solved using stochastic collocation and intrusive PCE, referred to as stochastic Galerkin
method. These were applied to a 1-D material loaded cavity with uncertainties in the domain-filling
electric permittivity, and positions of boundaries and material interface. A 2-D circular cylinder with
uncertainties in the source term, electric permittivity, and geometric dimensions was also investigated.
The geometry uncertainties were modeled by embedding the uncertainty in the mesh parameters and
the PCE simulation was able to run with a single generated mesh. All simulations were run for a single
random parameter at a time. The simulations have demonstrated the efficiency of the the polynomial
chaos expansion compared to Monte Carlo method.
A FDTD formulation of the polynomial chaos expansion method (PCE-FDTD) was first presented
in [20]. In this method, the fields at each mesh cell and time step are expanded in terms of polynomial
chaos basis, and the Yee’s algorithm are reformulated to solve for the expansion coefficients. This method
was used to study the electromagnetic compatibility problem, where a PCB is placed inside a shielded
enclosure with an aperture located on one side. The reflection coefficient of the PCB is modeled as
a random variable, to account for the variations in the absorption of impinging electric field due to
material uncertainties. The mean and variance of a probed electric field were then determined from the
PCE-FDTD simulation. A second numerical experiment was conducted to study the scattering from a
dielectric sphere with uncertainties in its radius, electric permittivity, and magnetic permeability.
In both numerical experiments, the PCE-FDTD method showed around 90% reduction in computa-
tion time over the MCM method. However, in the case of the second experiment where three random
parameters are involved, the memory requirement of the PCE-FDTD method is also considerably larger.
Furthermore, unlike the stochastic-FDTD method where only the mean and variance of the electric and
magnetic fields are determined, the PCE-FDTD method solves for the PCE of the fields which allows
the extraction of the complete statistical information of the random field quantities.
The approach to modelling geometric uncertainties in PCE-FDTD given in [20] requires the geometric
uncertainty to be transformed into an equivalent material uncertainty, e.g., variations in the radius of the
sphere were related to an equivalent uncertainty in the magnetic permeability and electric permittivity.
This approach is not viable for all geometric variations, such as those involving microstrip dimensions of
Chapter 1. Introduction 9
microwave circuits. This problem was resolved by a new formulation of the PCE-FDTD for geometric
uncertainty given in [22], where geometric uncertainties are embedded in the mesh cell dimensions, giving
rise to a direct modelling of geometric uncertainties in FDTD. The proposed method used a curvilinear
mesh in the regions with dimension uncertainty and the dimension variations are imposed by distorting
the curvilinear mesh. This method was used to investigate uncertainties in stub length of a microstrip
cascaded stub filters as well as separation distances between coupled lines in directional couplers. It was
reported that the PCE simulation of the cascaded stub filter again reported roughly 10 times speed up
over the Monte Carlo method.
1.2 Thesis Motivation and Objectives
The recent developments in the PCE-FDTD method have created one of the most versatile and effi-
cient tool for quantifying uncertainty in electromagnetic structures, capable of modelling both uncertain
material and geometric parameters directly within the simulation domain. There are, however, two short-
comings of the PCE-FDTD that have yet to be addressed. First, in random media where uncertainties
in the material parameters are present at the FDTD domain boundaries, the resulting random wave
impedance must be accounted for by the boundary conditions used to terminate the domain. Previous
work on this problem has employed Mur’s first order boundary conditions [20]. Hence, the state-of-
the-art in PCE-FDTD absorbing boundary conditions lags behind the corresponding state-of-the-art in
general FDTD, which is defined by the perfectly matched layer absorber (PML [28, 29]). Second, the
efficiency of the PCE method has been established in various numerical experiments where a small num-
ber of random inputs are involved. As the number of random parameters increases, however, the number
of expansion coefficients P +1 grows rapidly, a symptom of the “curse of dimensionality.” Coupled with
the fact that the computation time of the PCE-FDTD is proportional to (P + 1)2, the efficiency of the
PCE method for investigating multi-parametric analysis can be significantly compromised.
The objective of this thesis is to resolve these two shortcomings and to further advance the PCE-
FDTD formulation for quantifying uncertainty propagation in electromagnetic engineering. To that end,
two formulations of the PCE-FDTD are introduced.
The first issue is addressed with the formulation of a perfectly matched layer absorber for terminating
random media, thus bridging the gap between the state of the art boundary conditions used in FDTD and
the PCE-FDTDmethod. The second issue is addressed with a hybrid Monte Carlo / PCE-FDTDMethod
for analyzing multi-parametric uncertainties in electromagnetic simulations, where the polynomial chaos
expansion is employed as a “control variate”, similar to [30]. The term “control variate” refers to
Chapter 1. Introduction 10
a random variable used to transform the Monte-Carlo estimator of the undetermined output random
variable to a form with an improved convergence rate. Instead of using PCE with an increasingly higher
order to improve accuracy, a PCE-based control variate (CV-PCE) method can achieve a similar effect
with a lower order PCE as the control variate combined with a small number of Monte-Carlo samples.
1.3 Thesis Outline
The chapters in this thesis are outlined as follows.
Chapter 2 provides a review of the FDTD method and uncertain quantification methods pertaining to
the topic of this thesis. The polynomial chaos expansion-based FDTD method is outlined for modelling
material and geometric uncertainty. A numerical case study is carried out on microstrip stub filters with
uncertain stub lengths.
Chapter 3 presents the formulation of the convolutional perfectly matched layer within the framework
of PCE-FDTD. This is used to study the termination of the random media on a 2-D domain filled
with random electric permittivity as well as to a microstrip circuit with uncertain substrate electric
permittivity.
Chapter 4 presents the hybrid Monte-Carlo / PCE-FDTD method for multi-parametric analysis.
The performance of this method is demonstrated on a study of a 1-D Bragg reflector exhibiting slab
permittivity uncertainties.
And finally Chapter 5 concludes this thesis.
Chapter 2
Background
2.1 Introduction
This chapter reviews the finite-difference time domain (FDTD) method and the uncertainty propaga-
tion and quantification based on the Monte Carlo method and the polynomial chaos expansion method.
Previous formulations for the polynomial chaos expansion-based FDTD (PCE-FDTD) for electromag-
netic structures exhibiting material and geometric uncertainties are outlined. Finally, the PCE-FDTD
is demonstrated on a numerical example involving a cascaded network of microstrip stub filter with
uncertain stub lengths.
2.2 Finite-difference time-domain method
In this section the FDTD method is outlined. The update procedure of the electric and magnetic fields
are derived for the Yee’s mesh cell configuration. The FDTD numerical stability and numerical dispersion
conditions, boundary conditions are discussed.
11
Chapter 2. Background 12
2.2.1 Maxwell’s Equations
The governing laws of electrodynamics are given by a set of equations collectively known as Maxwell’s
equations. In their differential form, they are stated as:
∂D
∂t= ∇×H− J (2.1a)
∂B
∂t= −∇×E−M (2.1b)
∇ ·D = ρ0 (2.1c)
∇ ·B = 0 (2.1d)
where D is the electric flux density, H is the magnetic field intensity, E is the electric field intensity, B
is the magnetic flux density, J is the electric current density, M is the magnetic current density, and ρ0
is the charge density in free space. In addition, the constitutive equations relating the field quantities
are given by:
D = εE = ε0εrE (2.2a)
B = µH = µ0µrH (2.2b)
J = εE = ε0εrE (2.2c)
M = εE = ε0εrE (2.2d)
where ε is the electric permittivity, ε0 is the free space permittivity, εr is the relative permittivity, µ is
the magnetic permeability, µ0 is the free space permeability, µr is the relative permeability. The electric
and magnetic currents are defined as:
J = Jsource + σE (2.3a)
M = Msource + σ∗H (2.3b)
where σ and σ∗ are the electric and magnetic conductivity, respectively, and Jsource and Msource are
the external sources of J and M.
2.2.2 Central Finite-difference
The finite-difference time-domain method solves for the electric and magnetic fields in time and space by
approximating the partial derivatives in Faraday’s law (2.1a) and Ampere’s law (2.1b) with central finite
Chapter 2. Background 13
differences. The central finite difference scheme discretizes a continuous function and approximates its
derivative at each discrete point by the quantities at the two adjacent points. For instance, applying
Taylor expansion to the electric field E(x, tn) at a fixed time tn about the points x0 +∆x and x0 −∆x:
E(x0+∆x
2)
∣
∣
∣
∣
tn
= E(x0)
∣
∣
∣
∣
tn
+∆x
2
∂
∂xE(x)
∣
∣
∣
∣
x0,tn
+1
2
(
∆x
2
)2∂2
∂x2E(x)
∣
∣
∣
∣
x0,tn
+1
6
(
∆x
2
)3∂3
∂x3E(x)
∣
∣
∣
∣
x0,tn
+ ...
(2.4)
E(x0−∆x
2)
∣
∣
∣
∣
tn
= E(x0)
∣
∣
∣
∣
tn
+∆x
2
∂
∂xE(x)
∣
∣
∣
∣
x0,tn
− 1
2
(
∆x
2
)2∂2
∂x2E(x)
∣
∣
∣
∣
x0,tn
+1
6
(
∆x
2
)3∂3
∂x3E(x)
∣
∣
∣
∣
x0,tn
+ ...
(2.5)
Subtracting the two equations and isolating for ∂∂xE(x):
∂
∂xE(x)
∣
∣
∣
∣
x0,tn
=E(x0 +
∆x2 )− E(x0 − ∆x
2 )
∆x+O
[
(∆x)2]
(2.6)
≈ E(x0 +∆x2 )− E(x0 − ∆x
2 )
∆x(2.7)
The higher order terms O[
(∆x)2]
is a function of the square of the discretization size ∆x. Truncating all
higher order terms leads to a discretization error that is of second order, i.e., reducing the discretization
size ∆x by two reduces the error by four.
2.2.3 The Yee’s Algorithm
In the Yee’s algorithm, the time and spatial derivatives in the Maxwell’s curl equations are approximated
by the central finite-difference scheme, and an explicit form of the field quantities is derived in terms of
the fields in the previous time steps [31]. The discretized field quantities are arranged in a spatial grid of
mesh cells in a staggered configuration. Each mesh cell is referred to as a Yee’s cell, as shown in Fig.2.1.
By staggering the electric and magnetic fields by half cell size, each electric field component is surrounded
by four circulating magnetic fields and vice versa for the magnetic field components. In this manner, the
flux of one field is associated with the circulation of the other, which we can use to approximate both
the integral form and the differential form of Faraday’s law and Ampere’s law. In addition, the Gauss’s
laws for the magnetic and the electric fields are also satisfied, as can be demonstrated by evaluating the
total flux of the electric or magnetic field over the closed surface formed by a single Yee’s cell. In time,
the electric fields and magnetic fields are also staggered by a half step size.
The derivation of the explicit forms of the fields is applied to Ex, as an example. Starting with the
Chapter 2. Background 14
Ez
i+½, j, kEx
i, j, k+½ Hyi+½, j+½, k+½
Ex
Ey
Ez
Ez
Ey i, j+½, k+1 Ey
Ex
Hz
Hxi+1,j+1, k+½
i+1, j+1, k+½
i+1, j, k+½
i+½, j, k+1
i+1, j+½, k+1
i+½, j+1, k+1
i+½, j+½, k+1
i+1, j+½, k
∆x
∆y
∆z
x
yz
Figure 2.1: Configuration of the electric and magnetic fields in the Yee’s cell. The indices i, j, and kindicate the Yee’s cell’s location on the grid in the x, y, and z direction, respectively.
x−component of Ampere’s law:
∂Ex
∂t=
1
ε
[
∂Hz
∂y− ∂Hy
∂z− σEx − Jsource
]
(2.8)
Applying the central difference approximation to the time and spatial derivatives in (2.8) to obtain:
Ex
∣
∣
∣
n+1
i+ 12,j,k
− Ex
∣
∣
∣
n
i+ 12,j,k
∆t
=1
ε
(Hz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y−Hy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z−σi+ 1
2,j,kEx
∣
∣
∣
n+ 12
i+ 12,j,k
−Jsource
∣
∣
∣
n+ 12
i+ 12,j,k
)
(2.9)
where n is the time step index and i, j, and k indicate the Yee’s cell position in the x, y, and z direction,
respectively. The field E at the (n + 12 )-th time step is given by a semi-implicit approximation of the
form:
Ex
∣
∣
∣
n+ 12
i+ 12,j,k
≈Ex
∣
∣
∣
n+1
i+ 12,j,k
+ Ex
∣
∣
∣
n
i+ 12,j,k
2(2.10)
Assuming there is no external current source Jsource, and substituting (2.10) into (2.9), Ex at time
(n+ 1)∆t can be expressed as a function of the electric field in the previous time step and the adjacent
Chapter 2. Background 15
magnetic fields in the previous half time step by:
Ex
∣
∣
∣
n+1
i+ 12,j,k
=
1−σi+ 1
2,j,k∆t
2εi+ 12,j,k
1 +σi+ 1
2,j,k∆t
2εi+ 12,j,k
Ex
∣
∣
∣
n
i+ 12,j,k
+
∆t
εi+ 12,j,k
1 +σi+ 1
2,j,k∆t
2εi+ 12,j,k
Hz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y−
Hy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
(2.11)
The Ey and Ez components are similarly derived to be:
Ey
∣
∣
∣
n+1
i,j+ 12,k
=
1−σi,j+ 1
2,k∆t
2εi,j+ 12,k
1 +σi,j+ 1
2,k∆t
2εi,j+ 12,k
Ey
∣
∣
∣
n
i,j+ 12,k
+
∆t
εi,j+ 12,k
1 +σi,j+ 1
2,k∆t
2εi,j+ 12,k
Hx
∣
∣
∣
n+ 12
i,j+ 12,k+ 1
2
−Hx
∣
∣
∣
n+ 12
i,j+ 12,k− 1
2
∆z−
Hz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
∣
n+ 12
i− 12,j+ 1
2,k
∆x
(2.12)
Ez
∣
∣
∣
n+1
i,j,k+ 12
=
1−σi,j,k+ 1
2∆t
2εi,j,k+ 12
1 +σi,j,k+ 1
2∆t
2εi,j,k+ 12
Ez
∣
∣
∣
n
i,j,k+ 12
+
∆t
εi,j,k+ 12
1 +σi,j,k+ 1
2∆t
2εi,j,k+ 12
Hx
∣
∣
∣
n+ 12
i,j+ 12,k+ 1
2
−Hx
∣
∣
∣
n+ 12
i,j− 12,k+ 1
2
∆y−
Hy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
∣
n+ 12
i− 12,j,k+ 1
2
∆x
(2.13)
Likewise, by applying central difference scheme to Faraday’s law, the update equations for the magnetic
Chapter 2. Background 16
field H are given by:
Hx
∣
∣
∣
n+ 12
i,j+ 12,k+ 1
2
=
1−σ∗
i,j+ 12,k+ 1
2
∆t
2µi,j+ 12,k+ 1
2
1 +σ∗
i,j+ 12,k+ 1
2
∆t
2µi,j+ 12,k+ 1
2
Hx
∣
∣
∣
n− 12
i,j+ 12,k+ 1
2
+
∆t
µi,j+ 12,k+ 1
2
1 +σ∗
i,j+ 12,k+ 1
2
∆t
2µi,j+ 12,k+ 1
2
Ey
∣
∣
∣
n
i,j+ 12,k+1
− Ey
∣
∣
∣
n
i,j+ 12,k
∆z−
Ez
∣
∣
∣
n
i,j+1,k+ 12
− Ez
∣
∣
∣
n
i,j,k+ 12
∆y
(2.14)
Hy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
=
1−σ∗
i+ 12,j,k+ 1
2
∆t
2µi+ 12,j,k+ 1
2
1 +σ∗
i+ 12,j,k+ 1
2
∆t
2µi+ 12,j,k+ 1
2
Hy
∣
∣
∣
n− 12
i+ 12,j,k+ 1
2
+
∆t
µi+ 12,j,k+ 1
2
1 +σ∗
i+ 12,j,k+ 1
2
∆t
2µi+ 12,j,k+ 1
2
Ez
∣
∣
∣
n
i+1,j,k+ 12
− Ez
∣
∣
∣
n
i,j,k+ 12
∆x−
Ex
∣
∣
∣
n
i+ 12,j,k+1
− Ex
∣
∣
∣
n
i+ 12,j,k
∆z
(2.15)
Hz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k
=
1−σ∗
i+ 12,j+ 1
2,k∆t
2µi+ 12,j+ 1
2,k
1 +σ∗
i+ 12,j+ 1
2,k∆t
2µi+ 12,j+ 1
2,k
Hz
∣
∣
∣
n− 12
i+ 12,j+ 1
2,k
+
∆t
µi+ 12,j+ 1
2,k
1 +σ∗
i+ 12,j+ 1
2,k∆t
2µi+ 12,j+ 1
2,k
Ex
∣
∣
∣
n
i+ 12,j+1,k
− Ex
∣
∣
∣
n
i+ 12,j,k
∆y−
Ey
∣
∣
∣
n
i+1,j+ 12,k− Ey
∣
∣
∣
n
i,j+ 12,k
∆x
(2.16)
Within each Yee’s cell the material parameters σ, ε, and µ are assumed to be homogeneous. At each
field node position, the value of the material parameters can be found from an arithmetic averaging of
the material parameters in the Yee’s cells bordering the field node.
