Conference Agenda

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).

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Session Overview
Session
MS 34: non-local operators for scattering problems: analysis and low-rank approximation
Time:
Friday, 16/July/2021:
12:00pm - 2:00pm

Session Chair: Steffen Börm
Session Chair: Jens Markus Melenk
Session Chair: Stefan Sauter
Virtual location: Zoom 4


Session Abstract

A major bottleneck for the modern numerical treatment of scattering problems, in particular in a

time-harmonic setting, lies in the approximation of non-local operators. Typical applications are

boundary integral equations, solution operators for finite element discretization (either realized by

an (iterative) "solver" or by a data-sparse approximation of the discrete solution operator). In

contrast to the low-frequency case, the analytical properties of these operators are much more

complex due, for example, to the strong impact material properties and the geometry may have. In

turn, the efficient numerical treatment of such operators is challenging.

The minisymposium brings together specialists working in analysis and in numerics to discuss recent

advances in the understanding of the analytical properties of non-local operators arising for

scattering problems, the potential to represent discretized operators in data-sparse formats, and

their algorithm realization.


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Presentations
12:00pm - 12:30pm

Mathematical Studies of Extraordinary Light Transmission through Subwavelength Holes in metallic slab

Hai Zhang

HKUST, Hong Kong S.A.R. (China)

Since the discovery of extraordinary optical transmission through nanohole arrays

in metallic films by Ebbesen, a wealth of research has been sparked in the experimental and

theoretical investigation of the transmission enhancement in subwavelength nanostructures. In this

talk, using two-dimensional periodic slits as a prototype, I will present mathematical studies of the

transmission enhancement in such subwavelength structures. Based upon the layer potential

technique, asymptotic analysis, different types of enhancement mechanism are unveiled, which

includes non-resonance effect, resonance effect and surface waves.



12:30pm - 1:00pm

Matrix compression for the high-frequency Helmholtz BEM

Steffen Börm, Christina Börst

Christian-Albrechts-Universität zu Kiel, Germany

Fast algorithms for boundary element matrices frequently rely on low-rank approximations of submatrices that are frequently derived via polynomial approximation of the kernel function, e.g., by interpolation or quadrature. In the case of the high-frequency Helmholtz equation, the rapid oscillations of the kernel function can be taken care of by splitting the kernel function into a plane wave that can be written as a tensor product and a smooth part that can be approximated by standard techniques.

The resulting approximations are only accurate in subdomains, and in order to guarantee sufficiently fast convergence, the number of subdomains has to be quite large. We can reduce the computational complexity by employing a multi-level representation for the subdomains, but this introduces additional approximation errors that have to be handled appropriately.

Our contribution gives an overview of the resulting matrix approximation and the corresponding error analysis.



1:00pm - 1:30pm

Nearly linear scaling matvec and preconditioners for highly oscillatory kernels

Haizhao Yang

Purdue University, United States of America

This talk introduces a nearly linear scaling unified framework for evaluating the matvec g=Kf, where K is the discretization of an oscillatory integral transform g(x)=\int K(x,y)f(y)dy with a kernel function K(x,y)=a(x,y)e^{2\pi i P(x,y)}, where a(x,y) is a smooth amplitude function, and P(x,y) is a piecewise smooth phase function with O(1) discontinuous points in x and y. This unified framework is based on either the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an O(N) fast algorithm to determine whether NUFFT or BF is more suitable.



1:30pm - 2:00pm

On coercivity of combined boundary integral equations in high-frequency scattering by smooth non-convex obstacles

Valery P SMYSHLYAEV

University College London (UCL), United Kingdom

Coercivity properties play an important role in numerical analysis of Galerkin methods for combined boundary integral equations in surface scattering. It was shown by E. Spence, I. Kamotski and the author (CPAM 2015) that a wave-number $k$-uniform coercivity holds for high-frequency acoustic scattering by smooth uniformly convex Dirichlet scatterers, which in particular completed the numerical analysis of high frequency numerical-asymptotic BEMs for the case of such convex obstacles. Little is known however on whether the coercivity may still hold for any non-convex obstacles. An example of absence of a $k$-uniform coercivity was given by Chandler-Wilde, Spence, Gibbs and the author (SIMA 2020) for a non-trapping obstacle although trapping in a certain generalised sense.

We show in this talk that no $k$-uniform coercivity can in fact hold for any smooth non-convex 2-D obstacles. The construction essentially mimics a whispering gallery high-frequency asymptotic mode along a concave part of the boundary. A numerical evidence (Betcke & Spence, SINUM 2011) may suggest that coercivity, at high frequencies, still holds for some “strongly non-trapping" (non-convex) domains like e.g. smooth star-shaped ones, although the coercivity constant would then mildly degenerate as $k$ tends to infinity. (This is in stark contrast with the bounds for the inverse operator which are known to be $k$-uniform for non-trapping obstacles.) A key for determining the rate of such a degeneracy may be held by the problem of whispering gallery wave scattering by a boundary inflection. The latter is a canonical diffraction problem with some history, see https://arxiv.org/abs/2103.04734 (to appear in “St Petersburg Mathematical Journal”, 2021) for a review and some recent advances.



 
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