Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique

Inverse Problems in Wave Scattering

The workshop treated inverse problems for partial differential equations, especially inverse scattering problems, and their applications in technology. While special attention was paid to sampling methods, decomposition methods, Newton methods and questions of unique determination were also investigated

Research in applied mathematics, numerical analysis, and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering (ICASE) in applied mathematics, numerical analysis, and computer science is summarized and abstracts of published reports are presented. The major categories of the ICASE research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers

A robust and high precision algorithm for elastic scattering problems from cornered domains

The Navier equation is the governing equation of elastic waves, and computing its solution accurately and rapidly has a wide range of applications in geophysical exploration, materials science, etc. In this paper, we focus on the efficient and high-precision numerical algorithm for the time harmonic elastic wave scattering problems from cornered domains via the boundary integral equations in two dimensions. The approach is based on the combination of Nystr\"om discretization, analytical singular integrals and kernel-splitting method, which results in a high-order solver for smooth boundaries. It is then combined with the recursively compressed inverse preconditioning (RCIP) method to solve elastic scattering problems from cornered domains. Numerical experiments demonstrate that the proposed approach achieves high accuracy, with stabilized errors close to machine precision in various geometric configurations. The algorithm is further applied to investigate the asymptotic behavior of density functions associated with boundary integral operators near corners, and the numerical results are highly consistent with the theoretical formulas

Inverse Problems in Wave Scattering and Impedance Tomography

[no abstract available

Dimer Models and Conformal Structures

Dimer models have been the focus of intense research efforts over the last years. This paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere. Our classification of the geometries of frozen boundaries can be seen as geometric universality result for all dimer models. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realised already by the lozenge model. To achieve this we present a new boundary regularity study for a class of Monge-Amp\`ere equations in non-strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers. Our results are related to the seminal paper "Limit shapes and the complex Burgers" equation where R. Kenyon and A. Okounkov studied the asymptotic height function in the special class of lozenge tilings and domains. Part of the motivation for development of the new methods in this paper is that it seems difficult to extend those methods to cover more general dimer models, in particular domino tilings, as we do in the present paper. Indeed, our methods prove new and sharper results already for the lozenge model.Comment: 98 pages. Typos corrected and some clarifications adde

Numerical aspects of enriched and high-order boundary element basis functions for Helmholtz problems.

In this thesis several aspects of the Partition of Unity Boundary Element Method (PUBEM) are investigated, with novel results in three main areas: 1. Enriched modelling of wave scattering from polygonal obstacles. The plane waves are augmented by a set of enrichment functions formed from fractional order Bessel functions, as informed by classical asymptotic solutions for wave fields in the vicinity of sharp corners. It is shown that the solution accuracy can be improved markedly by the addition of a very small number of these enrichment functions, with very little effect on the run time. 2. High-order formulations. Plane waves are not the only effective means of introducing oscillatory approximation spaces. High-Order Lagrange polynomials and high-order Non-Uniform Rational B-Splines (NURBS) also exhibit oscillation and these are tested and compared against PUBEM. It is found that these high-order functions significantly outperform the corresponding low-order (typically quadratic) polynomials and NURBS that are commonly used, and that for large problems the highest order tested (11th) has potential to be competitive with PUBEM without the associated ill-conditioning. 3. Integration. The accuracy of PUBEM traditionally comes at the cost of the requirement to evaluate many highly-oscillatory integrals. Several candidate integration strategies are investigated with the aim of find- ing a robust, accurate and efficient approach. Schemes tested include the Filon and asymptotic methods, as well as the Method of Stationary Phase (MSP). Although these schemes are found to be spectacularly successful for many cases, they fail for a sufficient number of situations to cause a complete PUBEM analysis based on these methods to lack robustness. Conclusions are drawn about the effective use of more traditional quadrature for robust implementations