The set of equations (2.11) to (2.16) form the FDTD update equations. The fields at the current
time step are evaluated from the fields from the previous time steps in a leap-frog manner, shown in Fig.
(2.2.3): the magnetic fields at the (n + 12 )-th time step are updated from the electric fields at n-time
step, then the electric fields at the (n + 1)-th time step are updated by the magnetic fields (n + 12 )-th
time step. This is repeated until the fields settle into a steady state.
Chapter 2. Background 17
t
x
n∆t
(n+1)∆t
(n+½)∆t
(i-1)∆x i∆x(i-½)∆x (i+½)∆x (i+1)∆x
E
H
Figure 2.2: The update procedure of the electric and magnetic fields staggered in space and time by leapfrogging. The arrows pointing from one field to another indicate the directions the updates proceed.
2.2.4 Numerical Dispersion and Stability
The numerical dispersion relationship of a 3-D structure is given by [31]:
[
1
v∆tsin
(
ω∆t
2
)]2
=
[
1
∆xsin
(
kx∆x
2
)]2
+
[
1
∆ysin
(
ky∆y
2
)]2
+
[
1
∆zsin
(
kz∆z
2
)]2
(2.17)
where kx, ky, kz denote the components of numerical wave number.
To ensure numerical stability, the stable time step ∆t must satisfy [31]:
∆t ≤ 1
c√
1∆x2 + 1
∆y2 + 1∆z2
(2.18)
2.2.5 Perfectly Matched Layer
Many common applications of the FDTD require simulation within an unbounded environment. To
accomplish this with a finite domain of reasonable size, outward propagating waves must be absorbed
at the boundaries with minimal reflections back into the domain. To this end, a variety of methods
have been developed, of which the perfectly-matched layers (PML) is the most robust and effective
boundary condition currently [32]. The PML attenuates outward propagating waves with a layer of
lossy absorber appended to the domain boundary. The absorbers are matched to the wave impedance of
the domain such that impinging waves of all frequency, polarization, and incidence angles are effectively
admitted. Berenger first presented a split-field formulation of the absorber, where each polarization
of the fields in the PML is separated into two orthogonal sub-components and matching is achieved
by properly adjusting the conductivity parameters assigned to each sub-component [33]. It was later
Chapter 2. Background 18
shown that the split field method can be equivalently represented by mapping the Maxwell’s equations
into a complex stretched-coordinate [34]. This form provides even further improvement over Berenger’s
original formulation with the addition of the complex frequency shifted coefficients, where late-time
frequency errors and evanescent waves can be mitigated [35]. An approach to efficiently form the
convolution terms present in the Maxwell’s equations in the complex stretched coordinate, using the
recursive convolution method, was used to develop a numerically efficient implementation of the PML,
known as the convolutional PML (CPML) [36].
In this section, the formulation of the PML based on the recursive convolution method, as outlined
in [31], is reviewed. Beginning with Maxwell’s curl equations in the complex stretched-coordinate form
given by :
∂D
∂t=(
sy ∗∂
∂yHz − sz ∗
∂
∂zHy
)
x+(
sz ∗∂
∂zHx − sx ∗ ∂
∂xHz
)
y +(
sx ∗ ∂
∂xHy − sy ∗
∂
∂yHx
)
z (2.19)
∂B
∂t=(
sy ∗∂
∂yEz − sz ∗
∂
∂zEy
)
x+(
sz ∗∂
∂zEx − sx ∗ ∂
∂xEz
)
y +(
sx ∗ ∂
∂xEy − sy ∗
∂
∂yEx
)
z (2.20)
where s is the complex frequency shifted (CFS) tensor coefficient. In the frequency domain it takes on
the form:
sw = κw +σw
aw + jωε0(2.21)
where σw is the PML conductivity profile in the w-direction, and along with the parameters κw and
aw, their values are used to shape the performance of the PML. While a large σPML is desired in order
to maximize the wave attenuation within the PML, it can also lead to a larger spurious reflection due
to increased conductivity discontinuity at the PML interface. This can be reduced by spatially grading
σw from zero at the PML interface to a larger value along the direction normal to the interface. As an
example, a polynomial-graded profile can be used with the form:
σx(x) =(x
d
)m
σx,max (2.22)
For a fixed number of PML mesh cells, the discretization error can still be incurred from the grading of
σx(x) within the PML. The order of the polynomial profile m determines how rapidly σw rises within
the PML. A rapidly rising σw can lead to large discretization errors as a result of an increase in the σpml
difference between adjacent Yee’s cells. Typically, m is set to 3 ≤ m ≤ 4. An optimal value of σx,max
which minimizes spurious reflection at the PML boundary is found by analyzing the reflection factor R
from the PML interface. For a PML layer of thickness d and a σ graded by a polynomial profile, R is
Chapter 2. Background 19
given by:
R(θ) = e−2ησx,maxd cos θ/(m+1) (2.23)
where θ is the angle of incidence and η is the wave impedance. σx,max is given by:
σx,max = − (m+ 1) ln[R(0)]
2ηd(2.24)
From numerical experiments, the optimal value of R(0) for a 10-cell PML can be found to be around
e−16, which gives the expression for σx,max in terms of a desired m:
σx,opt = − (m+ 1)(−16)
(2η)(10∆)=
0.8(m+ 1)
η∆(2.25)
The real part of the CFS coefficient κ can be interpreted as a direct scaling of the spatial coordinate
along the PML. This is to further reduce the discretization error caused by conductivity difference. The
spatial profile of κ is also set to a polynomial profile of the form:
κx(x) = 1 + (κx,max − 1)(x
d
)m
(2.26)
The value of parameter a is selected to ensure that the pole is shifted into the upper-half complex
plane to produce a causal and stable s and reduce spurious reflections in the low frequencies. However,
if a is too large then the attenuation of low-frequency wave propagating inside PML can be greatly
diminished. Therefore, the value of a is spatially scaled within the PML such that it attains a maximum
value at the PML interface and is gradually decreased to zero at the end of the PML. The spatial profile
a is expressed by:
ax(x) = ax,max
(d− x
d
)ma
(2.27)
Numerical experiments have demonstrated that the optimal values usually lie between 0 and 20 for κmax
and usually lie between 0 and 0.4 for ax,max [31].
The time-domain expression of the tensor coefficient sw , w = x, y, z, is defined as:
sw = F−1
1
κw + σw
aw+jωε0
, w = x, y, z
=δ(t)
κw+
σw
ε0κ2w
e−
σw
ε0κ2w+ aw
ε0
tu(t)
=δ(t)
κw+ χw(t) (2.28)
Chapter 2. Background 20
where η is the free-space wave impedance, ∆w is the mesh cell dimension in the w-direction, and εr,eff
and µr,eff are the electric permittivity and magnetic permeability, respectively. (2.19) and (2.20) are
rewritten as:
∂D
∂t=
(
1
κy
∂
∂yHz −
1
κz
∂
∂zHy + χy ∗
∂
∂yHz − χz ∗
∂
∂zHy
)
x
+
(
1
κz
∂
∂zHx − 1
κx
∂
∂xHz + χz ∗
∂
∂zHx − χx ∗ ∂
∂xHz
)
y
+
(
1
κx
∂
∂xHy −
1
κy
∂
∂yHx + χx ∗ ∂
∂xHy − χy ∗
∂
∂yHx
)
z (2.29)
−∂B
∂t=
(
1
κy
∂
∂yEz −
1
κz
∂
∂zEy + χy ∗
∂
∂yEz − χz ∗
∂
∂zEy
)
x
+
(
1
κz
∂
∂zEx − 1
κx
∂
∂xEz + χz ∗
∂
∂zEx − χx ∗ ∂
∂xEz
)
y
+
(
1
κx
∂
∂xEy −
1
κy
∂
∂yEx + χx ∗ ∂
∂xEy − χy ∗
∂
∂yEx
)
z (2.30)
The convolution terms in (2.29) and (2.30) are efficiently implemented using the recursive convolution
method. First, the convolution terms are discretized by a piecewise constant approximation:
ζw,v = χw(t) ∗∂
∂wEv(t)
∣
∣
∣
∣
t=n∆t
≈n−1∑
m=0
Zw(m)∂
∂wEv(n−m) (2.31)
where Zw(m) is defined by:
Zw(m) =
∫ (m+1)∆t
m∆t
χw(τ)dτ = − σw
ǫ0κ2w
∫ (m+1)∆t
m∆t
e−(
σwǫ0κw
+ awǫ0
)
τdτ = cwe−
(
σwǫ0κw
+ awǫ0
)
m∆t (2.32)
with cw defined by:
cw =σw
σwκw + κ2waw
[
e
(
σwε0κw
−awε0
)
∆t − 1
]
(2.33)
The discrete convolution in (2.31) is implemented recursively by:
ζw,v(n) = bwζw,v(n− 1) + cw∂
∂wEx(n) (2.34)
where bw is:
bw = e
(
σwε0κw
−awε0
)
∆t(2.35)
With the recursive discrete convolution, (2.29) and (2.30) can be discretized by finite-difference. For the
Chapter 2. Background 21
Ex component:
∂
∂t(ǫEx) + σEx =
(
1
κy
∂
∂yHz −
1
κz
∂
∂zHy + χy ∗
∂
∂yHz − χz ∗
∂
∂zHy
)
(2.36)
Applying finite-difference to (2.36)
ǫi+ 12,j,k
Ex
∣
∣
n+1
i+ 12,j,k
− Ex
∣
∣
n
i+ 12,j,k
∆t+ σi+ 1
2,j,k
Ex
∣
∣
n+1
i+ 12,j,k
− Ex
∣
∣
n
i+ 12,j,k
2
=Hz
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
n+ 12
i+ 12,j− 1
2,k
κy,j∆y−
Hy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
n+ 12
i+ 12,j,k− 1
2
κz,k∆z.
+ ζEx,y
∣
∣
n+ 12
i+ 12,j,k
− ζEx,z
∣
∣
n+ 12
i+ 12,j,k
(2.37)
Rearranging (2.37) for Ex
∣
∣
n+1
i+ 12,j,k
, the update equation of Ex within the PML is given by:
Ex
∣
∣
n+1
i+ 12,j,k
= C1
∣
∣
i+ 12,j,k
Ex
∣
∣
n
i+ 12,j,k
+ C2
∣
∣
i+ 12,j,k
Hz
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
n+ 12
i+ 12,j− 1
2,k
κy,j∆y−
Hy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
n+ 12
i+ 12,j,k− 1
2
κz,k∆z
+ ζEx,y
∣
∣
n+ 12
i+ 12,j,k
− ζEx,z
∣
∣
n+ 12
i+ 12,j,k
(2.38)
C1
∣
∣
i+ 12,j,k
=
1−σi+1
2,j,k
∆t
2ǫi+1
2,j,k
1 +σi+1
2,j,k
∆t
2ǫi+1
2,j,k
(2.39)
C2
∣
∣
i+ 12,j,k
=
∆tǫi+1
2,j,k
1 +σi+1
2,j,k
∆t
2ǫi+1
2,j,k
(2.40)
Applying finite-difference to (2.34), the discrete convolution terms ζ are updated at each step by:
ζEx,y
∣
∣
n+ 12
i+ 12,j,k
= by,jζEx,y
∣
∣
n− 12
i+ 12,j,k
+ cy,j
Hz
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
(2.41)
ζEx,z
∣
∣
n+ 12
i+ 12,j,k
= bz,kζEx,z
∣
∣
n− 12
i+ 12,j,k
+ cz,k
Hy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
(2.42)
Chapter 2. Background 22
The magnetic update equations are derived in a similar manner. For the Hx component:
Hx
∣
∣
n+ 12
i,j+ 12,k
= D1
∣
∣
i+ 12,j,k
Hx
∣
∣
n− 12
i,j+ 12,k
+D2
∣
∣
i+ 12,j,k
Ez
∣
∣
n
i,j+ 12,k+ 1
2
− Ez
∣
∣
n
i,j− 12,k+ 1
2
κy,j∆y−
Ey
∣
∣
n
i,j+ 12,k+ 1
2
− Ey
∣
∣
n
i,j+ 12,k− 1
2
κz,k∆z
+ ζHx,y
∣
∣
n
i,j+ 12,k+ 1
2
− ζHx,z
∣
∣
n
i,j+ 12,k+ 1
2
(2.43)
D1
∣
∣
i,j+ 12,k
=
1−σ∗
i,j+ 12,k∆t
2ǫi,j+ 12,k
1 +σ∗
i,j+ 12,k∆t
2ǫi,j+ 12,k
(2.44)
D2
∣
∣
i,j+ 12,k
=
∆tǫi,j+1
2,k
1 +σ∗
i+ 12,j,k
∆t
2ǫi+ 12,j,k
(2.45)
ζHx,y
∣
∣
n
i,j+ 12,k+ 1
2
= by,j+ 12ζHx,y
∣
∣
n−1
i,j+ 12,k+ 1
2
+ cy,j+ 12
Ez
∣
∣
n
i,j+1,k+ 12
− Ez
∣
∣
n
i,j,k+ 12
∆y
(2.46)
ζHx,z
∣
∣
n
i,j+ 12,k+ 1
2
= bz,k+ 12ζHx,z
∣
∣
n−1
i,j+ 12,k+ 1
2
+ cz,k+ 12
Ey
∣
∣
n
i,j+ 12,k+1
− Ey
∣
∣
n
i,j+1,k
∆z
(2.47)
Within the framework of a general FDTD algorithm, the implementation of the electric field update
procedure in (2.38) is separated into two stages. First, the electric fields are updated directly from the
magnetic fields, which is equivalent to applying the general FDTD update equations, with the κ scaling
factors applied to the corresponding mesh dimension. Finally, the electric fields are updated from the ζ
terms determined from (2.41) and (2.42). This is repeated for all other field components.
