(PHD, 2023)

Abstract:

This thesis presents a novel *Interpolated Factored Green Function* (IFGF) method for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of *O*(*N* log *N*) operations for an *N*-point surface mesh. The IFGF strategy capitalizes on slow variations inherent in a certain Green function *analytic factor*, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. In fact, a (hybrid MPI-OpenMP) parallel implementation of the IFGF algorithm is proposed in this thesis which results in highly efficient data communication, and which exhibits in practice excellent parallel scaling up to large numbers of cores – without any hard limitations on the number of cores concurrently employed with high efficiency. Moreover, on any given number of cores, the proposed parallel approach preserves the linearithmic (*O*(*N* log *N*)) computing cost inherent in the sequential version of the IFGF algorithm. This thesis additionally introduces a complete acoustic scattering solver that incorporates the IFGF method in conjunction with a suitable singular integration scheme. A variety of numerical results presented in this thesis illustrate the character of the proposed parallel IFGF-accelerated acoustic solver. These results include applications to several highly relevant engineering problems, e.g., problems concerning acoustic scattering by structures such as a submarine and an aircraft-nacelle geometry, thus establishing the suitability of the IFGF method in the context of real-world engineering problems. The theoretical properties of the IFGF method, finally, are demonstrated by means of a variety of numerical experiments which display the method’s serial and parallel linearithmic scaling as well as its excellent weak and strong parallel scaling – for problems of up to 4,096 wavelengths in acoustic size, and scaling tests spanning from 1 compute core to all 1,680 cores available in the High Performance Computing cluster used.

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(PHD, 2021)

Abstract:

This thesis focuses on the solution of causal, time-dependent wave propagation and scattering problems, in two- and three-dimensional spatial domains. This important and long-lasting problem has attracted a great deal of interest reflecting not only its use as a model problem but also the prevalence of wave phenomena in diverse areas of modern science, technology and engineering. Essentially all prior methods rely on “time-stepping” in one form or another, which involves local-in-time approximation of the evolution of the solution of the partial differential equation (PDE) based on the immediate time history and temporal finite-difference approximation. In addition to the need to manage the accumulation of (dispersion) error and the burdensome increase in computational cost over time, there are additionally difficult issues of stability, time-domain boundary conditions, and absorbing boundary conditions which often need to be addressed.

To sidestep many of these problems, this thesis develops a novel highly-efficient approach for time-dependent wave scattering problems employing the global-in-time techniques of Fourier transformation and leading to a frequency/time hybrid method for the time-dependent wave equation. Thus, relying on Fourier Transformation in time and utilizing a fixed (time-independent) number of frequency-domain solutions, the method evaluates the desired time-domain evolution with errors that both, decay faster than any negative power of the temporal sampling rate, and that, for a given sampling rate, are additionally uniform in time for all time. The fast error decay guarantees that high accuracies can be attained on the basis of relatively coarse temporal and frequency discretizations. The uniformity of the error for all time with fixed sampling rate, a property known as dispersionlessness, plays a crucial role, together with other properties of the Fourier transform, in enabling the evaluation of solutions for long times at *O*(1) cost. In particular, this thesis demonstrates the significant advantages enjoyed by the proposed methods over alternative approaches based on volumetric discretizations, time-domain integral equations, and convolution-quadrature.

The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider in detail the Laplace-based method, and only briefly point out its essential features and associated challenges.

The proposed frequency/time Fourier-transform methods for obstacle scattering problems are easily generalizable to any linear partial differential equation in the time domain for which frequency-domain solutions can readily be obtained, including e.g. the time-domain Maxwell equations, the linear elasticity equations, inhomogeneous and/or frequency-dependent dispersive media, etc. Further, the proposed approach can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping—that is, solution sampling at any given time *T* at *O*(1)-bounded sampling cost, for arbitrarily large values of *T*, and without requirement of evaluation of the solution at intermediate times. In particular, effective algorithms are introduced that, relying on use of time-asymptotics, compute two-dimensional solutions at *O*(1) cost despite the very slow time-decay that takes place in the two-dimensional case.

