ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
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).
Please note that all times are shown in the time zone of the conference. The current conference time is: 28th Nov 2022, 08:29:32pm CET

Session Overview 
Session  
MS 34: nonlocal operators for scattering problems: analysis and lowrank approximation
 
Session Abstract  
A major bottleneck for the modern numerical treatment of scattering problems, in particular in a timeharmonic setting, lies in the approximation of nonlocal operators. Typical applications are boundary integral equations, solution operators for finite element discretization (either realized by an (iterative) "solver" or by a datasparse approximation of the discrete solution operator). In contrast to the lowfrequency 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 nonlocal operators arising for scattering problems, the potential to represent discretized operators in datasparse formats, and their algorithm realization.  
Presentations  
12:00pm  12:30pm
Mathematical Studies of Extraordinary Light Transmission through Subwavelength Holes in metallic slab 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 twodimensional 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 nonresonance effect, resonance effect and surface waves. 12:30pm  1:00pm
Matrix compression for the highfrequency Helmholtz BEM ChristianAlbrechtsUniversität zu Kiel, Germany Fast algorithms for boundary element matrices frequently rely on lowrank 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 highfrequency 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 multilevel 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 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 nonuniform 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 highfrequency scattering by smooth nonconvex obstacles 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 wavenumber $k$uniform coercivity holds for highfrequency acoustic scattering by smooth uniformly convex Dirichlet scatterers, which in particular completed the numerical analysis of high frequency numericalasymptotic BEMs for the case of such convex obstacles. Little is known however on whether the coercivity may still hold for any nonconvex obstacles. An example of absence of a $k$uniform coercivity was given by ChandlerWilde, Spence, Gibbs and the author (SIMA 2020) for a nontrapping 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 nonconvex 2D obstacles. The construction essentially mimics a whispering gallery highfrequency 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 nontrapping" (nonconvex) domains like e.g. smooth starshaped 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 nontrapping 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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