2.3 Quantifying Output Uncertainty by Uncertainty Propaga-
tion
This section reviews the Monte Carlo method and the polynomial chaos expansion method, as outlined
in [23, 11] for solving numerical models with uncertain parameters, such as a stochastic differential
equation of the form:
L(x, t, ξ;u(ξ)) = f(x, t; ξ) (2.48)
Chapter 2. Background 23
where u(ξ) is the output as a function of M input random variables ξ = ξ1, ξ2, ξ3, ..., ξM and f is
some forcing function. The problems considered in this thesis will contain only statistically-independent
random parameters.
2.3.1 Monte Carlo Method
The Monte Carlo method is a sampling-based method with a straightforward implementation. First, N
samples of the random parameters are generated from experiments or with a random number gen-
erator according to their probability distribution. For each sample of the random parameter ξi=
ξi1, ξi2, ξi3, ..., ξiM, i = 1, ..., N , the stochastic differential equation reduces to a deterministic system:
L(x, t, ξi;u(ξi)) = f(x, t; ξi) (2.49)
which can be solved using conventional numerical methods. This process is repeated to generate the N
samples of the output, which are used to estimate the statistical moments of the output variable.
εr, 1 X1
εr, 2
εr, 3
εr, N
Numerical Solver
Numerical Solver
Numerical Solver
Numerical Solver
X2
X3
XN
Figure 2.3: The Monte Carlo algorithm applied to quantifying uncertainty in some output of interest X,given a random parameter εr.
For example, let the output of interest Xi = u(ξi) and the Monte Carlo method is used to generate
the samples X1, X2, ..., XN . Then the sample mean of X1, X2, ..., XN , used to approximate the exact
value of the mean of X, is defined by:
µ[X] =1
N
N∑
i=1
Xi (2.50)
Similarly, its sample variance:
σ2[X] =1
N − 1
N∑
i=1
Xi − µ[X]2
(2.51)
Chapter 2. Background 24
The sample means are also sometimes referred to as Monte Carlo estimators within the Monte Carlo
method. The accuracy of the Monte Carlo estimators are specified by two mathematical theorems: The
law of large numbers and the central limit theorem [37].
The law of large numbers states that the limit of the Monte Carlo estimator approaches the population
mean.
limN→∞
1
N
N∑
i=1
Xi = E[X] =
∫
S
X(ξ)P (ξ)dξ (2.52)
The existence of this limit ensures that the estimate of the mean will give us an accurate estimate as
the number of Monte Carlo iterations becomes sufficiently large.
The central limit theorem allows us to quantify the uncertainty in the Monte Carlo estimator. For
an N -sampled Monte Carlo estimate µ[X] of a random variable with the exact mean E[X] and exact
standard deviation STD(X), the central limit theorem is stated as:
limN→∞
|µ[X]− E[X]|STD(X)/
√N
≤ λ
=1√2π
∫ λ
−λ
e−u2/2du (2.53)
This indicates that the asymptotic distribution of the estimated means from an N -iterated Monte Carlo
is normal-distributed, i.e., if M instances of the N -iterated Monte Carlo simulations are run, then the
resulting M estimators of the mean would form a sampling distribution which approaches a normal
distribution centered on the exact value of the mean, shown in Fig. (2.4). The standard deviation of
the normal distribution is commonly referred to as the standard error of the Monte Carlo estimate SE,
given by:
SE =STD(X)√
N(2.54)
For a single Monte Carlo simulation, its standard error is an indication of the range of values the estimate
will land in with a 68.3% probability. The likelihood of obtaining an accurate estimate is inversely
proportional to the square root of number of Monte Carlo iterations, e.g., improving the accuracy of
the estimate by a decimal point requires 100-times the number of Monte Carlo samples. Therefore,
the computational cost associated with obtaining an accurate estimate of the statistical moments from
Monte Carlo can be substantial.
Chapter 2. Background 25
µ[X] E[X] SESE
34.1% 34.1%
Figure 2.4: Probability distribution of an N -iterated Monte Carlo mean estimate µ[X]. The distributionis centered on the exact value of the mean of X, with a standard deviation defined by SE.
2.3.2 Generalized Polynomial Chaos Expansion
The polynomial chaos expansion seeks a solution of the stochastic system X(ξ) of the form:
X(ξ) =
P∑
m=0
amΨm(ξ) (2.55)
where Ψm(ξ) is the m−th order orthogonal polynomial basis function with the corresponding expansion
coefficient am. The number of terms is given by P +1. The polynomial basis functions Ψ(ξ) satisfy the
orthogonality relationship defined by:
〈Ψl(ξ)Ψm(ξ)〉 =∫
Ψl(ξ)Ψm(ξ)P (ξ)dξ = 〈Ψ2l (ξ)〉δlm (2.56)
where δlm is the Kronecker delta and P (ξ) is the joint probability density functions of the input random
variables. The expansion coefficients am are evaluated by projecting X(ξ) with the corresponding order
polynomial basis function. For example, the l-th expansion coefficient al is given by:
al = 〈X(ξ),Ψl(ξ)〉 =1
〈Ψ2l (ξ)〉
∫
X(ξ)Ψl(ξ)P (ξ)dξ (2.57)
Chapter 2. Background 26
Given that the uncertainties are cast into the random polynomial basis functions, the orthogonality
relationship allows the polynomial chaos coefficients to be evaluated from a set of P + 1 deterministic
equations given by (2.57). Substituting the (2.55) into the stochastic system (2.48):
L(x, t, ξ;P∑
m=0
amΨm(ξ)) = f(x, t; ξ) (2.58)
Each coefficient is determined by projection:
〈L(x, t, ξ;P∑
m=0
amΨm(ξ)),Ψl(ξ)〉 = 〈f(x, t; ξ),Ψl(ξ)〉 (2.59)
Therefore, the stochastic system is reduced to P+1 coupled deterministic systems used to evaluate P+1
PCE coefficients. This is usually referred to as a spectral Galerkin or an “intrusive” approach, as the
implementation requires the reformulation of the numerical solver for the stochastic system. Another
common approach is the “non-intrusive” where the output expansion coefficient is determined using
numerical quadrature rule:
al =
∫
X(ξ)Ψl(ξ)ρ(ξ)dξ ≈Q∑
q=1
X(ξq)Ψl(ξq)ρ(ξq)wq (2.60)
where ξq and wq are the quadrature points and weights, respectively. Numerical solvers are used to obtain
X(ξq), q = 1, ..., Q, by running Q deterministic simulations. This approach has the implementation
advantage compared to the intrusive method, as no modification of existing numerical solvers for solving
the equivalent deterministic systems is necessary. Non-intrusive approach is also easy to parallelize, as
each simulation can be run independently.
However, the accuracy of the non-intrusive approach is affected by the error introduced by the
discretizing of the output function with quadrature nodes, which is classified as an aliasing error. On
the other hand, the errors in the intrusive approach are minimized as the residue of the stochastic
equations is orthogonal to the linear space spanned by orthogonal polynomial basis functions. For large
multi-dimensional random parameter space, the aliasing error from numerical quadrature may be much
larger than the error accumulated from the intrusive approach and as a result the intrusive approach
may require less number of equations than the non-intrusive approach to achieve the same accuracy.
Polynomial basis functions are selected from a class of orthogonal polynomials classified as the Askey-
scheme, shown in Table (2.1). This has been shown numerically to yield an optimal convergence rate
of the polynomial chaos expansion with respect to the polynomial chaos order [26]. Unfortunately, the
Chapter 2. Background 27
Orthogonal Polynomial Probability Distribution Function Support
Gaussian Hermite (-∞, ∞)Uniform Legendre [a, b]Gamma Laguerre [0, ∞)Poisson Charlier 0, 1, 2, ...Binomial Krawtchouk 0, 1, 2, ..., k
Negative Binomial Meixner 0, 1, 2, ...Hypergeometric Hahn 0, 1, 2, ..., k
Table 2.1: Askey-scheme polynomials and the probability distribution function corresponding to theirweight functions.
Ψ(ξ
)
ξ
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
l = 0l = 1l = 2l = 3
Figure 2.5: The first four order of Legendre polynomials.
probability distribution of the output random variable is generally not available before the stochastic
system is solved. Instead, the polynomial chaos basis functions are selected based on the probability
distributions of the random parameters. The optimal convergence is not guaranteed, however, unless the
output is a linear function of the input [11]. In general, the approach of selecting the polynomial basis
functions based on the input probability distribution has produced excellent convergence with respect
to PCE order.
For problems with N statistically-independent random parameters, the basis functions are formed
Chapter 2. Background 28
from the products of the uni-variate basis functions of each individual random parameter:
Ψ(ξ) =
N∏
i=1
Ψ(ξi) (2.61)
The basis functions are grouped together based on their total-order index D, which denotes the sum
of the orders of their constituent univariate basis. Given a PCE up to a total-order D, the number of
polynomial chaos expansion terms P + 1 is found from:
P + 1 =
D∑
s=1
1
s!
s−1∏
r=0
(N + r) =(N +D)!
N !D!(2.62)
For example, given the random Hermite polynomials defined in Table 2.2, the basis functions for a PCE
up to a total order D = 3 with 2 random parameters are shown in Table 2.3.
Basis Index j Basis Function Hermite Polynomial
0 Ψ0(ξ) 11 Ψ1(ξ) ξ2 Ψ2(ξ) ξ2 − 13 Ψ3(ξ) ξ3 − 3ξ4 Ψ4(ξ) ξ4 − 6ξ2 + 35 Ψ5(ξ) ξ5 − 10ξ3 + 15ξ
Table 2.2: Hermite polynomials up to the first five order.
Total Order D PCE basis ordered by single index j Uni-variate Basis Product Hermite Basis Product
0 Ψ1(ξ1, ξ2) Ψ0(ξ1)Ψ0(ξ2) 11 Ψ2(ξ1, ξ2) Ψ1(ξ1)Ψ0(ξ2) ξ1
Ψ3(ξ1, ξ2) Ψ0(ξ1)Ψ1(ξ2) ξ22 Ψ4(ξ1, ξ2) Ψ1(ξ1)Ψ1(ξ2) ξ1ξ2
Ψ5(ξ1, ξ2) Ψ2(ξ1)Ψ0(ξ2) ξ21 − 1Ψ6(ξ1, ξ2) Ψ0(ξ1)Ψ2(ξ2) ξ22 − 1
3 Ψ7(ξ1, ξ2) Ψ2(ξ1)Ψ1(ξ2) ξ22ξ1 − ξ1Ψ8(ξ1, ξ2) Ψ1(ξ1)Ψ2(ξ2) ξ21ξ2 − ξ2Ψ9(ξ1, ξ2) Ψ3(ξ1)Ψ0(ξ2) ξ31 − 3ξ1Ψ10(ξ1, ξ2) Ψ0(ξ1)Ψ3(ξ2) ξ32 − 3ξ2
Table 2.3: Hermite polynomial basis for two random parameters up to a total order D = 3.
The statistical moments, e.g. the mean and the variance, can be directly evaluated from the PCE
Chapter 2. Background 29
[23]. From the definition of the mean:
E[X] =
∫
X(ξ)P (ξ)dξ
=
∫ P∑
j=0
ajΨj(ξ)P (ξ)dξ
= a0 (2.63)
V AR(X) = E[X2]− E2[X]
=
∫
(
P∑
i=0
aiΨi(ξ)
)(
P∑
j=0
ajΨj(ξ)
)
P (ξ)dξ − a20
=
P∑
i=0
P∑
j=0
aiaj
∫
Ψi(ξ)Ψj(ξ)P (ξ)dξ − a20
=
P∑
i=0
P∑
j=0
aiaj〈Ψi(ξ),Ψj(ξ)〉 − a20
=
P∑
i=1
a2i 〈Ψi(ξ))Ψi(ξ))〉 (2.64)
Since the polynomial chaos expansion gives an analytical form of the output as a function of the random
parameters, a statistical ensemble of the output can be found by evaluating its PCE in a Monte Carlo
approach.
E1
E2
EN
ƒ( E1)
ƒ( E2)
ƒ( EN)
Y1
Y2
YN
Figure 2.6: Evaluating uncertainties in the field-derived output of interest X, using the PCE as asurrogate model.
Chapter 2. Background 30
2.4 Intrusive Polynomial Chaos Expansion-based Finite-difference
Time-Domain Method
In this section, the PCE-FDTD methods presented in [20] and [22] are reviewed. The modelling of
geometric and material uncertainties within the FDTD simulation domain is given and the derivation of
the FDTD update equations for the polynomial chaos expansion coefficients is outlined. As a numerical
example, the PCE-FDTD is applied to a cascaded stub filter with random variations in its stub lengths.
In this section and the rest of the thesis, PCE-FDTD may be simply shortened to PCE method.
