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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Density interpolation methods for volume Integral
equations - Carlos Perez-Arancibia (Universiteit T
wente)
DTSTART;TZID=Europe/London:20230418T144500
DTEND;TZID=Europe/London:20230418T153000
UID:TALK198721AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/198721
DESCRIPTION:\n\n\nVolume potentials (VPs) are essential tools
to solve and analyze partial differential equation
s (PDEs). These operators provide &rdquo\;inverses
&rdquo\; of linear elliptic partial differential o
perators with known fundamental solutions\, making
it possible to recast linear and non-linear PDEs
as simpler (sometimes derivative-free) equations t
hat are easier to solve and analyze. Although VPs
have broad applicability\, their computation has b
een relatively neglected in the context of complex
geometries until recently. This neglect is due to
several challenges that must be addressed to effi
ciently and accurately evaluate these operators\,
including the singular nature of the fundamental s
olution\, nearly singular integrals\, and high com
putational costs associated with slow kernel decay
and dense-matrix computations.\nThis talk outline
s a novel class of high-order methods for the effi
cient numerical evaluation of Helmholtz VPs define
d by volume integrals over complex geometries. Ins
pired by the Density Interpolation Method (DIM) fo
r boundary integral operators\, the proposed metho
dology leverages Green&rsquo\;s third identity and
a local polynomial interpolation of the density f
unction to recast a given VP as a linear combinati
on of surface-layer potentials and a volume integr
al with a regularized (bounded or smoother) integr
and. The layer potentials can be accurately and ef
ficiently evaluated inside and outside the integra
tion domain using existing methods (e.g. DIM)\, wh
ile the regularized volume integral can be accurat
ely evaluated by applying elementary quadrature ru
les to integrate over structured or unstructured d
omain decompositions without local numerical treat
ment at and around the kernel singularity. The pro
posed methodology is flexible\, easy to implement\
, and fully compatible with well-established fast
algorithms such as the Fast Multipole Method and H
-matrices\, enabling VP evaluations to achieve lin
earithmic computational complexity. To demonstrate
the merits of the proposed methodology\, we appli
ed it to the Nyströ\;m discretization of the L
ippmann-Schwinger volume integral equation for fre
quency-domain scattering problems in piecewise-smo
oth variable media.\nThis is joint work with Thoma
s G. Anderson (U. Michigan)\, Marc Bonnet (POEMS l
ab/CNRS/INRIA/ENSTA Paris)\, and Luiz M. Faria (PO
EMS lab/INRIA/ENSTA Paris).\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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