arXiv subjects

Thomas G. Anderson

Publications and source records attributed to Thomas G. Anderson.

8 recordsLinked to original sources

arXiv2026

Fast high-order solvers for the Lippmann--Schwinger equation in piecewise-smooth heterogeneous media

This article presents a fast, high-order Nyström solver for the two-dimensional Lippmann--Schwinger equation arising from time-harmonic scattering by penetrable, piecewise-smooth heterogeneous media. Relying on high-order evaluation of the Newtonian potential on unstructured grids adapted to interfaces of discontinuity, the methodology achieves high-order accuracy using existing fast algorithms such as the fast multipole method. As an iterative method the solver exhibits rapid convergence when coupled to a preconditioning strategy that exploits a class of structured-grid solvers---fast solution methods offering quasi-linear time and memory complexity and nearly-constant iteration counts, but long limited in accuracy. The preconditioning strategy couples the Nyström discretization---given by high-order quadrature nodes over a (curved) unstructured mesh conforming to the support of the spatially varying contrast---to a uniform Cartesian grid underlying the fast preconditioner via a pair of transfer operators. The resulting preconditioner inherits the frequency-robust behavior of its Cartesian counterpart without sacrificing the geometric flexibility and high-order accuracy of the unstructured discretization. We prove that invertibility of the proposed preconditioner holds under explicit conditions on the mesh sizes and on the Cartesian preconditioner. Numerical experiments demonstrate that the preconditioned system requires significantly fewer GMRES iterations than its unpreconditioned counterpart, with iteration counts almost independent of mesh size and wavenumber, and illustrate the method's robustness for inhomogeneities with piecewise-smooth refractive indices and jump discontinuities across interfaces.

math.NA↗

arXiv2026

A general-purpose global regularization method for 3D volume integral operators

Singular volume integral operators associated with constant-coefficient partial differential operators extend the applicability of potential theory to inhomogeneous problems, for example arising from nonlinearities or variable coefficients. Typically the PDE kernels in these operators give rise to singularities at all $\mathcal{O}(1/h^3)$ volume discretization/evaluation points in a mesh of characteristic size $h$, while the slowly-decaying nature of such kernels give rise to long-range interactions that require coupling to fast summation algorithms. The presented method uses Green's identities to regularize a wide variety of both scalar-valued and vector-valued volume integral operators by use of a certain regularizing volume density interpolant. The analysis shows how the regularizing effect of the interpolant is global in the sense that the interpolation quality increases in an exactly compensatory fashion as the distance to the Green's function singularity decreases. High-order convergence estimates with tabulated simplex quadratures are established, including with exact representation of curved domains.

math.NA↗

arXiv2024

Can Efficient Fourier-Transform Techniques Favorably Impact on Broadband Computational Electromagnetism?

In view of recently demonstrated joint use of novel Fourier-transform techniques and effective high-accuracy frequency domain solvers related to the Method of Moments, it is argued that a set of transformative innovations could be developed for the effective, accurate and efficient simulation of problems of wave propagation and scattering of broadband, time-dependent wavefields. This contribution aims to convey the character of these methods and to highlight their applicability in computational modeling of electromagnetic configurations across various fields of science and engineering.

physics.comp-ph↗

arXiv2024

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.

math.NA↗

arXiv2023

Construction of polynomial particular solutions of linear constant-coefficient partial differential equations

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, \texttt{ElementaryPDESolutions.jl}, which implements the proposed methodology in an efficient and user-friendly format.

math.NA↗

arXiv2022

"Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation

This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--including certain types of "trapping" obstacles whose periodic trapped orbits span a set of positive volumetric measure, and for which no previous fast-decay theory was available. The results, which do not rely on consideration of the Lax-Phillips complex-variables scattering framework and associated resonance-free regions in the complex plane, utilize only real-valued frequencies, and follow from use of Green functions and boundary integral equation representations in the frequency and time domains, together with a certain $q$-growth condition on the frequency-domain operator resolvent.

math.AP↗

arXiv2022

A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to line integrals bounding a reference volume cell using cell mappings and elements of the Poincaré lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. The scheme achieves compatibility with fast multipole methods (FMMs) and thereby optimal asymptotic complexity by coupling global rules for target-independent quadrature of smooth functions to local target-dependent singular quadrature corrections, and it relies on orthogonal polynomial systems on each cell for well-conditioned, high-order and efficient (with respect to number of required volume function evaluations) approximation of arbitrary volumetric sources. Our domain discretization scheme is naturally compatible with standard meshing software such as Gmsh, which are employed to discretize a narrow region surrounding the domain boundaries. We present 8th-order accurate results, demonstrate the success of the method with examples showing up to 12-digit accuracy on complex geometries, and, for static geometries, our numerical examples show well over $99\%$ of evaluation time of the particular solution is spent in the FMM step.

math.NA↗

arXiv2020

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: $\mathcal{O}(1)$ Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. 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 the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it 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 $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.

math.NA↗