2.4.1 PCE-FDTD Formulation for Modelling Material Uncertainties
Uncertainties in material parameters, such as electric permittivity, electric conductivity, and magnetic
permeability are represented as functions of the random variables ξ, which may uniformly or beta dis-
tributed, for example. The Maxwell’s curl equations become stochastic equations where, as an example,
the Ampere’s law for the Ex component is written as:
∂Ex(ξ)
∂t=
1
ε(ξ)
[
∂Hz(ξ)
∂y− ∂Hy(ξ)
∂z− σ(ξ)Ex(ξ)
]
(2.65)
In the same manner the deterministic FDTD algorithm is discretized, the finite-difference scheme is
applied to (2.65) to obtain the update equation for Ex(ξ):
Ex(ξ)∣
∣
∣
n+1
i+ 12,j,k
= C1(ξ)∣
∣
∣
i+ 12,j,k
Ex(ξ)∣
∣
∣
n
i+ 12,j,k
+ C2(ξ)∣
∣
∣
i+ 12,j,k
Hz(ξ)∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz(ξ)
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
−Hy(ξ)
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy(ξ)∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
(2.66)
C1
∣
∣
∣
i+ 12,j,k
(ξ) =
1−σi+ 1
2,j,k∆t
2ǫi+ 12,j,k(ξ)
1 +σi+ 1
2,j,k(ξ)∆t
2ǫi+ 12,j,k(ξ)
(2.67)
Chapter 2. Background 31
C2
∣
∣
∣
i+ 12,j,k
(ξ) =
∆t
ǫi+ 12,j,k(ξ)
1 +σi+ 1
2,j,k(ξ)∆t
2ǫi+ 12,j,k(ξ)
(2.68)
where the FDTD update coefficients C1(ξ) and C2(ξ) contain the uncertain electric permittivities and
conductivities. The random electric and magnetic field quantities at each Yee’s cell and time step are
expanded in terms of orthogonal polynomial basis functions Ψl(ξ)
Ex(ξ) =
P∑
l=0
elxΨl(ξ) (2.69)
Hx(ξ) =
P∑
l=0
hlxΨl(ξ) (2.70)
The update equation with the PCE of the fields becomes:
P∑
m=0
emx
∣
∣
∣
n+1
i+ 12,j,k
Ψm(ξ) = C1
∣
∣
∣
i+ 12,j,k
(ξ)
P∑
m=0
emx
∣
∣
∣
n
i+ 12,j,k
Ψm(ξ)
+ C2
∣
∣
∣
i+ 12,j,k
(ξ)
P∑
m=0
hmz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
−hmy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hmy
∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
Ψm(ξ) (2.71)
The l-th coefficient el is evaluated by projecting Ψl(ξ) on both sides of (2.71) :
elx∣
∣
n+1
i+ 12,j,k
=1
〈Ψ2l (ξ)〉
P∑
m=0
emx∣
∣
n
i+ 12,j,k
〈C1(ξ)Ψm(ξ),Ψl(ξ)〉
+1
〈Ψ2l (ξ)〉
P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
−hmy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hmy
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
〈C2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.72)
Chapter 2. Background 32
The update equations for the coefficients ey and ez are:
ely∣
∣
n+1
i+ 12,j,k
=1
〈Ψ2l (ξ)〉
P∑
m=0
emy∣
∣
n
i+ 12,j,k
〈C1(ξ)Ψm(ξ),Ψl(ξ)〉
+1
〈Ψ2l (ξ)〉
P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
−hmy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hmy
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
〈C2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.73)
elz∣
∣
n+1
i,j,k+ 12
=1
〈Ψ2l (ξ)〉
P∑
m=0
emz∣
∣
n
i,j,k+ 12
〈C1(ξ)Ψm(ξ),Ψl(ξ)〉
+1
〈Ψ2l (ξ)〉
P∑
m=0
hmx
∣
∣
n+ 12
i,j+ 12,k+ 1
2
− hmx
∣
∣
n+ 12
i,j− 12,k+ 1
2
∆y
−hmy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hmy
∣
∣
n+ 12
i− 12,j,k+ 1
2
∆x
〈C2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.74)
Likewise, the update equations of the PCE coefficients of the magnetic fields are given by:
hlx
∣
∣
n+ 12
i,j+ 12,k+ 1
2
=1
〈Ψ2l (ξ)〉
P∑
m=0
hmx
∣
∣
n− 12
i,j+ 12,k+ 1
2
+1
〈Ψ2l (ξ)〉
P∑
m=0
emy∣
∣
n
i,j+ 12,k+1
− emy∣
∣
n
i,j+ 12,k
∆z
−emz∣
∣
n
i,j+1,k+ 12
− emz∣
∣
n
i,j,k+ 12
∆y
〈D2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.75)
hly
∣
∣
n+ 12
i+ 12,j,k+ 1
2
=1
〈Ψ2l (ξ)〉
P∑
m=0
hmy
∣
∣
n− 12
i+ 12,j,k+ 1
2
+1
〈Ψ2l (ξ)〉
P∑
m=0
emz∣
∣
n
i+1,j,k+ 12
− emz∣
∣
n
i,j,k+ 12
∆x
−emx∣
∣
n
i+ 12,j,k+1
− emx∣
∣
n
i+ 12,j,k
∆z
〈D2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.76)
Chapter 2. Background 33
hlz
∣
∣
n+ 12
i+ 12,j+ 1
2,k
=1
〈Ψ2l (ξ)〉
P∑
m=0
hmz
∣
∣
n− 12
i+ 12,j+ 1
2,k
+1
〈Ψ2l (ξ)〉
P∑
m=0
emx∣
∣
n
i+ 12,j+1,k
− emx∣
∣
n
i+ 12,j,k
∆y
−emy∣
∣
n
i+1,j+ 12,k− emy
∣
∣
n
i,j+ 12,k
∆x
〈D2(ξ)Ψm(ξ),Ψl(ξ)〉 (2.77)
and D2 is defined as:
D2
∣
∣
∣
i,j+ 12,k+ 1
2
(ξ) =∆t
µi,j+ 12,k+ 1
2(ξ)
(2.78)
The inner product terms 〈Ψ2l (ξ)〉, 〈C1(ξ)Ψm(ξ),Ψl(ξ)〉, 〈C2(ξ)Ψm(ξ),Ψl(ξ)〉 and 〈D2(ξ)Ψm(ξ),Ψl(ξ)〉
are computed prior to FDTD time stepping. If the field node is surrounded by two mesh cells with
no randomness in permittivity and two mesh cells with permittivities following independent random
variable ξ1 and ξ2, as shown in Fig. 2.7, and an arithmetic averaging of the permittivities is used, the
inner product integral is given by:
〈C2(ξ1, ξ2)Ψm(ξ1, ξ2),Ψl(ξ1, ξ2)〉 =∫
Ω
2∆t
2ε0(2 + εr(ξ1)εr(ξ2))/4 + σ∆t
Ψm(ξ1, ξ2)Ψl(ξ1, ξ2)ρ(ξ1, ξ2)dξ1dξ2 (2.79)
E
ε(ξ1) ε(ξ2)
ε(ξ) ε(ξ)
Figure 2.7: An electric field node surrounded by four Yee’s cells with statistically-independent permit-tivity distributions.
FDTD time-stepping is used to evaluate the PCE coefficients at each time step and mesh cell, which
are then used to reconstruct the field random processes. In regions of domain where uncertain parameters
are located, updating the PCE coefficients requires the evaluation of a projection in the manner of (2.72).
While the uncertainties in the material parameters may be limited to a particular region, uncertainties
Chapter 2. Background 34
in the fields are assumed over all regions that can be reached by fields propagating from the sources
of parameter uncertainties. In regions where no uncertain parameters exist, the orthogonal projection
decouples the update equations and each coefficient is updated independently.
Given the uncertainty is simulated for a range of electric permittivity values, the mesh cell size
must account for the maximum electrical permittivity and magnetic permeability in order to ensure the
dispersion error is minimized. For a maximum excitation frequency fmax, and a number of mesh cells
per wavelength Nλ, the mesh size ∆ is given by:
∆ =λmin
Nλ=
c
Nλfmax√µr,maxǫr,max
(2.80)
The time step size ∆t which satisfies the FDTD stability condition is given by:
∆t ≤ 1
c√
1∆x2 + 1
∆y2 + 1∆z2
(2.81)
where c is the speed of light, and ∆x, ∆y, and ∆z are the mesh cell dimensions in the x, y, z direction,
respectively.
2.4.2 PCE-FDTD Update Equations for Geometric Uncertainties
Modelling uncertainties in the geometric features in FDTD requires the mesh dimensions to vary arbi-
trarily within a local region of the mesh. A method based on mesh distortion was proposed to directly
model geometric uncertainties [22]. An approach was first introduced where distortion was directly ap-
plied to a rectilinear mesh. A major drawback of this is that the mesh distortion cannot be contained
in the directions orthogonal to the direction of varying mesh dimension. Therefore, it is not viable for
modeling statistically-independent uncertainties aligned in the same direction. A more versatile method
based on curvilinear mesh distortion was therefore introduced in the same reference which is capable of
localizing the distortion in every dimension.
In simple problems without aligned geometric uncertainties, the rectilinear mesh distortion is often
sufficient. In this section, the rectilinear mesh distortion is outlined by demonstrating an example of
microstrip stub filters with stub length uncertainty.
Single-stub Microstrip Filter with Stub Length Uncertainties
Consider the single stub filter, shown in Fig. (2.8), with a uniformly distributed stub length lstub(ξ)
= 3.7 ± 0.5 mm. The variation in the stub length is distributed over a group of mesh cells such that
Chapter 2. Background 35
2.5 mm
lstub (ξ1)
3.7 mmPort 1
1.3 mm
0.75 mm
εr = 2.2
Port 2
1.3 mm
PML
Figure 2.8: The planar geometry of single-stub microstrip filter with stub length uncertainties.
only small changes are induced in each mesh cell. This effectively transforms the FDTD domain into
a nonuniform mesh with local cell distortion in region with geometric uncertainties. The local mesh
distortion cannot be arbitrarily large without introducing significant error. Therefore, we need to find
an equivalent uncertainty model for the dimension of the mesh cells, e.g. ∆, of the form:
∆(ξ) = ∆nominal ±∆var (2.82)
and a number of mesh cells n∆varfor which the stub variations are distributed over, such that ∆var is
small.
Consider the distorted mesh which must be compressed to model the maximum extent of the stub
length variation along with the nominal mesh, shown in Fig. (2.9). The maximum stub length variation
can be written as:
∆lstub,max = n∆var(∆nominal −∆min) (2.83)
An expression for n∆varcan then be found for :
n∆var=
lstub,max
(∆nominal −∆min)=
lstub,max
∆nominal
1− ∆min
∆nominal
(2.84)
The number of mesh cells in the compression region should be selected to ensure the maximum com-
pression of the mesh cell, i.e the ratio ∆min
∆nominal, is close to 1 to reduce errors associated with the change
Chapter 2. Background 36
of mesh dimensions at the boundary between distorted mesh region and nominal mesh region. As a rule
of thumb, ∆(ξ) should be selected such that 0.5∆nominal ≤ ∆(ξ) ≤ 2∆nominal to avoid large errors [31].
lstub,minimum
∆lstub,max
lstub,nominaln∆var∆min
Nominal Mesh Distorted Mesh
∆nominal ∆min
Figure 2.9: Modelling stub length variations by varying the mesh cell dimensions.
With the equivalent distribution of the mesh cell size the PCE-FDTD can be formulated to account
for geometric uncertainties. For brevity, only a single-variate case where variations in the y-direction is
considered here. The FDTD update equation for Ex, as an example, becomes:
Ex(ξ)∣
∣
∣
n+1
i+ 12,j,k
= C1
∣
∣
∣
i+ 12,j,k
Ex(ξ)∣
∣
∣
n
i+ 12,j,k
+ C2
∣
∣
∣
i+ 12,j,k
Hz(ξ)∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz(ξ)
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y(ξ)
−Hy(ξ)
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy(ξ)∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
(2.85)
Expanding the field quantities in terms of (ξ):
Ex(ξ) =P∑
l=0
elxΨl(ξ) (2.86)
Hx(ξ) =P∑
l=0
hlxΨl(ξ) (2.87)
Chapter 2. Background 37
and substitute into the FDTD update equations with ∆y(ξ). :
P∑
m=0
emx
∣
∣
∣
n+1
i+ 12,j,k
Ψm(ξ) = C1
∣
∣
∣
i+ 12,j,k
P∑
m=0
emx
∣
∣
∣
n
i+ 12,j,k
Ψm(ξ)
+ C2
∣
∣
∣
i+ 12,j,k
P∑
m=0
hmz
∣
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y(ξ)
−hmy
∣
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hmy
∣
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
Ψm(ξ) (2.88)
The l-th coefficient el is evaluated by projecting Ψl(ξ) on both sides of (2.88):
elx∣
∣
n+1
i+ 12,j,k
= C1
∣
∣
∣
i+ 12,j,k
elx∣
∣
n
i+ 12,j,k
+1
〈Ψ2l (ξ)〉
C2
∣
∣
∣
i+ 12,j,k
P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
⟨ 1
∆y(ξ)Ψm(ξ),Ψl(ξ)
⟩
−hly
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hly
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z(2.89)
Similarly for the Hx:
hlx
∣
∣
n+ 12
i,j+ 12,k+ 1
2
=1
〈Ψ2l (ξ)〉
P∑
m=0
hmx
∣
∣
n− 12
i,j+ 12,k+ 1
2
+1
〈Ψ2l (ξ)〉
D2
∣
∣
∣
i,j+ 12,k+ 1
2
P∑
m=0
emz∣
∣
n
i,j+1,k+ 12
− emz∣
∣
n
i,j,k+ 12
⟨ 1
∆y(ξ)Ψm(ξ),Ψl(ξ)
⟩
−ely∣
∣
n
i,j+ 12,k+1
− ely∣
∣
n
i,j+ 12,k
∆z(2.90)
For a maximum excitation frequency fmax, and a number of mesh cells per wavelength Nλ, the mesh
size ∆ is given by:
∆ =λmin
Nλ=
c
Nλfmax√µrǫr
(2.91)
Since the FDTD grid now contains mesh sizes of varying dimensions, the CFL stability condition
must be satisfied for all mesh sizes to ensure numerical stability. This is achieved when the CFL stability
condition is met for the minimum mesh size, and the resulting time step is given by:
∆t ≤ 1
c√
1∆x2
min
+ 1∆y2
min
+ 1∆z2
min
(2.92)
Chapter 2. Background 38
where c is the speed of light, and ∆xmin, ∆ymin, and ∆zmin are the minimum mesh cell dimensions in
the x, y, z direction, respectively, according to their probability distribution.
For the single-stub microstrip filter geometry given in Fig. (2.8), its FDTD domain is discretized
by a 50x50x20 mesh with a nominal mesh cell dimension ∆x = ∆y = ∆z = 0.25 mm. The mesh is
excited with a modulated Gaussian current source with a pulse width t0 = 0.6246 ps and a delay tw
of 3t0 centered at 10 GHz along the cross-section of the microstrip at port 1. The simulation is run
for 8000 time steps. The computational domain is terminated by a 10-cell CPML layer on all sides
with a polynomial-graded PML conductivity profile. The polynomially-graded conductivity profile of
the CPML has a maximum conductivity equal to the σopt of (2.25). The parameter profiles a and κ
have maxima amax and κmax of 0.0 and 3.0, respectively. The polynomial grading orders for the three
parameters are mσ = mκ = 3 and ma = 1.
Given the stub length lstub(ξ) = 3.7± 0.5 mm, the mesh variation is distributed over a 10 cell-thick
layer over the microstrip stub with a mesh size variation of ∆y = 0.25 ± 0.05 mm, which corresponds
to a maximum mesh distortion of 80% of the nominal mesh size. The PCE-FDTD simulation is run
with the electric and magnetic fields expanded in terms of PCE order P = 0, 1, 2, 3. The scattering
parameters S11 and S21 are then found from:
S11(ξ) =
F
P∑
j=0
Eport1,j(t)Ψj(ξ)− Einc(t)
FEinc(t)(2.93)
S21(ξ) =
P∑
j=0
FEport2,j(t)Ψj(ξ)
FEinc(t)(2.94)
The statistic moments of the scattering parameters, the mean µ|S11| and µ|S21| and standard deviation
σ|S11| and σ|S21|, are determined and are shown in Fig. (2.10 - 2.13), along with the difference between
the Monte Carlo result and the PCE-FDTD results.
