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SUMMARY:Density interpolation methods for volume Integral equations - Carl
 os Perez-Arancibia (Universiteit Twente)
DTSTART:20230418T134500Z
DTEND:20230418T143000Z
UID:TALK198721@talks.cam.ac.uk
DESCRIPTION:\n\n\nVolume potentials (VPs) are essential tools to solve and
  analyze partial differential equations (PDEs). These operators provide &r
 dquo\;inverses&rdquo\; of linear elliptic partial differential operators w
 ith known fundamental solutions\, making it possible to recast linear and 
 non-linear PDEs as simpler (sometimes derivative-free) equations that are 
 easier to solve and analyze. Although VPs have broad applicability\, their
  computation has been relatively neglected in the context of complex geome
 tries until recently. This neglect is due to several challenges that must 
 be addressed to efficiently and accurately evaluate these operators\, incl
 uding the singular nature of the fundamental solution\, nearly singular in
 tegrals\, and high computational costs associated with slow kernel decay a
 nd dense-matrix computations.\nThis talk outlines a novel class of high-or
 der methods for the efficient numerical evaluation of Helmholtz VPs define
 d by volume integrals over complex geometries. Inspired by the Density Int
 erpolation Method (DIM) for boundary integral operators\, the proposed met
 hodology leverages Green&rsquo\;s third identity and a local polynomial in
 terpolation of the density function to recast a given VP as a linear combi
 nation of surface-layer potentials and a volume integral with a regularize
 d (bounded or smoother) integrand. The layer potentials can be accurately 
 and efficiently evaluated inside and outside the integration domain using 
 existing methods (e.g. DIM)\, while the regularized volume integral can be
  accurately evaluated by applying elementary quadrature rules to integrate
  over structured or unstructured domain decompositions without local numer
 ical treatment at and around the kernel singularity. The proposed methodol
 ogy is flexible\, easy to implement\, and fully compatible with well-estab
 lished fast algorithms such as the Fast Multipole Method and H-matrices\, 
 enabling VP evaluations to achieve linearithmic computational complexity. 
 To demonstrate the merits of the proposed methodology\, we applied it to t
 he Nystr&ouml\;m discretization of the Lippmann-Schwinger volume integral 
 equation for frequency-domain scattering problems in piecewise-smooth vari
 able media.\nThis is joint work with Thomas G. Anderson (U. Michigan)\, Ma
 rc Bonnet (POEMS lab/CNRS/INRIA/ENSTA Paris)\, and Luiz M. Faria (POEMS la
 b/INRIA/ENSTA Paris).\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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