A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

A complex-scaled boundary integral equation for time-harmonic water waves

This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent

Learned infinite elements for helioseismology

This thesis presents efficient techniques for integrating the information contained in the Dirichlet-to-Neumann (DtN) map of time-harmonic waves propagating in a stratified medium into finite element discretizations. This task arises in the context of domain decomposition methods, e.g. when reducing a problem posed on an unbounded domain to a bounded computational domain on which the problem can then be discretized. Our focus is on stratified media like the Sun, that allow for strong reflection of waves and for which suitable methods are lacking. We present learned infinite elements as a possible approach to deal with such media utilizing the assumption of a separable geometry. In this case, the DtN map is separable, however, it remains a non-local operator with a dense matrix representation, which renders its direct use computationally inefficient. Therefore, we approximate the DtN only indirectly by adding additional degrees of freedom to the linear system in such a way that the Schur complement w.r.t. the latter provides an optimal approximation of DtN and sparsity of the linear system is preserved. This optimality is ensured via the solution of a small minimization problem, which incorporates solutions of one-dimensional time-harmonic wave equations and allows for great flexibility w.r.t. properties of the medium. In the first half of the thesis we provide an error analysis of the proposed method in a generic framework which demonstrates that exponentially fast convergence rates can be expected. Numerical experiments for the Helmholtz equation and an in-depth study on modelling the solar atmosphere with learned infinite elements demonstrate the high accuracy and flexibility of the proposed method in practical applications. In the second half of the thesis, the potential of learned infinite elements in the context of sweeping preconditioners for the efficient iterative solution of large linear systems is investigated. Even though learned infinite elements are very suitable for separable media, they can only be used for tiny perturbations thereof since the corresponding DtN maps turn out to be extremely sensitive to perturbations in the presence of strong reflections.2021-12-2

A model for calculating EM field in layered medium with application to biological implants

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Modern wireless telecommunication devices (GSM Mobile system) (cellular telephones and wireless modems on laptop computers) have the potential to interfere with implantable medical devices/prostheses and cause possible malfunction. An implant of resonant dimensions within a homogeneous dielectric lossy sphere can enhance local values of SAR (the specific absorption rate). Also antenna radiation pattern and other characteristics are significantly altered by the presence of the composite dielectric entities such as the human body. Besides, the current safety limits do not take into account the possible effect of hot spots arising from metallic implants resonant at mobile phone frequencies. Although considerable attention has been given to study and measurement of scattering from a dielectric sphere, no rigorous treatment using electromagnetic theory has been given to the implanted dielectric spherical head/cylindrical body. This thesis aims to deal with the scattering of a plane electromagnetic wave from a perfectly conducting or dielectric spherical/cylindrical implant of electrically small radius (of resonant length), embedded eccentrically into a dielectric spherical head model. The method of dyadic Green's function (DGF) for spherical vector wave functions is used. Analytical expressions for the scattered fields of both cylindrical and spherical implants as well as layered spherical head and cylindrical torso models are obtained separately in different chapters. The whole structure is assumed to be uniform along the propagation direction. To further check the accuracy of the proposed solution, the numerical results from the analytical expressions computed for the problem of implanted head/body are compared with the numerical results from the Finite-Difference Time-Domain (FDTD) method using the EMU-FDTD Electromagnetic simulator. Good agreement is observed between the numerical results based on the proposed method and the FDTD numerical technique. This research presents a new approach, away from simulation work, to the study of exact computation of EM fields in biological systems. Its salient characteristics are its simplicity, the saving in memory and CPU computational time and speed.Cochlear UK Limited and EPSR

On the constant constitutive parameter (e.g., mass density) assumption in integral equation approaches to (acoustic) wave scattering

In 2D acoustic and elastodynamic problems the spatial variability of a constitutive parameter such as the mass density makes it difficult to employ boundary integral and domain integral techniques to solve the forward and inverse wave scattering problems. The oft-employed method for avoiding this problem is to assume this constitutive parameter (which is chosen herein to be the mass density) to be spatially-invariant throughout all space. The reliability of this assumption is evaluated both theoretically and numerically and it is shown, in the example of a canonical-shaped scattering obstacle, that the scattered field can be obtained in the form of a series of powers of the mass density contrast (the latter vanishing for constant mass density). The first term of this series is the solution for the scattered field corresponding to the constant density assumption and it is shown that taking into account only two more terms in the series enables to correct for practically all the errors incurred by the constant mass density assumption for a wide range of the other constitutive parameters and frequencies. It is shown how to apply this result for obstacles of non-canonical shape

Scattering of electromagnetic waves by two- and three-dimensional dielectric bodies

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