The single-stub filter exhibits a resonance near 8 GHz and the largest standard deviation for both
S11 and S21 are observed just near the resonance. As the varying stub length shifts the resonance, the
relatively steep slope of the scattering parameters as a function of frequency near the resonance leads to a
large variation of scattering parameter values. The statistical moments determined from the PCE-FDTD
method also show rapid convergence with respect to the PEC order, as the mean and standard deviation
of the S11 and S21 for P = 1 show excellent agreement with th 1000-iterated Monte Carlo simulation.
The probability distribution of S11 and S21 at 7 GHz, corresponding to the large standard deviations
Chapter 2. Background 39
near the near the resonance, is shown in Fig. (2.14) and demonstrates relatively good agreement.
The PCE-FDTD time-stepping and Monte Carlo simulations are implemented in C++. MATLAB
is used to evaluate the inner product integrals with the quadgk function and to generate statistics of the
outputs. All simulations are run on a Intel Core i7 CPU @ 2.4 GHz. The PCE-FDTD time-stepping
takes 53 min while the Monte Carlo simulations require a total of 111 hrs. The computational costs
of evaluating the inner products and the statistical moments are usually insignificant relative to the
PCE-FDTD time-stepping.
Chapter 2. Background 40
Frequency [GHz]
|S11
| [dB
]
0 5 10 15−30
−25
−20
−15
−10
−5
0
µ, PCE, P = 0µ, PCE, P = 1µ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|µ|
S11
| PC
E −
µ|S
11| M
CM
| [dB
]
0 5 10 15−100
−90
−80
−70
−60
−50
−40
−30
−20
PCE, P = 0PCE, P = 1
(b)
Figure 2.10: (a) The mean of |S11| of the single-stub microstrip filter with a uniformly-distributed stublength for increasing PCE order, compared with Monte Carlo result. (b) The difference in the mean of|S11| between Monte Carlo and PCE results.
Chapter 2. Background 41
Frequency [GHz]
|S11
| [dB
]
0 5 10 15−50
−45
−40
−35
−30
−25
−20
σ, PCE, P = 1σ, PCE, P = 2σ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|σ|
S11
| PC
E −
σ|S
11| M
CM
| [dB
]
0 5 10 15−120
−110
−100
−90
−80
−70
−60
−50
PCE, P = 1PCE, P = 2
(b)
Figure 2.11: (a) The standard deviation of |S11| of the single-stub microstrip filter with a uniformly-distributed stub length for increasing PCE order, compared with Monte Carlo result. (b) The differencein the standard deviation of |S11| between Monte Carlo and PCE results.
Chapter 2. Background 42
Frequency [GHz]
|S21
| [dB
]
0 5 10 15−30
−25
−20
−15
−10
−5
0
µ, PCE, P = 0µ, PCE, P = 1µ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|µ|
S21
| PC
E −
µ|S
21| M
CM
| [dB
]
0 5 10 15−100
−90
−80
−70
−60
−50
−40
−30
−20
PCE, P = 0PCE, P = 1
(b)
Figure 2.12: (a) The mean of |S21| of the single-stub microstrip filter with a uniformly-distributed stublength for increasing PCE order, compared with Monte Carlo result. (b) The difference in the mean of|S21| between Monte Carlo and PCE results.
Chapter 2. Background 43
Frequency [GHz]
|S21
| [dB
]
0 5 10 15−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
σ, PCE, P = 1σ, PCE, P = 2σ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|σ|
S21
| PC
E −
σ|S
21| M
CM
| [dB
]
0 5 10 15−140
−120
−100
−80
−60
−40
PCE, P = 1PCE, P = 2
(b)
Figure 2.13: (a) The standard deviation of |S21| of the single-stub microstrip filter with a uniformly-distributed stub length for increasing PCE order, compared with Monte Carlo result. (b) The differencein the standard deviation of |S21| between Monte Carlo and PCE results.
Chapter 2. Background 44
|S11
|
Pro
babi
lity
Den
sity
0.75 0.8 0.85 0.9 0.950
0.01
0.02
0.03
0.04
0.05
0.067 GHz, PCE P = 27 GHz, MC
(a)
|S21
|
Pro
babi
lity
Den
sity
0.35 0.4 0.45 0.5 0.55 0.6 0.650
0.01
0.02
0.03
0.04
0.05
0.06
0.077 GHz, PCE P = 27 GHz, MC
(b)
Figure 2.14: The probability density distribution of (a) |S11| and (b) |S21| at 7 GHz.
Chapter 2. Background 45
Cascaded Network Representation of a Two-stub Microstrip Filter with Stub Length Un-
certainties
2.5 mm 3.7 mm
lstub (ξ1)
2.5 mm3.7 mmPort 1
1.3 mm
0.75 mm
εr = 2.2
Port 2
lstub (ξ2)
PML
1.3 mm 1.3 mm
Unit Cell 1 Unit Cell 2
Figure 2.15: The planar geometry of two-stub microstrip filter with statistically-independent stub lengthsuncertainties.
Consider the two-stub microstrip filter with uncertain stub lengths, shown in Fig. (2.15) with identi-
cally and independently distributed stub lengths. Direct simulation of the two-stub microstrip requires
the mesh distortion of the stub lengths to be localized such that the variations in the stub remain sta-
tistically independent. This can be accomplished by the curvilinear mesh approach presented in [22]
for modelling geometric uncertainties. Here a cascaded network model is used to obtain the statistical
moments of the scattering parameters of the two-stub filter from the polynomial chaos expansion of the
fields obtained from the single-stub filter in the last section. This example demonstrates the capability
of PCE-FDTD to extend the uncertainty propagation to other output of interest from post-processing
the field PCE as well as circumvents the issue of localizing geometric uncertainties.
To determine the scattering parameters of the cascaded network, we use the surrogate model approach
described in Section 2.3.2 The scattering parameters S11 and S21 are determined from a PCE-FDTD
simulation with an excitation at port 1, in the same manner described in the single-stub microstrip
example, and S12 and S22 are generated by running a second PCE-FDTD with an excitation source
at port 2. Monte Carlo simulations are used to generate 1000 samples of the scattering parameters by
evaluating their PCE, such as (2.93) and (2.94), with ξi, i = 1, 2, ..., 1000. For each set of scattering
Chapter 2. Background 46
parameters evaluated from a particular ξi, a series of network transformations is used to obtain the
equivalent scattering matrix of the two-stub filter. To maintain statistical-independence between the
two stub lengths, the same ABCD network model is evaluated by two sets of random inputs ξi1 and ξi2
that are drawn independently of each other, such that the cascaded network ABCD matrix is found by
evaluating:
A(ξ(i)1 , ξ
(i)2 ) B(ξ
(i)1 , ξ
(i)2 )
C(ξ(i)1 , ξ
(i)2 ) D(ξ
(i)1 , ξ
(i)2 )
=
A(ξ(i)1 ) B(ξ
(i)1 )
C(ξ(i)1 ) D(ξ
(i)1 )
A(ξ(i)2 ) B(ξ
(i)2 )
C(ξ(i)2 ) D(ξ
(i)2 )
(2.95)
The mean and the variance of the resulting |S11| and |S21| of the 2 cell stub filter are then tallied from
the ensemble of scattering parameters transformed from the cascaded ABCD matrix from (2.95).
In Fig. (2.16 - 2.19), the mean µ[|S11|] and µ[|S21|] and standard deviations σ[|S11|] and σ[|S21|] are
shown for varying PCE order P , along with the differences between the PCE-FDTD and Monte Carlo
results. A direct 1000-iterated Monte Carlo simulation of the 2 cell stub filter, using a sub-cell approach
to model the varying stub lengths, is used to generate the reference mean and variance. In the case of
the mean, the PCE results have reached convergence by PCE order P = 1. For µ[|S21|], discrepancy
can be observed at 8 GHz, which can be attributed to the fact that he cascade network model does not
account for mutual coupling between adjacent unit cells. This discrepancy is very small with an absolute
error roughly around 0.02. The standard deviation σ[|S11|] required P = 2 to reach agreement with the
Monte Carlo result. σ[|S11|] reaches convergence by P = 1.
Chapter 2. Background 47
Frequency [GHz]
|S11
| [dB
]
0 5 10 15−30
−25
−20
−15
−10
−5
0
µ, PCE, P = 0µ, PCE, P = 1µ, PCE, P = 2µ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|µ|
S11
| PC
E −
µ|S
11| M
CM
| [dB
]
0 5 10 15−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
PCE, P = 0PCE, P = 1PCE, P = 2
(b)
Figure 2.16: (a) The mean of |S11| of the two-stub microstrip filter with a uniformly-distributed stublength for increasing PCE order, compared with Monte Carlo result. (b) The difference in the mean of|S11| between Monte Carlo and PCE results.
Chapter 2. Background 48
Frequency [GHz]
|S11
| [dB
]
0 5 10 15−70
−60
−50
−40
−30
−20
−10
0σ, PCE, P = 1σ, PCE, P = 2σ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|σ|
S11
| PC
E −
σ|S
11| M
CM
| [dB
]
0 5 10 15−100
−90
−80
−70
−60
−50
−40
−30PCE, P = 1PCE, P = 2
(b)
Figure 2.17: (a) The standard deviation of |S11| of the two-stub microstrip filter with a uniformly-distributed stub length for increasing PCE order, compared with Monte Carlo result. (b) The differencein the standard deviation of |S11| between Monte Carlo and PCE results.
Chapter 2. Background 49
Frequency [GHz]
|S21
| [dB
]
0 5 10 15−60
−50
−40
−30
−20
−10
0
µ, PCE, P = 0µ, PCE, P = 1µ, PCE, P = 2µ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|µ|
S21
| PC
E −
µ|S
21| M
CM
| [dB
]
0 5 10 15−70
−60
−50
−40
−30
−20PCE, P = 0PCE, P = 1PCE, P = 2
(b)
Figure 2.18: (a) The mean of |S21| of the two-stub microstrip filter with a uniformly-distributed stublength for increasing PCE order, compared with Monte Carlo result. (b) The difference in the mean of|S21| between Monte Carlo and PCE results.
Chapter 2. Background 50
Frequency [GHz]
|S21
| [dB
]
0 5 10 15−70
−60
−50
−40
−30
−20
−10
0σ, PCE, P = 1σ, PCE, P = 2σ, MC, 1000 iterations
(a)
Frequency [GHz]
20lo
g|σ|
S21
| PC
E −
σ|S
21| M
CM
| [dB
]
0 5 10 15−100
−90
−80
−70
−60
−50
−40
−30PCE, P = 1PCE, P = 2
(b)
Figure 2.19: (a) The standard deviation of |S21| of the two-stub microstrip filter with a uniformly-distributed stub length for increasing PCE order, compared with Monte Carlo result. (b) The differencein the standard deviation of |S21| between Monte Carlo and PCE results.
Chapter 2. Background 51
These results show the capability of the PCE-FDTD method to achieve an accurate approximation of
the mean and the standard deviation with only a few order P . The direct 1000-iterated Monte Carlo of
the two-stub filter FDTD simulations requires 155 hrs, where as a PCE-FDTD simulation of the single-
stub filter with a PCE order P = 4 requires 53 min. The cascaded network approach also exploits the fact
that running Monte Carlo using the PCE of the fields as surrogate models is much faster than running a
direct Monte Carlo on the FDTD simulations. Therefore, the exceptional efficiency of the PCE-FDTD
for uncertainty propagation and quantification for electromagnetic problems is demonstrated.
2.5 Conclusion
A review of the finite-difference time-domain method and the uncertainty quantification techniques using
the Monte Carlo method and the polynomial chaos expansion method were presented in this section. The
extension of the polynomial chaos expansion method to FDTD with material and geometric uncertainties
was outlined. A numerical example of the PCE-FDTD was presented for a two-stub filter with uncertain
stub lengths, demonstrating excellent computational efficiency compared with conventional Monte Carlo
method.
Chapter 3
A PML Absorber for the
Termination of Random Media
3.1 Introduction
A form of uncertainty that practically arises in many microwave circuit and antenna applications is
related to substrate material parameters, namely dielectric permittivity and loss tangent. Manufacturers
specify these parameters by providing their expected minimum and maximum values, implying that
their actual value is merely a sample of a random variable. Embedding this form of uncertainty into
the “intrusive” PCE-FDTD requires the use of boundary conditions with respect to uncertain working
volume complex permittivities. Previous work on this problem has employed Mur’s first order boundary
conditions [20]. Hence, the state-of-the-art in PCE-FDTD absorbing boundary conditions lags the
corresponding state-of-the-art in general FDTD, which is defined by the perfectly matched layer absorber
(PML [28, 29]).
In this chapter, a PML absorber for random lossy, dielectric media is introduced for enhancing the
accuracy of PCE-FDTD for modelling random media and related applications. First, a formulation
of the convolutional PML absorber (CPML [36]) is provided for PCE-FDTD. The accuracy of this
absorber is evaluated with respect to various important parameters (such as the order of the polynomial
approximation in PCE-FDTD and the variance of the input random variables). Finally, the proposed
absorber is applied to three-dimensional printed circuit geometries with random media substrates.
