International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 9th Dec 2022, 12:35:50am CET
MS 27b: fast and high order solution techniques for boundary integral equations
Boundary integral equations for the solution of boundary value problems for partial differential
equations such as Stokes, Helmholtz, and (frequency-domain) Maxwell’s equations have a computa-
tional cost that scales linearly with respect to the number of boundary degrees of freedom. Due the
complexity of accurately discretizing and dealing with the resulting dense linear system, boundary
integral equations have not been computational tractable until recently. Key contributions to mak-
ing integral equations useful for large scale applications include the development of representations
that yield second-kind equations, techniques for handling complex geometries, fast algorithms, fast
direct solvers, and high order singular quadrature methods. The contributions highlighted in this
minisymposium represent some of the significant advances towards this effort.
As these methods are utilize high order discretizations and have the corresponding accuracy, cost
profile and capabilities, this session naturally fits into the ICOSAHOM program. Based on a small
but significant presence of integral equation researchers at the past ICOSAHOM in London, we hope
to deepen and grow the connection between the two communities. To this end, this minisymposium
brings together a highly international and predominantly junior group of researchers working at the
forefront of research on integral equation methods.
4:00pm - 4:30pm
Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects
Flatiron Institute, United States of America
A common use of boundary integral equation methods is for simulations involving a large number of relatively simple objects that may come close to each other. Examples include homogenization of material composites, complex fluids, sedimentation, etc. Such simulations are often dominated by the cost of a large number of near-singular potential evaluations. We show that precomputing an operator mapping surface density to an effective source representation renders this highly efficient in 2D or 3D, for the common case of smooth objects each needing a moderate number of unknowns (<10^4). The resulting scheme is kernel independent, needs only a smooth upsampled surface quadrature rule, and is drop-in compatible with FMM acceleration without near-vs-far bookkeepping.
4:30pm - 5:00pm
Numerical aspects of relative Krein spectral shift function in acoustic scattering and Casimir energy computation
1University College London, United Kingdom; 2University of Leeds, Leeds, United Kingdom
Computing the Casimir energy is a classical problem of quantum electrodynamics going back to the 1940s. In the early two thousands a significant breakthrough was achieved by representing the Casimir energy as the computation of the integral of the log-determinant of certain boundary integral operators in the complex plane [Reid et al]. In more recent work this log-determinant formula was investigated in the context of the Krein spectral shift function, which suggests a potential alternative computational approach based on the numerical evaluation of scattering matrices [Hanisch et. al.].
In this talk we will give an overview of these computational techniques and demonstrate a number of numerical experiments comparing these different approaches, and finally discuss future avenues of how to speed up Casimir computations for large-scale practical problems.
5:00pm - 5:30pm
Fast Multipole Methods for Continuous Charge Distributions
1Flatiron Institute/Simons Foundation, United States of America; 2New Jersey Institute of Technology, United States of America; 3New York University, United States of America; 4Cornell University, United States of America
Applications with continuous charge distributions or continuously varying material properties require the calculation of so-called volume integrals involving the Green's function of the governing PDE. These integrals can be both singular and nearly singular and thus require special quadrature. We present a method for generating such quadrature rules that is efficient enough to be done on-the-fly. We also discuss how these quadrature rules are incorporated in a fast multipole method and demonstrate some applications of the scheme to optical scattering problems.
5:30pm - 6:00pm
An adaptive discretization technique for boundary integral equations in the plane
1University of Colorado at Boulder, United States of America; 2University of Michigan, United States of America
Typically the discretization of integral equations on two dimensional complex geometries involves the use of a panel based quadrature (such as variants of Gaussian quadrature). The placement of the panels is often ad hoc and based on being able to integrate quantities such as arc-length and/or curvature to a desired accuracy. These quantities do not necessarily correspond to what is needed to achieve accuracy in the solution to a partial differential equation. Alternatively, a refinement strategy based on looking at relative error and wisely choosing which part of the geometry to refine can be done but this involves global solves which can be prohibitively expensive. In this talk, we will present an adaptive discretization technique which is guaranteed to achieve the desired accuracy and does not require the inversion of a full discretized integral equation at each step in the refinement process. Numerical results will illustrate the performance of the method.
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