A significant portion of this thesis is devoted to a theoretical study of the validity of a certain stopping criterion used by the algorithm, which guarantees that certain field contributions can safely be neglected after certain stopping times. Roughly speaking, the theoretical results guarantee that, after the incident field is turned off, the magnitude of the future scattering density (and thus the magnitudes of the fields) can be estimated by the magnitude of the integral density *over a time period comparable to the time required by a wave to travel a distance equal to the diameter of the scatterer*. The criterion, which is crucial in ensuring the *O*(1) computational cost of the algorithm, is closely related to the well-known scattering theory developed in the 1960s and ’70s by Lax, Morawetz, Phillips, Strauss and others. Our approach to the decay problem is based on use of frequency-domain estimates (developed previously in the context of numerical analysis of frequency-domain problems) on integral operators in the high-frequency regime for obstacles of various trapping classes. In particular, our theory yields, for the first time, decay estimates for a class of connected trapping obstacles: all previous estimates of scattered-field decay for connected obstacles are restricted to nontrapping structures.

In all, the proposed approach leverages the power of the Fourier transformation together with a range of newly developed spectrally convergent numerical methods in both the frequency and time domain and a variety of novel theoretical results in the general area of scattering theory to produce a radically-new framework for the solution of time-dependent wave propagation and scattering problems.

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(PHD, 2020)

Abstract:

This thesis presents novel boundary integral equation (BIE) and associated optimization methodologies for photonic devices. The simulation and optimization of such structures is a vast and rapidly growing engineering area, which impacts on design of optical devices such as waveguide splitters, tapers, grating couplers, and metamaterial structures, all of which are commonly used as elements in the field of integrated photonics. The design process has been significantly facilitated in recent years on the basis of a variety of methods in computational electromagnetic (EM) simulation and design. Unfortunately, however, the expense required by previous simulation tools has limited the extent and complexity of the structures that can be treated. The methods presented in this thesis represent the results of our efforts towards accomplishing the dual goals of 1) Accurate and efficient EM simulation for general, highly-complex three-dimensional problems, and 2) Development of effective optimization methods leading to an improved state of the art in EM design.

One of the main proposed elements utilizes BIE in conjunction with a modified-search algorithm to obtain the modes of uniform waveguides with arbitrary cross sections. This method avoids spurious solutions by means of a certain normalization procedure for the fields within the waveguides. In order to handle problems including nonuniform waveguide structures, we introduce the windowed Green function (WGF) method, which used in conjunction with auxiliary integral representations for bound mode excitations, has enabled accurate simulation of a wide variety of waveguide problems on the basis of highly accurate and efficient BIE, in two and three spatial dimensions. The “rectangular-polar” method provides the basic high-order singular-integration engine. Based on non-overlapping Chebyshev-discretized patches, the rectangular-polar method underlies the accuracy and efficiency of the proposed general-geometry three-dimensional BIE approach. Finally, we introduce a three-dimensional BIE framework for the efficient computation of sensitivities — i.e. gradients with respect to design parameters — via adjoint techniques. This methodology is then applied to the design of metalenses including up to a thousand parameters, where the overall optimization process takes in the order of three hours using five hundred computing cores. Forthcoming work along the lines of this effort seeks to extend and apply these methodologies to some of the most challenging and exciting design problems in electromagnetics in general, and photonics in particular.

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(PHD, 2020)

Abstract:

This thesis presents a full-spectrum, well-conditioned, Green-function methodology for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies, where most existing methods break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell’s and elasticity equations. Ultimately, the thorough understanding of RW-anomalies this thesis provides yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings, and other related structures. In particular, the methods developed in this thesis can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations. A variety of computational examples are presented which demonstrate the flexibility of the overall approach, including, in particular, a problem of diffraction by a double-helix structure, for which numerical simulations did not previously exist, and for which the scattering pattern presented in this thesis closely resembles those obtained in crystallography experiments for DNA molecules.

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(PHD, 2017)

Abstract:

This thesis concerns development of efficient high-order boundary integral equation methods for the numerical solution of problems of acoustic and electromagnetic scattering in the presence of planar layered media in two and three spatial dimensions. The interest in such problems arises from application areas that benefit from accurate numerical modeling of the layered media scattering phenomena, such as electronics, near-field optics, plasmonics and photonics as well as communications, radar and remote sensing.