52
Chapter 3. A PML Absorber for the Termination of Random Media 53
3.2 The PCE-FDTD PML absorber: Formulation
Under the presence of uncertain material parameters at the domain boundary, the PML must be matched
to the random wave impedance in the domain to ensure minimal spurious reflection. This is accomplished
by reformulating the field update equations within the PML of the PCE-FDTD method, presented in
Section 2.2.5 to model uncertain material parameters as well as the fields and the ζ-variables random
processes. For instance, with uncertainty, the update equation for Ex within the PML becomes:
Ex(ξ)∣
∣
n+1
i,j+ 12,k
= C1(ξ)∣
∣
i+ 12,j,k
Ex(ξ)∣
∣
n
i,j+ 12,k
+ C2(ξ)∣
∣
i+ 12,j,k
Hz(ξ)∣
∣
n+ 12
i+ 12,j+ 1
2,k−Hz(ξ)
∣
∣
n+ 12
i+ 12,j− 1
2,k
κy,j∆y−
Hy(ξ)∣
∣
n+ 12
i+ 12,j,k+ 1
2
−Hy(ξ)∣
∣
n+ 12
i+ 12,j,k− 1
2
κz,k∆z
+ ζEx,y(ξ)∣
∣
n+ 12
i+ 12,j,k
− ζEx,z(ξ)∣
∣
n+ 12
i+ 12,j,k
(3.1)
Expanding the electric and magnetic fields, as well as the ζ functions in terms of polynomial chaos basis
functions and substituting into (3.1) yields:
P∑
m=0
emx∣
∣
n+1
i+ 12,j,k
Ψm(ξ) = C1
∣
∣
i+ 12,j,k
(ξ)
P∑
m=0
emx∣
∣
n
i− 12,j,k
Ψm(ξ)
+ C2
∣
∣
i+ 12,j,k
(ξ)P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
κy,j∆y−
hmy
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
y
∣
∣
n+ 12
i− 12,j+ 1
2,k
κz,k∆z
+ ζmEx,y
∣
∣
n+ 12
i+ 12,j,k
− ζmEx,z
∣
∣
n+ 12
i+ 12,j,k
Ψm(ξ) (3.2)
Similarly, the update equations of the auxiliary variable ζ from (2.41) and (2.42) are written as:
P∑
m=0
ζmEx,y
∣
∣
n+ 12
i+ 12,j,k
Ψm(ξ) = by,j
P∑
m=0
ζmEx,y
∣
∣
n+ 12
i+ 12,j,k
Ψm(ξ) + cy,j
P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
Ψm(ξ)
(3.3)
P∑
m=0
ζEx,z
∣
∣
n+ 12
i+ 12,j,k
Ψm(ξ) = bz,k
P∑
m=0
ζEx,z
∣
∣
n− 12
i+ 12,j,k
Ψm(ξ) + cz,k
P∑
m=0
hy
∣
∣
n+ 12
i+ 12,j,k+ 1
2
− hy
∣
∣
n+ 12
i+ 12,j,k− 1
2
∆z
Ψm(ξ)
(3.4)
Chapter 3. A PML Absorber for the Termination of Random Media 54
The update equation of each field coefficient is found from the inner product of (3.2) with the corre-
sponding basis function. Each coefficient is first updated from the magnetic fields by:
elx∣
∣
n+1
i+ 12,j,k
=1
〈Ψ2l (ξ)〉
P∑
m=0
emx∣
∣
n
i− 12,j,k
〈C1(ξ)Ψm(ξ),Ψl(ξ)〉
+1
〈Ψ2l (ξ)〉
P∑
m=0
hmz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
κy,j∆y−
hmy
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hm
y
∣
∣
n+ 12
i− 12,j+ 1
2,k
κz,k∆z
〈C2(ξ)Ψm(ξ),Ψl(ξ)〉
(3.5)
The update equation of ζ coefficients is obtained by applying the same process to (2.41) and (2.42).
Although the parameters b and c are functions of electric permittivity through the expression for de-
termining the optimal value of the PML conductivity, it will be shown that the optimal b and c for
the mean value of the permittivity is sufficient for permittivity distributions even with relatively large
variances. As a result, the ζ expansion coefficients can be updated independently of each other by:
ζlEx,y
∣
∣
n+1
i+ 12,j,k
= by,jζlEx,y
∣
∣
n
i+ 12,j,k
+ cy,j
hlz
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hl
z
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆y
(3.6)
ζlEx,z
∣
∣
n+1
i+ 12,j,k
= bz,kζlEx,z
∣
∣
n
i+ 12,j,k
+ cz,k
hly
∣
∣
n+ 12
i+ 12,j+ 1
2,k− hl
y
∣
∣
n+ 12
i+ 12,j− 1
2,k
∆z
(3.7)
The updated ζ coefficients are used to complete the update procedure for the electric field coefficients
within the PML:
elx∣
∣
n+1
i+ 12,j,k
= elx∣
∣
n
i+ 12,j,k
+P∑
m=0
ζmEx,y
∣
∣
n+ 12
i+ 12,j,k
+ ζmEx,z
∣
∣
n+ 12
i+ 12,j,k
〈C2(ξ)Ψl(ξ),Ψm(ξ)〉 (3.8)
Similarly, the magnetic fields are updated by:
P∑
m=0
hmx
∣
∣
n+ 12
i,j+ 12,k
= D1
∣
∣
i+ 12,j,k
P∑
m=0
hmx
∣
∣
n− 12
i,j+ 12,k
+D2
∣
∣
i+ 12,j,k
P∑
m=0
emz∣
∣
n
i,j+ 12,k+ 1
2
− emz∣
∣
n
i,j− 12,k+ 1
2
κy,j∆y−
emy∣
∣
n
i,j+ 12,k+ 1
2
− emy∣
∣
n
i,j+ 12,k− 1
2
κz,k∆z
+ ζHx,y
∣
∣
n
i,j+ 12,k+ 1
2
− ζHx,z
∣
∣
n
i,j+ 12,k+ 1
2
Ψm(ξ) (3.9)
ζHx,y
∣
∣
n
i,j+ 12,k+ 1
2
= by,j+ 12ζHx,y
∣
∣
n−1
i,j+ 12,k+ 1
2
+ cy,j+ 12
Ez
∣
∣
n
i,j+1,k+ 12
− Ez
∣
∣
n
i,j,k+ 12
∆y
(3.10)
Chapter 3. A PML Absorber for the Termination of Random Media 55
ζHx,z
∣
∣
n
i,j+ 12,k+ 1
2
= bz,k+ 12ζHx,z
∣
∣
n−1
i,j+ 12,k+ 1
2
+ cz,k+ 12
Ey
∣
∣
n
i,j+ 12,k+1
− Ey
∣
∣
n
i,j+1,k
∆z
(3.11)
The update of the other four field components are derived in the same manner. This completes the
formulation of the convolution PML for terminating random media.
3.3 Numerical Results
3.3.1 Random dielectric-filled two-dimensional domain
source
sample point
PML
2 mm
2 mm
40 mm
40 mm
Figure 3.1: Two dimensional unbounded domain, filled with a random dielectric medium, with a currentsource excitation at the center of the domain and the electric field sampled at the corner point close tothe PML.
The presented PCE-FDTD formulation is implemented for a 2-D TEz dielectric domain, shown in
Fig. (3.1) discretized by a 40x40-cells mesh with cell dimensions ∆x = ∆z = 0.1 cm. The mesh is excited
with a differentiated Gaussian current source at the center of the mesh with a pulse width t0 = 26.53
ps and a delay tw of 4t0. The simulation includes 1024 time steps with a step size equal to 0.99 of the
Courant limit calculated in free space. The computational domain is terminated by a 10-cell CPML on
all sides. The polynomial-graded conductivity profile of the PML has a maximum conductivity given
by σopt. The parameter profiles a and κ have maxima amax and κmax of 0.2 and 1.0, respectively. The
polynomial grading orders for the three parameters are ma = mσ = 3 and mκ = 1. The fields are
sampled at a point near the corner 2 mm from the two adjacent boundaries, as shown in Fig. (3.1). A
Chapter 3. A PML Absorber for the Termination of Random Media 56
reference domain with 1024x1024 mesh cells with the same cell dimensions and the same current source
excitation is used to obtain a reflection-free waveform at the same position relative to the excitation.
The reflection from the PML is measured by the relative error En at the n-th time step between the
sampled electric field EnCPML in the PML-terminated domain and its reference simulation counterpart
EnRef :
En =
∣
∣
∣En
CPML − EnRef
∣
∣
∣
max∣
∣
∣En
Ref
∣
∣
∣
(3.12)
For a uniformly distributed relative permittivity with a mean value µ[ε] = 2.2 and σ2[ε] = 0.05 µ[ε],
or equivalently ε = 2.2±0.5745, a Legendre polynomial-based PCE is employed. The electric field PCE
coefficients are sampled at the corner sample point of Fig. (3.1). The sampled PCE coefficients are
used to determine the electric field and the corresponding relative error with respect to the reference
simulation (using the same random dielectric to fill the enlarged domain) defined as:
En(ξ) =
∣
∣
∣
∣
P∑
m=0(enCPML,m − enRef,m)Ψm(ξ)
∣
∣
∣
∣
max
∣
∣
∣
∣
P∑
m=0enRef,mΨm(ξ)
∣
∣
∣
∣
(3.13)
is found. Note that this error is itself a random process; hence, it will be characterized by its mean
value and variance as opposed to its value at each time step (a standard metric in PML benchmarking
studies).
The mean and standard deviation of the relative error are found by evaluating (3.13) at random
samples of the variable ξ ∼ U[-1 1] (uniformly distributed). The results are compared to those computed
with 10,000 Monte Carlo trials of deterministic simulations, generated by assigning random samples to
εr (uniformly distributed in the interval 2.2±0.5745). The standard CPML is used to terminate the
domain in these cases.
Fig. (3.2) shows excellent agreement between the Monte Carlo and the PCE-FDTD results, in the
sense that the mean value and variance of the relative error of the CPML are the same in both cases.
For comparison purposes, results employing Mur’s first order absorbing boundary condition (ABC), as
formulated for PCE-FDTD in [20], are appended. As expected, the mean and variance of the relative
error for the CPML is significantly smaller than the one produced by Mur’s ABC.
Increasing the variance of the relative dielectric permittivity to 0.1 and 0.2 of its mean value does
not compromise the performance of the proposed CPML. Relevant results are shown in Fig. (3.3). As
expected, σ[En] is higher as the σ[ε] is increased. Both µ[En] and σ[En] remain small for both cases;
Chapter 3. A PML Absorber for the Termination of Random Media 57
hence, the CPML remains well-matched to the random medium domain, even in cases with relatively
large variance of the dielectric permittivity.
Fig. (3.4) shows the convergence of the statistical moments with respect to the polynomial chaos
order P . Both the mean and standard deviation of En converged for P = 2. The MCM, on the other
hand, requires about 600 iterations for convergence, as shown in Fig. (3.5). With a 3.47 GHz Intel Xeon
processor, this translates to a total simulation time of 20 mins for PCE-FDTD to reach convergence,
compared to 3 hrs for MCM.
Chapter 3. A PML Absorber for the Termination of Random Media 58
Time Step
µ [
] [d
B]
En
200 400 600 800 1000−200
−150
−100
−50
0CPML PCE, P = 3CPML MC, 1000 iterationsMur’s ABC PCE, P = 3
(a)
Time Step
σ [
] [d
B]
En
200 400 600 800 1000−200
−150
−100
−50
0CPML PCE, P = 3CPML MC, 1000 iterationsMur’s ABC PCE, P = 3
(b)
Figure 3.2: The (a) mean and (b) standard deviation of the relative error En for a uniformly-distributedrelative permittivity with σ2[ε] = 0.05 µ[ε].
Chapter 3. A PML Absorber for the Termination of Random Media 59
Time Step
µ [
] [d
B]
En
200 400 600 800 1000−160
−150
−140
−130
−120
−110
−100
−90
−80
σ2 [ε] = 0.05 µ [ε]
σ2 [ε] = 0.1 µ [ε]
σ2 [ε] = 0.2 µ [ε]
(a)
Time Step
σ [
] [d
B]
En
200 400 600 800 1000−160
−150
−140
−130
−120
−110
−100
−90
−80
σ2 [ε] = 0.05 µ [ε]
σ2 [ε] = 0.1 µ [ε]
σ2 [ε] = 0.2 µ [ε]
(b)
Figure 3.3: The (a) mean and (b) standard deviation of the relative error En for uniformly-distributedpermittivity with varying σ2[ε].
Chapter 3. A PML Absorber for the Termination of Random Media 60
Time Step
µ [
] [d
B]
En
100 120 140 160 180 200−120
−115
−110
−105
−100
−95
−90
−85
−80
PCE, P = 0PCE, P = 1PCE, P = 2PCE, P = 3
(a)
Time Step
σ [
] [d
B]
En
100 120 140 160 180 200−130
−125
−120
−115
−110
−105
−100
−95
−90
PCE, P = 1PCE, P = 2PCE, P = 3
(b)
Figure 3.4: The convergence of (a) mean and (b) standard deviation of the relative error En with respectto PCE orders P .
Chapter 3. A PML Absorber for the Termination of Random Media 61
number of MC iterations
∈
0 200 400 600 800 1000
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3x 10
−5
µ [∈ ]σ [∈ ]
Figure 3.5: The convergence of mean and standard deviation of the relative error En with respect tonumber of Monte Carlo simulations at n = 100.
3.3.2 Microstrip Low-pass Filter
5.69 mm
2.44 mm
2.54 mm
Port 2
Port 1
0.795 mm εr (ξ)
4.23 mm
4.23 mm
5.69 mm
PML
4.06 mm
Figure 3.6: The planar geometry of the microstrip low-pass filter with substrate permittivity uncertainty.
The PCE-FDTD CPML is used to terminate the substrate of a microstrip low-pass filter with uncer-
tain substrate permittivity. The geometry of the structure (previously considered with a fixed substrate
Chapter 3. A PML Absorber for the Termination of Random Media 62
permittivity in [38]), shown in Fig. (3.6), is discretized with 80x100x16 mesh cells with cell dimensions
Frequency [GHz]
|S11
| [dB
]
0 5 10 15−80
−70
−60
−50
−40
−30
−20
−10
0
µ, PCE, P = 4µ, MC, 1000 iterationsσ, PCE, P = 4σ, MC, 1000 iterations
(a)
Frequency [GHz]
[dB
]
0 5 10 15
−120
−110
−100
−90
−80
−70
−60
−50
20log|µ|S11
|PCE
− µ|S11
|MCM
|
20log|σ|S11
|PCE
− σ|S11
|MCM
|
(b)
Figure 3.7: (a) The mean and standard deviation of |S11| for a uniformly distributed substrate relativepermittivity ε with µ[ε] = 2.2 and σ2[ε] = 0.05 µ[ε]. (b) The difference in the statistical momentsdetermined from the PCE-FDTD method and the Monte Carlo method.
∆x = 0.4064 mm, ∆y = 0.4233 mm, ∆z = 0.265 mm. The microstrip is excited by a Gaussian current
source with a pulse width t0 = 6.2464 ps and a delay tw of 4t0. The simulation runs for 8000 time steps
Chapter 3. A PML Absorber for the Termination of Random Media 63
Frequency [GHz]
|S21
| [dB
]
0 5 10 15−80
−70
−60
−50
−40
−30
−20
−10
0
µ, PCE, P = 4µ, MC, 1000 iterationsσ, PCE, P = 4σ, MC, 1000 iterations
(a)
Frequency [GHz]
[dB
]
0 5 10 15−120
−110
−100
−90
−80
−70
−60
−50
20log|µ|S21
|PCE
− µ|S21
|MCM
|
20log|σ|S21
|PCE
− σ|S21
|MCM
|
(b)
Figure 3.8: (a) The mean and standard deviation of |S21| for a uniformly distributed substrate relativepermittivity ε with µ[ε] = 2.2 and σ2[ε] = 0.05 µ[ε]. (b) The difference in the statistical momentsdetermined from the PCE-FDTD method and the Monte Carlo method.
with a step size equivalent to 0.99 of the Courant limit in free-space. The computational domain is
terminated by a 10-cell CPML on all sides. The polynomially-graded conductivity profile of the CPML
has a maximum conductivity equal to the σopt. The parameter profiles a and κ have maxima amax and
Chapter 3. A PML Absorber for the Termination of Random Media 64
κmax of 0.0 and 7.0, respectively. The polynomial grading orders for the three parameters are mσ =
mκ = 3 and ma = 1. The substrate relative permittivity is uniformly distributed with µ[ε] = 2.2 and
σ2[ε] = 0.05 µ[ε]. The field quantities are expanded in terms of Legendre polynomials up to 4-th order.
