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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Analysis of oversampled collocation methods for wa
ve scattering problems - Georg Maierhofer (Sorbonn
e Université)
DTSTART;TZID=Europe/London:20230209T090000
DTEND;TZID=Europe/London:20230209T094500
UID:TALK194794AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/194794
DESCRIPTION:Collocation methods for boundary integral formulat
ions of wave scattering problems are often simpler
and cheaper to implement than Galerkin methods be
cause the elements of the discretisation matrix ar
e given by lower-dimensional integrals. However\,
in general\, their application is a delicate matte
r: in contrast to Galerkin methods\, there is no s
tandardised convergence theory for collocation met
hods\, and their success is highly sensitive to th
e choice of collocation points. Moreover\, in the
integral equation setting\, collocation methods ty
pically lead to slower convergence rates than Gale
rkin methods.\nIn this talk\, we will explore the
extent to which the convergence properties of coll
ocation methods for Fredholm integral equations ca
n be improved by least-squares oversampling. We pr
ovide rigorous analysis to show that superlinear o
versampling can enhance the convergence rates of t
he collocation method and reduce its sensitivity t
o the distribution of collocation points. In addit
ion\, we prove that linear oversampling can still
lead to a substantial improvement in the error con
stant\, even though the asymptotic convergence rat
e is not improved. Indeed\, we will see an example
where oversampling by a constant factor leads to
an improvement of the error at a cubic rate in thi
s constant\, whilst incurring only a linear increa
se in cost. We support our analysis with several n
umerical examples for the two-dimensional Helmholt
z equation. This is joint work with Daan Huybrechs
(KU Leuven).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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