A number of efficient algorithms applicable to various problems in these areas are pre- sented in this thesis, including (i) A Sommerfeld integral based high-order integral equation method for problems of scattering by defects in presence of infinite ground and other layered media, (ii) Studies of resonances and near resonances and their impact on the absorptive properties of rough surfaces, and (iii) A novel *Window Green Function Method* (WGF) for problems of scattering by obstacles and defects in the presence of layered media. The WGF approach makes it possible to completely avoid use of expensive Sommerfeld integrals that are typically utilized in layer-media simulations. In fact, the methods and studies referred in points (i) and (ii) above motivated the development of the markedly more efficient WGF alternative.

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(PHD, 2016)

Abstract: This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain “Zaremba” boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

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(PHD, 2015)

Abstract: This thesis presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional spatial domains. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of higher-order backward differentiation formulae (BDF) and the alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. In fact this thesis presents, for the first time in the literature, high-order time-convergence curves for Navier-Stokes solvers based on the ADI strategy—previous ADI solvers for the Navier-Stokes equations have not demonstrated orders of temporal accuracy higher than one. An extended discussion is presented in this thesis which places on a solid theoretical basis the observed quasi-unconditional stability of the methods of orders two through six. The performance of the proposed solvers is favorable. For example, a two-dimensional rough-surface configuration including boundary layer effects at Reynolds number equal to one million and Mach number 0.85 (with a well-resolved boundary layer, run up to a sufficiently long time that single vortices travel the entire spatial extent of the domain, and with spatial mesh sizes near the wall of the order of one hundred-thousandth the length of the domain) was successfully tackled in a relatively short (approximately thirty-hour) single-core run; for such discretizations an explicit solver would require truly prohibitive computing times. As demonstrated via a variety of numerical experiments in two- and three-dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, useful stability properties, limited dispersion, and high parallel efficiency.

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(PHD, 2014)

Abstract: This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.

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(PHD, 2013)

Abstract:

Many important engineering problems, ranging from antenna design to seismic imaging, require the numerical solution of problems of time-domain propagation and scattering of acoustic, electromagnetic, elastic waves, etc. These problems present several key difficulties, including numerical dispersion, the need for computational boundary conditions, and the extensive computational cost that arises from the extremely large number of unknowns that are often required for adequate spatial resolution of the underlying three-dimensional space. In this thesis a new class of numerical methods is developed. Based on the recently introduced Fourier continuation (FC) methodology (which eliminates the Gibbs phenomenon and thus facilitates accurate Fourier expansion of nonperiodic functions), these new methods enable fast spectral solution of wave propagation problems in the time domain. In particular, unlike finite difference or finite element approaches, these methods are very nearly dispersionless—a highly desirable property indeed, which guarantees that fixed numbers of points per wavelength suffice to solve problems of arbitrarily large extent. This thesis further puts forth the mathematical and algorithmic elements necessary to produce highly scalable implementations of these algorithms in challenging parallel computing environments—such as those arising in GPU architectures—while preserving their useful properties regarding convergence and dispersion.

Additionally, this thesis develops a fast method for evaluation of computational boundary conditions which is based on Kirchhoff’s integral formula in conjunction with the FC methodology and an accelerated equivalent source integration method introduced recently for solution of integral equation problems. The combination of these ideas gives rise to a physically exact radiating boundary condition that is nonlocal but fast. The only known alternatives that provide all three of these features are only applicable to a highly restrictive class of domains such as spheres or cylinders, whereas the Kirchhoff-based approach considered here only requires a bounded domain with nonvanishing thickness. As is the case with the FC scattering solvers mentioned above, the boundary-conditions algorithm is modified into a formulation that admits efficient implementation in GPU and other parallel infrastructures.

Finally, this thesis illustrates the character of the newly developed algorithms, in both GPU and parallel CPU infrastructures, with a variety of numerical examples. In particular, it is shown that the GPU implementations result in thirty- to fiftyfold speedups over the corresponding single CPU implementations. An extension of the boundary-condition algorithm, further, is demonstrated, which enables for propagation of time-domain solutions over arbitrarily large spans of empty space at essentially null computational cost. Finally, a hybridization of the FC and boundary condition algorithm is presented, which is also part of this thesis work, and which provides an interface of the newly developed algorithms with legacy finite-element representations of geometries and engineering structures. Thus, combining spectral and classical PDE solvers and propagation methods with novel GPU and parallel CPU implementations, this thesis demonstrates a computational capability that enables solution, in novel computational architectures, of some of the most challenging problems in the broad field of computational wave propagation and scattering.