The electric field coefficients are sampled at port 1 and 2 (defined in Fig. (3.6)) and are used to find
the S11 and S21 parameters by post-processing. The mean and standard deviation of the magnitude of
the scattering parameters |S11| and |S21|, given in Fig. (3.7) and Fig. (3.7), respectively, along with the
difference between their values determined from Monte Carlo and PCE-FDTD. The results from MCM
and PCE-FDTD are in excellent agreement. The PCE-FDTD simulation requires 9 hrs whereas 1000
FDTD-based MCM iterations require 666 hrs. The means of |S11| and |S21| are similar to those of a
deterministic simulation with substrate permittivity equal to µ[ε] . As expected, the standard deviation
is large near the stopband edge frequencies and relatively small in the pass-band and stop-band regions.
This is expected, as variations in the substrate permittivity change the line impedance of the microstrip,
which subsequently shifts the resonance frequencies of the filter. For small permittivity variations, the
largest change in the magnitude of the scattering parameters occurs near the edge of the stop-band,
since the steep slopes at those frequency regions means that a small shift in frequency can result in a
large change in magnitude. Similar statistical behaviour of the scattering parameters was also observed
for the same structure when uncertainties in the stub lengths were considered [22].
3.4 Conclusion
A PML absorber for random media, simulated via the PCE-FDTD method, has been presented. This
method is applied to a 2-D domain filled with random electric permittivity. The statistic moments of
the reflection error from the CPML boundary are considerably smaller than that from the previous
Mur’s absorbing boundary condition formulation of PCE-FDTD. The convergence of the PCE-FDTD
with CPML with respect to the PCE order converges rapidly, compared with the Monte Carlo method.
The PCE-based CPML is applied to a microstrip circuit with uncertain substrate permittivity and
demonstrated excellent computational efficiency compared with the Monte Carlo method.
Chapter 4
Hybrid Monte Carlo / Polynomial
Chaos Expansion FDTD Method
4.1 Introduction
The efficiency of the PCE method has been observed in various numerical experiments where random
outputs with a small number of random inputs are sufficiently approximated with a few PCE terms,
such as in the numerical example of the cascaded microstrip stub filter, as well as in [20, 22]. As the
number of random variable increases, however, the number of expansion terms rises rapidly for the same
total order D. For example, a PCE simulation with 2 random inputs for D = 1, 2, 3 requires 3, 6, 10
PCE terms, respectively. By comparison, for the same total orders, a simulation with 10 random inputs
requires 11, 66, and 286 PCE terms. This rapid growth in the number of PCE terms for increasing
number of random inputs, a symptom of the “curse of dimensionality,” can significantly diminish the
efficiency of the PCE method.
To overcome the potentially colossal computation time required for multi-parametric PCE simula-
tions, we take advantage of a particular Monte Carlo method known as control variate. The control
variate method is one of a group of methods collectively known as variance reduction method[37]. As
was discussed in Chapter 2, the accuracy of an Monte Carlo estimate is estimated by its standard error
SE = V AR/√N . Where as conventional Monte Carlo method increases the accuracy of the estimate by
simply increasing the number of iterations, a class of methods have been developed for reducing the vari-
ance of the system involved in the Monte Carlo, collectively known as variance reduction method. The
control variate method is a particular of which using a known approximate random function, referred to
65
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 66
as control variate, to achieve a reduction of variance.
In this chapter, a method using a polynomial chaos expansion-based control variate is presented,
similar to the proposal by [30], with the aim to reduce the number of PCE terms required for analyzing
multi-parametric uncertainties in electromagnetic simulations. Instead of using PCE with an increasingly
higher order to improve accuracy, a PCE-based control variate (CV-PCE) method can achieve a similar
effect with a lower order PCE as the control variate combined with a small number of Monte Carlo
samples. This is shown with the example of an FDTD analysis of a Bragg reflector with uncertain slab
permittivities. A clear improvement in computation efficiency, compared to PCE, in the presence of an
increasing number of slabs with random permittivities is demonstrated.
4.2 Polynomial Chaos Expansion as Control Variate
The control variate method was originally introduced for improving Monte Carlo performance, as dis-
cussed in [39, 37]. We briefly outline the method, using PCE as the control variate. For a random
variable X(ξ) with an exact mean E[X], exact standard deviation STD[X], exact variance V AR[X],
and probability distribution P (ξ), its Monte Carlo (MC) mean estimator µ[X], for N iterations, is
defined as:
µ[X] =1
N
N∑
i=1
X(ξ(i)) ≈∫
X(ξ)P (ξ)dξ = E[X] (4.1)
Let the “low order” PCE approximation of X, G(ξ), be the control variate:
G(ξ) =
P∑
l=0
alΨl(ξ) (4.2)
The central limit theorem prescribes the probabilistic behaviour of µ[X(ξ)] as a normal random
variable with a standard deviation given by the standard error SEX = V AR[X]/√N [37]. Hence, the
accuracy of the MC estimator can be improved either by increasing the number of iterations or by
reducing σ [37]. To this end, (4.1) is rewritten as:
E[X] =
∫
(X(ξ)− λG(ξ) + λG(ξ))P (ξ)dξ
=
∫
(X(ξ)− λG(ξ)P (ξ)dξ + λ
∫
G(ξ))P (ξ)dξ
≈ µ[X − λG] + λµ[G]
(4.3)
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 67
where µ[X − λG] is given by:
µ[X − λG] =1
N
N∑
i=1
(
X(ξ(i))− λG(ξ(i))
(4.4)
The standard error of (4.4) is given by:
SEX−λG =STD[X − λG]√
N(4.5)
This allows us to solve µ[X] by solving µ[X−λG], which generally has a smaller SE ifX is well correlated
with G. Furthermore, we can find the value of λ that minimizes V AR[X − λG] by setting:
d
dλV AR[X − λG]
∣
∣
∣
λ=λopt
= 0 (4.6)
and finding λopt to be:
λopt =Cov(X,G)
V AR[G](4.7)
with the corresponding minimum:
min V AR[X − λoptG] = V AR[X](
1− ρ2(X,G))
(4.8)
Notably, the effectiveness of the control variate G in reducing the SE of the MC estimator depends on
the correlation ρ between X and G, given as :
ρ(X,G) =Cov(X,G)
STD[X]STD[G](4.9)
The covariance Cov(X,G) is estimated by:
Cov(X,G) =1
N
N∑
i=1
(G(ξ(i))− µ[G])(X(ξ(i))− µ[X]) (4.10)
The introduction of the control variate effectively transforms the mean estimator µ[X] into two integrals,
one solved by a low order PCE-FDTD and one by a Monte Carlo simulation with a faster convergence
rate than conventional Monte Carlo. The variance estimate σ2[X] is obtained by applying the same
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 68
procedure to the integral:
σ2[X] =1
N − 1
N∑
i=1
(X(ξ(i))− E[X(ξ)])2 ≈∫
(X(ξ)− µ[X(ξ)])2P (ξ)dξ = V AR[X] (4.11)
and using the corresponding control variate function J(ξ):
J(ξ) = (G(ξ)− µ[G(ξ)])2 (4.12)
In this case, µ[X] must be determined before σ2[X] can be estimated.
4.3 Numerical Results
d1
<ε1> ε0
d2 d1
<ε2> ε0
d2 d1
<ε3> ε0
d2 d1
<εN> ε0
d2
<ε1> ε0 ε0 ε0<ε2> <ε3> <εN> ε0
Figure 4.1: An N -unit cell Bragg reflector with d1 = 1.268 cm, d2 = 4.1 cm, and a uniformly distributedε1 with µ[ε1] = 2.7556 with σ[ε1] = 0.05µ[ε1].
The proposed formulation is implemented on a 1-D Bragg reflector with 8 unit cells, where the
geometry of a single unit cell is shown in Fig. 4.1. The PCE-FDTD domain is discretized by 1584 mesh
cells with a cell size ∆x = 0.36265 mm. The domain boundaries are terminated by Mur’s ABC. The mesh
is excited with a transparent Gaussian source before the Bragg reflector slab. The simulation runs for
100,000 time steps with a step size equal to 0.90 of the Courant limit. The permittivity of each dielectric
slab is independently and uniformly distributed with a mean µ[ε] = 2.7756 and a σ[ε] = 0.05µ[ε]. For a
total of 8 independent random parameters, PCE-FDTD simulations of total orders D = 1, 2, 3 require
P + 1 = 9, 45, 165 expansion terms, respectively. The PCE-FDTD time-stepping and Monte Carlo
simulations are implemented in C++. MATLAB is used to evaluate the inner product integrals with
the quadgk function and to generate statistics of the outputs. All simulations are run on a Intel Core i7
CPU @ 2.4 GHz.
The mean and the variance of |S21|, σ2[|S21|] are shown in Fig. (4.2) for increasing numbers of PCE
order D. A 10,000 trial MCM is used to generate the reference solution. The mean converges rapidly,
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 69
Frequency [GHz]
µ [|S
21|]
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PCE D = 1PCE D = 2MC N = 10000
(a)
Frequency [GHz]
σ2 [|S
21|]
1.9 1.95 2 2.05 2.10
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018PCE D = 1PCE D = 2MC N = 10000
(b)
Figure 4.2: The (a) mean and (b) variance of |S21| of an 8 cell Bragg reflector for increasing PCE orderD, compared with Monte Carlo.
reaching excellent agreement with the reference by D = 1. Since varying the slab permittivity causes the
S21 to shift in frequency, the largest variance is usually present around the edge of the band-gap regions
where the slope of the curve is steep. Thus we focus only on the variance near the band-gap around 2
GHz. For D = 1, discrepancies are observed between the PCE and the reference result. A PCE order
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 70
of D = 2 is required for the variance of |S21| to reach complete agreement with the reference.
Given the rapid convergence of the mean of this structure, this study only considers the variance of
|S21|, σ2[|S21|]. To investigate the effectiveness of the PCE as control variate, the σ2[|S21|] around 2.0
GHz from MCM and CV-PCE with PCE D = 1 as a control variate are shown in Fig. (4.3), along with
the reference as well as the PCE result. After 800 MC iterations, the MCM result displays noticeable
discrepancies from the reference whereas applying the PCE D = 1 as CV results in better accuracy at
the frequency range higher than 2 GHz and below 1.92 GHz. In between these two frequencies the two
methods yield similar values. CV-PCE with D = 1 shows considerable improvement over PCE D = 1.
Frequency [GHz]
σ2 [|S
21|]
1.9 1.95 2 2.05 2.10
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018CV: (PCE D = 1, MC N = 800)PCE D = 1MC N = 800MC N = 10000
Figure 4.3: The frequency response of σ2[|S21|] around 2.0 GHz of the 8 cell Bragg reflector found fromCV-PCE. The results from the MCM and PCE with the same number of iterations and PCE order D,and a 10,000 MCM are shown for comparison.
From Fig. (4.3), two observations can be made. First, the accuracy of the CV-PCE result can do
no worse than Monte Carlo, as expected. Second, the CV-PCE does not improve upon the MCM over
the whole frequency range. This can be explained by the correlation between σ2[|S21|] from the PCE
method with total orders D = 1, 2, and the reference, shown in Fig. (4.4). The correlation coefficient
ρ(X,G) approaches unity as D increases, demonstrating the convergence of the PCE towards the exact
solution of σ2[|S21|]. For PCE D = 1, ρ is near or below 0.5 in the frequency between 1.92 to 1.97 GHz,
and the σ2[|S21|] from CV-PCE only performs as well as MCM for the same number of iterations. In
the frequency range where ρ(X,G) is large, the CV-PCE produces better result than both MCM and
PCE. This is apparent for frequencies larger than 2.0 GHz, where σ2[|S21|] from CV-PCE aligns with
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 71
Frequency [GHz]
ρ(X
, G)
1.8 1.85 1.9 1.95 2 2.05 2.1−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
PCE D = 1PCE D = 2
Figure 4.4: The correlation coefficient of σ2[|S21|] for varying PCE orders and the reference MCMsolution, for the 8 cell Bragg reflector.
the reference when neither the PCE method nor the MCM is able to achieved. Near 2 GHz, CV-PCE
also produced a more closer result than both the PCE method and the MCM. Thus, an effective control
variate must be well correlated with the exact solution of the estimator.
To quantify the accuracy of the CV-PCE method and the MCM in terms of the number of iterations,
the L2 relative error norm Rn at the n−th iteration with respect to the 10,000-iterated MCM result is
defined here as:
Rn =‖σ2
n[|S21|]− σ2Ref [|S21|]‖ω
‖σ2Ref [|S21|]‖ω
(4.13)
where the L2 norm of σ2[|S21|] is evaluated over the frequency range from 0 to 2.5 GHz, corresponding
to the frequency range up to the middle of the first stop-band. For the 8 cell Bragg reflector, the relative
error R of the CV and MCM are shown in Fig. (4.5), along with the relative error of the converged
PCE result. In this instance of MC and CV-PCE simulation, RMC requires 1310 iterations to fall below
0.05, compared to 604 iterations for CV-PCE with D = 1. The relative errors of the PCE results are
shown by comparison. For D = 1, RPCE is around 0.08, whereas RPCE for D = 2 is well below the 0.05
threshold. This error is due to the truncation of the PCE and cannot be reduced without increasing the
PCE order. The CV-PCE from D = 1 shows an appreciably smaller R than the PCE results with the
addition of a few hundred MC samples. This result shows quantitatively that the CV-PCE converges
faster than the MCM given the PCE control variate is reasonably correlated with the exact solution.
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 72
Furthermore, this also demonstrates that the CV-PCE is an alternative to improving the accuracy of
the PCE by generating Monte Carlo iterations, instead of re-running the PCE-FDTD simulation with
a higher PCE order. This is useful given that higher order PCE will inevitably require a large number
of PCE terms in a single simulation, where as the increment of Monte Carlo iterations can be finely
controlled.
Monte Carlo iterations
R
0 500 1000 1500 2000 2500 30000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
MCCV: (PCE D = 1)CV: (PCE D = 2)
PCE D = 1
PCE D = 2
Figure 4.5: The relative error of σ2[|S21|] of MCM and CV-PCE for increasing numbers of Monte Carlosamples, for an 8-cell Bragg reflector. The relative error of the PCE results for total orders D = 1, 2are shown for comparison.
The relative error of the 6, 10 unit cells with uncertain slab permittivity, corresponding to 6, 10
random inputs in their respective PCE-FDTD simulation, are shown in Fig. (4.6) and Fig. (4.7). In
the case of 6 unit cells, the PCE error is small for PCE D = 1, demonstrating the efficiency of the PCE
method for relatively small numbers of random parameter. As the number of unit cells increases to 10,
the relative error R is larger for the same order D. Thus, the PCE D = 1 becomes a less effective control
variate, with a relative error following a very similar convergence as the MCM.
