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.

ii

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

v

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

vii

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

viii

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

ix

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


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


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.


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.


“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.


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


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


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


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


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


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


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


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.


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


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


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)


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


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)


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)


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)


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→∞

PDF

|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.


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)


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


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


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


[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.


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)


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)


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


+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


+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)


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


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


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


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)


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)


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


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.


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.


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


(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.


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


(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.


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


(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.


|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.


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


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.


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.


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.


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.


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.


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)


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


+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


(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)


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


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;


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.


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].


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].


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 .


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


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


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


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)


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


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,


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


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


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.


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



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.


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,


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


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.

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Documents

Enhancing the Efficiency of the Polynomial Chaos Expansion