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(PHD, 2012)

Abstract:

This thesis presents a novel approach for the numerical solution of problems of diffraction by infinitely thin screens and apertures. The new methodology relies on combination of weighted versions of the classical operators associated with the Dirichlet and Neumann open-surface problems. In the two-dimensional case, a rigorous proof is presented, establishing that the new weighted formulations give rise to second-kind Fredholm integral equations, thus providing a generalization to open surfaces of the classical closed-surface Calderon formulae. High-order quadrature rules are introduced for the new weighted operators, both in the two-dimensional case as well as the scalar three-dimensional case. Used in conjunction with Krylov subspace iterative methods, these rules give rise to efficient and accurate numerical solvers which produce highly accurate solutions in small numbers of iterations, and whose performance is comparable to that arising from efficient high-order integral solvers recently introduced for closed-surface problems. Numerical results are presented for a wide range of frequencies and a variety of geometries in two- and three-dimensional space, including complex resonating structures as well as, for the first time, accurate numerical solutions of classical diffraction problems considered by the 19th-century pioneers: diffraction of high-frequency waves by the infinitely thin disc, the circular aperture, and the two-hole geometry inherent in Young’s experiment.

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(PHD, 2011)

Abstract: Advanced numerical solvers and associated simulation tools, such as, for example, numerical algorithms based on novel spectral methods, efficient time-stepping and domain meshing techniques for solution of Partial Differential Equations (PDEs) (enabling, in particular, effective resolution of extremely steep boundary layers in short computing times), can have a significant impact in the design of medical procedures. In this thesis we present three recently introduced numerical algorithms for medical problems whose performance improves significantly over those of earlier counterparts, and which can thereby provide solutions to a range of challenging computational problems for planning and design of medical treatments.

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(PHD, 2009)

Abstract: In this thesis we introduce an algorithm, based on the boundary integral equation method, for the numerical evaluation of singular solutions of the Laplace equation in three dimensional space, with singularities induced by a conical point on an otherwise smooth boundary surface. This is a model version of a fundamental problem in science and engineering: accurate evaluation of solutions of Partial Differential Equations in domains whose boundaries contain geometric singularities. For simplicity we assume a small region near the conical point coincides with a straight cone of given cross section; otherwise the boundary surface is not restricted in any way. Our numerical results demonstrate excellent convergence as discretizations are refined, even at the singular point where the solutions tend to infinity.

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(PHD, 2009)

Abstract:

A new methodology is introduced for the numerical solution of Partial Differential Equations in general spatial domains. The methodology is based on the use of the well-known Alternating Direction Implicit (ADI) approach of Peaceman and Rachford in conjunction with one-dimensional and high-order accurate Fourier representations of non-periodic data, obtained by way of a certain “continuation method” introduced recently for the resolution of the Gibbs phenomenon. We construct a number of high-order convergent PDE solvers on the basis of this strategy. Unlike previous alternating direction methods for general domains of order higher than one, the new algorithms possess the desirable property of unconditional stability for general spatial domains; the computational time required for these methods to advance one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points. In particular, the new methodology yields significant advantages over traditional low-order methods for computations involving wave propagation and “large domains,” as well as PDEs including diffusive terms. In all, we treat Dirichlet problems for the Heat Equation, the Poisson Equation, and the Wave Equation in two- and three-dimensional spatial domains with smooth boundaries. A stability analysis we present hinges upon the numerical evaluation of certain singular value decompositions. Numerical results arising from the implementations of two- and three-dimensional versions of the method for linear parabolic, hyperbolic and elliptic PDEs, exhibit unconditional stability and high-order convergence, in agreement with our theoretical results.

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(PHD, 2008)

Abstract:

We introduce a new second-kind integral equation method to solve direct rough surface scattering problems in two dimensions. This approach is based, in part, upon the bounded obstacle scattering method that was originally presented in Bruno et al. [2004] and is discussed in an appendix of this thesis. We restrict our attention to problems in which time-harmonic acoustic or electromagnetic plane waves scatter from rough surfaces that are perfectly reflecting, periodic and at least twice continuously differentiable; both sound-soft and sound-hard type acoustic scattering cases—correspondingly, transverse-electric and transverse-magnetic electromagnetic scattering cases—are treated. Key elements of our algorithm include the use of infinitely continuously differentiable windowing functions that comprise partitions of unity, analytical representations of the integral equation’s solution (taking into account either the absence or presence of multiple scattering) and spectral quadrature formulas. Together, they provide an efficient alternative to the use of the periodic Green’s function found in the kernel of most solvers’ integral operators, and they strongly mitigate the rapidly increasing computational complexity that is typically borne as the frequency of the incident field increases.