Fig. (4.8) shows the computation time for convergence for σ2[|S21|] of Bragg reflectors with 6, 8
and 10 unit cells with uncertain slab permittivity. The convergence threshold is defined to have been
reached when R is below 0.05. The computation time for CV-PCE is the sum of the computation time
needed to run PCE and to generate MCM samples. Overall, the CV-PCE performs better compared to
MCM. In the case of 6 random slabs, PCE method reaches convergence at D = 1. CV-PCE shows worse
or comparable performance to PCE as PCE already reached convergence at lower orders and does not
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 73
Monte Carlo iterations
R
0 500 1000 1500 2000 2500 30000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1MCCV: (PCE D = 1)CV: (PCE D = 2)
PCE D = 1
PCE D = 2
Figure 4.6: The relative error of σ2[|S21|] of MCM and CV-PCE for increasing numbers of Monte Carlosamples, for an 6-cell Bragg reflector. The relative error of the PCE results for total orders D = 1, 2are shown for comparison.
Monte Carlo iterations
R
0 500 1000 1500 2000 2500 30000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2MCCV: (PCE D = 1)CV: (PCE D = 2)
PCE D = 1
PCE D = 2
PCE D = 3
Figure 4.7: The relative error of σ2[|S21|] of MCM and CV-PCE for increasing numbers of Monte Carlosamples, for an 10-cell Bragg reflector. The relative error of the PCE results for total orders D = 1, 2,and 3 are shown for comparison.
benefit from the relatively computationally expensive Monte-Carlo samples. As the number of random
slabs increases to 8 and 10, PCE requires D = 3 to reach convergence. CV-PCE, on the other hand,
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 74
6 8 10
102
103
104
Tim
e [s
]
Number of Random Variables
MCMPCECV−PCE
Figure 4.8: The computation time required to reach R = 0.05 for increasing numbers of random dielectricslabs.
takes advantage of PCE D = 2 which results in significant reduction in computation time.
Number of Random Variables Method Monte Carlo iterations Computation Time (s)
6 Monte Carlo 1031 264PCE D = 1 13
CV-PCE D = 1 601 167 (MCM:154 + PCE:13)
8 Monte Carlo 1301 390PCE D = 1 16PCE D = 2 210
CV-PCE D = 1 531 192 (MCM:176 + PCE:16)
10 Monte Carlo 1202 468PCE D = 1 24PCE D = 2 1000PCE D = 3 10000
CV-PCE D = 1 1008 417 (MCM:393 + PCE:24)CV-PCE D = 2 514 1199 (MCM:199 + PCE: 1000)
Table 4.1: The computation time of the Monte Carlo method, PCE method, and the control variate-PCEmethod.
4.4 Conclusion
Quantifying multi-parametric uncertainties using polynomial chaos expansion as control variate has
demonstrated a superior performance compared to the conventional polynomial chaos method. The
Chapter 4. Hybrid Monte Carlo / Polynomial Chaos Expansion FDTD Method 75
proposed CV-PCE method was implemented on a simple 1-D Bragg reflector structure with uncertain
slab permittivities, where appreciable reduction in the number of PCE terms and computation time
was achieved. Furthermore, the use of a lower order PCE allows CV-PCE to maintain a relatively mild
increment in computation requirements as the number of random variables increases, mitigating the
curse of dimensionality of PCE simulations.
Chapter 5
Conclusions
The increasing complexity of electromagnetic structures demands efficient uncertainty quantification
techniques in order to make critical assessments of the impact of variability in the performance of the
design. The PCE-FDTD method as it stands is one of the most versatile and efficient method in
electromagnetic engineering with the capability to directly model geometric and material uncertainties.
The work of this thesis further improves upon the state-of-the-art PCE-FDTD method in order to
improve its efficiency.
First, a convolutional perfectly matched layer is formulated for the termination of a domain boundary
with uncertain material parameters, within the framework of PCE-FDTD. This method is applied to a 2-
D domain filled with random electric permittivity. The statistical moments of the reflection error from the
CPML boundary are considerably smaller compared with those found from the previous Mur’s absorbing
boundary condition formulation of PCE-FDTD. The PCE-based CPML is applied to a microstrip circuit
with uncertain substrate permittivity and demonstrated excellent computational efficiency compared
with the Monte Carlo method. The absorber is an important element to problems of interest, such as
printed microwave circuit and antenna structures, which involve constitutive parameter uncertainty for
unbounded media. Hence, incorporating CPML into the PCE-FDTD method addresses an important
gap in the relevant literature, advancing the state of the art of PCE-FDTD ABCs to the state of the art
of ABCs for regular FDTD.
Second, a Monte Carlo/polynomial chaos hybrid method based on the control variate-based PCE
(CV-PCE) approach is formulated. This method is applied to Bragg reflectors with large numbers of
random parameter in the form of uncertain slab electric permittivities in order to evaluate the variance
in their transmission coefficients. The effectiveness of a low order PCE as control variate is studied and
76
Chapter 5. Conclusions 77
shows that the PCE must be adequately correlated with the exact solution in order to act as an effective
control variate. The control variate method have demonstrated a convergence rate no worse than the
conventional Monte Carlo method, regardless of the control variate used, as expected. This allows the
improvement of the accuracy of the computed statistical moments to be calibrated by increasing the
Monte Carlo iterations, as opposed to increasing the PCE order, at a great reduction in computation
time. For this method to be effective, however, a balance must be found between the order of the PCE
sufficient to produce an effective control variate, and the number of Monte Carlo iterations required.
This is in general problem-specific. However, in this work, a first order or second order PCE appears
to improve upon the conventional PCE-FDTD result with the addition of a few hundred Monte Carlo
iterations. The conventional PCE-FDTD, on the other hand, requires higher order of PCE and a much
larger simulation to achieve similar accuracy. Therefore, the control variate have shown to be capable of
to mitigating the effect of the “curse of dimensionalit” associated with PCE-FDTD for multi-parametric
uncertainty analysis.
5.1 Contributions
• Z. Gu, C. Sarris, “A Perfectly Matched Layer Absorber for the Termination of Random Media in
the Polynomial Chaos Expansion Based Finite-Difference Time-Domain (PCE-FDTD) Method,”
in IEEE Trans. Antennas Propag. (Submitted)
• Z. Gu, L. Wang and C. D. Sarris, “Multi-Parametric Uncertainty Quantification with a Hybrid
Monte-Carlo / Polynomial Chaos Expansion,” in IEEE Antennas Wireless Propag. Lett. (Submit-
ted).
• L. Wang, Z. Gu, A. Austin and C. D. Sarris, “A Comparative Study of FDTD-based Uncertainty
Quantification Methods,” in Antennas and Propagation Society International Symposium, 2014
IEEE, July 2014.
5.2 Future Work
The coupling of the Monte Carlo method with the polynomial chaos method not only mitigates the
effect of the “curse of dimensionalit”, it also opens up the possibility to make additional improvement in
efficiency by incorporating known acceleration methods for PCE-FDTD and Monte Carlo method. For
example, in addition to control variate, a variety of variance reduction techniques for accelerating Monte
Chapter 5. Conclusions 78
Carlo method also exist [37]. Importance sampling is one such variance reduction method, where the
Monte Carlo estimator is transformed into a form with a probability distribution such that the sampling
is biased to produce a better estimate. Latin hypercube sampling is another popular accelerated Monte
Carlo method which can be incorporated into the CV-PCE method. Since the control variate method
makes no assumption of how sampling is done, the incorporation of sampling-based acceleration method
for Monte Carlo will not conflict with the control variate.
Acceleration methods for PCE-FDTD are also under investigation. For example, adaptive-schemes
for PCE-FDTD based on multi-resolution expansion of the uncertainty space in terms of wavelets ba-
sis functions have been proposed [40]. The relative amplitude of the wavelet coefficients are used to
determine the wavelet order required. The control variate method can be used to relax the threshold
condition to use a smaller wavelet basis order, and improve the accuracy using Monte Carlo iterations.
With a plethora of uncertainty tools available to be applied at once, in theory, it is not infeasible
to attain an uncertainty quantification method based on the hybrid Monte Carlo / PCE-FDTD with a
considerable speed up over conventional PCE-FDTD. The amount of speedup we can achieve will require
further study.
Bibliography
[1] G. E. Moore, “Cramming more components onto integrated circuits,” Electronics, vol. 38, no. 8,
April 1965.
[2] K. J. Kuhn, M. D. Giles, D. Becher, P. Kolar, A. Kornfeld, R. Kotlyar, S. T. Ma, A. Maheshwari,
and S. Mudanai, “Process technology variation,” IEEE Trans. Electron Devices, vol. 58, no. 8,
August 2011.
[3] M. Miranda, “The threat of semiconductor variability,” IEEE Spectrum, June 2012.
[4] Variability Expedition. [Online]. Available: http://www.variability.org/research
[5] IPC, “IPC-2141 Controlled Impedance Circuit Boards and High-speed Logic Design.”
[6] J. Coonrod, “Understanding when to use FR-4 or high frequency laminates,” OnBoard Technology,
September 2011.
[7] M. Fiszer and J. Gruszczynski, “Nimrod MRA4 still in trouble,” Journal of Electronic Defense,
April 2003.
[8] BBC. Scrapping RAF Nimrods ’perverse’ say military chiefs. [Online]. Available:
http://www.bbc.com/news/uk-england-12294766
[9] AIAA, “Guide for the verification and validation of computational fluid dynamics simulations,”
Report no. g-077-1998, American Institute of Aeronautics and Astronautics, 1998.
[10] P. Nair, Numerical Methods for Uncertainty Quantification Lecture Notes, 2014.
[11] O. L. Maıtre and O. Knio, Spectral Methods for Uncertainty Fluid Dynamics. Springer, 2010.
[12] National Hurricane Center. Definition of the NHC track forecast cone. [Online]. Available:
http://www.nhc.noaa.gov/aboutcone.html
79
Bibliography 80
[13] T. Olsson and A. Koptioug, “Statistical analysis of antenna robustness,” IEEE Trans. Antennas
Propag., vol. 53, no. 1, pp. 566–568, January 2005.
[14] J. F. Swidzinski and K. Chang, “Nonlinear statistical modeling and yield estimation technique
for use in Monte Carlo simulations,” IEEE Trans. Microw. Theory Techn., vol. 48, no. 12, pp.
2316–2324, December 2000.
[15] J. M. M. Pantoja, C. Lin, M. Shaalan, J. L. Sebastian, and H. L. Hartnagel, “Monte Carlo simulation
of microwave noise temperature in cooled GaAs and InP,” IEEE Trans. Microw. Theory Techn.,
vol. 48, no. 7, pp. 1275–1279, July 2000.
[16] S. M. Smith and C. Furse, “Stochastic FDTD for analysis of statistical variation in electromagnetic
fields,” IEEE Trans. Antennas Propag., vol. 60, no. 7, pp. 3343–3350, July 2012.
[17] T. Tan, A. Taflove, and V. Backman, “Single realization stochastic FDTD for weak scattering waves
in biological random media,” IEEE Trans. Antennas Propag., vol. 61, no. 2, pp. 818–828, February
2013.
[18] I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: process variation
via polynomial chaos,” IEEE Trans. Electromagn. Compat., vol. 54, no. 1, pp. 140–148, February
2013.
[19] ——, “Parameters variability effects on multiconductor interconnects via hermite polynomial
chaos,” IEEE Trans. Compon. Packag. Manuf. Technol., vol. 1, no. 8, pp. 1234–1239, August
2011.
[20] R. S. Edwards, A. C. Marvin, and S. J. Porter, “Uncertainty analyses in the finite difference time
domain method,” IEEE Trans. Electromagn. Compat., vol. 52, no. 1, pp. 155–163, February 2010.
[21] A. C. M. Austin, N. Sood, J. Siu, and C. D. Sarris, “Application of polynomial chaos to quantify
uncertainty in deterministic channel models,” IEEE Trans. Antennas Propag., vol. 61, no. 11, pp.
5754–5761, November 2013.
[22] A. C. M. Austin and C. D. Sarris, “Efficient analysis of geometrical uncertainty in the FDTD method
using polynomial chaos with application to microwave circuit,” IEEE Trans. Microw. Theory Techn.,
vol. 61, no. 12, pp. 4293–4301, December 2013.
[23] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton
University Press, 2010.
Bibliography 81
[24] N. Wiener, “The homogeneous chaos,” Amer. J. Math, vol. 60, 1938.
[25] R. H. Cameron and W. T. Martin, “The orthogonal development of nonlinear functionals in a series
of Fourier-Hermite functionals,” Ann. of Math., vol. 48, 1947.
[26] D. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential
equations,” SIAM J. Sci. Comput., vol. 24, no. 2, 2002.
[27] C. Chauviere, J. S. Hesthaven, and L. Lurati, “Computational modeling of uncertainty in time-
domain electromagnetics,” SIAM J. Sci. Comput., vol. 28, no. 2, pp. 751–775, May 2004.
[28] J. Berenger, “Evanescent waves in PML’s: Origin of the numerical reflection in wave-structure
interaction problems,” IEEE Trans. Antennas Propag., vol. 47, pp. 1497–1503, October 1999.
[29] ——, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,”
IEEE Trans. Antennas Propag., vol. 44, pp. 110–117, January 1996.
[30] X. Wan and G. E. Karniadakis, “Solving elliptic problems with non-Gaussian spatially-dependent
random coefficients,” Comput. Methods Appl. Mech. Engrg, vol. 198, pp. 1985–1995, 2009.
[31] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain
Method. Artech House, 2005.
[32] S. D. Gedney, Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromag-
netics. Morgan & Claypool, 2011.
[33] J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computa-
tional Physics, vol. 114, pp. 185–200, 1994.
[34] W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified maxwell’s equa-
tions with stretched coordinates,” IEEE Microwave Guided Wave Lett., vol. 7, pp. 599–604, 1994.
[35] J. Berenger, “Numerical reflection from FDTD PMLs: A comparison of the split PML with the
unsplit and CFS PMLs,” IEEE Trans. Antennas Propag., vol. 50, pp. 258–265, 2002.
[36] J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of
the CFS-PML for arbitrary media,” Microwave Optical Tech. Lett, vol. 27, pp. 334–339, December
2000.
[37] W. L. Dunn and J. K. Shultis, Exploring Monte Carlo Methods. Elsevier, 2012.
Bibliography 82
[38] D. Sheen, S. Ali, M. Abouzahra, and J. Kong, “Application of the three-dimensional finite-difference
time-domain method to the analysis of planar microstrip circuits,” IEEE Trans. Microw. Theory
Techn., vol. 38, no. 7, 1990.
[39] D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods. Wiley, 2011.
[40] L. Wang and C. D. Sarris, “A multi-resolution FDTD method for uncertainty quantification in the
time-domain modeling of microwave structures,” IEEE MTT-S International Microwave Sympo-
sium, June 2014.