After providing a complete description of our solver and illustrating its usefulness through some preliminary examples, we rigorously prove its convergence. In particular, the super-algebraic convergence of the method is established for problems with infinitely continuously differentiable scattering surfaces. We additionally show that accuracies within prescribed tolerances are achieved with fixed computational cost as the frequency increases without bound for cases in which no multiple reflections occur.

We present extensive numerical data demonstrating the convergence, accuracy and efficiency of our computational approach for a wide range of scattering configurations (sinusoidal, multi-scale and simulated ocean surfaces are considered). These results include favorable comparisons with other leading integral equation methods as well as the non-convergent Kirchhoff approximation. They also contain analyses of sets of cases in which the major physical parameters associated with these problems (i.e., surface height, wavenumber and incidence angle) are systematically varied. As a result of these tests, we conclude that the proposed algorithm is highly competitive and robust: it significantly outperforms other leading numerical methods in many cases of scientific and practical relevance, and it facilitates rapid analyses of a wide variety of scattering configurations.

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(PHD, 2008)

Abstract:

In many engineering applications, scattering of acoustic or electromagnetic waves from a body of arbitrary shape is considered in an infinite medium. Solving the underlying partial differential equations with a standard numerical method such as finite elements or finite differences requires truncating the unbounded domain of definition into a finite computational region. As a consequence, an appropriate boundary condition must be prescribed at the artificial boundary. Many approaches have been proposed for this fundamental problem in the field of wave scattering. All of them fall into one of three main categories.

The first class of methods is based on mathematical approximations or physical heuristics. These boundary conditions are easy to implement and run in short computing times. However, these approaches give rise to spurious reflections at the artificial boundary, which travel back into the computational domain and corrupt the solution.

A second group consists of accurate and convergent methods. However, these formulations are usually harder to implement and often more expensive than the computation of the interior scheme itself.

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Finally, there are methods which are accurate and fast. The drawback of these approaches lies in the fact that the outer boundary must be taken to be either a sphere, a plane, or a cylinder. For many applications of interest, this may require use of a computational domain much larger than actually needed, which leads to an expensive overall numerical scheme.

This work introduces a new methodology in order to compute the fields at the artificial boundary. Like the second class of methods described above, the proposed algorithm is accurate and numerically convergent, yet its computational cost is less than the underlying portion of the volumetric calculation. And, unlike the third category, this new approach allows us to choose the artificial boundary to be arbitrarily close to the scatterer. This method is based on a novel concept of "equivalent source’ representations which allows a highly accurate and fast evaluation of the boundary condition.

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(PHD, 2004)

Abstract: Optical coherence tomography (OCT) is a non-invasive imaging technique based on the use of light sources exhibiting a low degree of coherence. Low coherence interferometric microscopes have been successful in producing internal images of thin pieces of biological tissue; typically samples of the order of 1 milimeter in depth have been imaged, with a resolution of the order of 10 to 20 microns in some portions of the sample. In this thesis, I deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. In detail, I formulate and solve an inverse problem which, using the interference fringes that result as the back-scattering of low-coherence light is made to interfere with a reference beam, produces maps detailing the values of the refractive index within the imaged sample. Unlike previous approaches to this imaging problem, the solver I introduce does not require processing at data collection time, and it can therefore produce solutions for inverse problems of multi-layered structures containing thousands of layers from back-scattering interference fringes only. We expect that the approach presented in this work, which accounts fully for the statistical nature of the coherence phenomenon, should prove of interest in the fields of medicine, biology and materials science.

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(PHD, 2003)

Abstract:

Accurate results for the inherently nonlinear models involving moving boundaries can only be produced by sophisticated, high-quality numerical algorithms. However the most general approaches are usually not very accurate, while those producing accurate results for certain cases are hard to generalize. In an attempt to bridge this gap, we propose a general method, based on a unified framework for arbitrary parabolic operators, which, in particular, can accurately treat singular problems.

Our method consists of front-fixing and a Chebyshev series solution of the resulting nonlinear partial differential equation. An appropriate set of convergent smooth approximations is used in singular cases. For smooth problems, our method is very competitive in both speed and accuracy. At the same time, our method is able to produce accurate solutions in the most general setting, whenever existence theorems for moving boundary problems hold. We establish convergence of numerical solutions to the true solution for a large class of possibly singular initial conditions.

In addition to the general method, we introduce computational techniques which enhance its performance for singular problems. These include derivative evaluations with Padé approximations; prior integration in time; and domain decomposition.

We demonstrate the performance of our method with several regular and singular problems. A comparison with other methods shows that our algorithms produce more accurate results. The additional techniques, which do not use smoothing approximations, significantly shorten computing times while retaining reasonable accuracy.

We present a systematic study of the mathematical finance problem of pricing American options on a dividend-paying asset from the point of view of partial differential equations. A symmetry result, obtained via a simple change of variables, allows to reduce any American option problem to one of the two canonical cases, depending on the relation between the interest rate and the dividend yield. Each of these cases is equivalent to a singular Stefan problem, which can be solved by our method. We present calculations for the classical problems of options written on a single stock and the more complicated examples, such as index options and foreign currency options, thus demonstrating the remarkable practical scope of the proposed approach.

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(PHD, 2003)

Abstract:

In this thesis, we introduce a new, fast, high-order method for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low (O(N log N)) complexity of our integral equation method is obtained through extensive use of the fast Fourier transform (FFT) in evaluating the required convolutions. Unlike previous FFT-based methods, however, this method yields high-order accuracy, even for scatterers containing geometric singularities such as discontinuities, corners, and cusps.

We begin our discussion with a thorough theoretical analysis of an efficient, high-order method recently introduced by Bruno and Sei (IEEE Trans. in Antenn. Propag., 2000), which motivated the present work. This two-dimensional method is based on a Fourier approximation of the integral equation in polar coordinates and a related, generally low-order, Fourier smoothing of the scatterer. The claim that use of this low-order approximation of the scatterer leads to a high-order accurate numerical method generated considerable controversy. Our proofs establish that this method indeed yields high-order accurate solutions. We also introduce several substantial improvements to the numerical implementation of this two-dimensional algorithm, which lead to increased numerical stability with decreased computational cost.

We then present our new, fast, high-order method in three dimensions. An immediate generalization of the polar coordinate approach in two dimensions to a spherical coordinate approach in three dimensions appears less advantageous than our chosen approach: Fourier approximation and integration in Cartesian coordinates. To obtain smooth and periodic functons (which are approximated to high-order via Fourier series), we 1) decompose the Green’s function into a smooth part with infinite support and a singular part with compact support; and 2) replace, as in the two-dimensional approach, the (possibly discontinuous) scatterer with its truncated Cartesian Fourier series.

The accuracy of our three-dimensional method is approximately equal to that of the two-dimensional method mentioned above and, interestingly, is actually much simpler than the two-dimensional approach. In addition to our theoretical discussion of these high-order methods, we present a parallel implementation of our three-dimensional Cartesian approach. The efficiency, high-order accuracy, and overall performance of both the polar and Cartesian methods are demonstrated through several computational examples.

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(PHD, 2002)

Abstract: In endocrine glands, vigorous and coordinated responses are often elicited by modest changes in the concentration of the organist molecule. The mammalian parathyroid gland is a representative case. Small (5%) changes in serum calcium result in tenfold (1000%) changes in glandular parathyroid hormone (PTH) release. In vitro, single isolated cells are observed to secrete fewer hormones than cells residing within a connected group, suggesting that a network has emergent regulatory properties. In PTH secreting tumors however, the ability to quickly respond to changes in calcium is strongly damped. A unifying hypothesis that accounts for these phenomena is realized by extra-cellular modulation of calcium diffusivity. A theoretical model and computational experiments demonstrate qualitative agreement with published experimental results. Our results suggest that in addition to the cellular mechanisms, endocrine glandular networks may have regulatory prowess at the level of interstitial transport. The extra-cellular diffusional mechanism proposed provides a consistent argument for 1) higher secretion of single cells in a connected network compared to isolated cells, 2) the rapid nonlinear response seen in healthy glands as well as 3) the pathological responses seen in hyperplasia and adenoma. Since the proposed diffusional regulation strongly depends on the existence of a connected cell network (gland), it also suggests a rationale for the advantages of cell networks as organs versus a dispersed system of isolated cells (in the case of the parathyroid gland).

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