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SUMMARY:Applying GMRES to the Helmholtz equation with strong trapping: how
does the number of iterations depend on the frequency?
DTSTART:20230620T120000Z
DTEND:20230620T130000Z
DTSTAMP:20231002T214800Z
UID:indico-event-8871@indico.math.cnrs.fr
DESCRIPTION:Speakers: Pierre Marchand (Inria)\n\nWe are interested in solv
ing scattering problems with strong trapping using the Combined Field Int
egral Equation (CFIE) and the Generalized Minimal Residual method (GMRes).
In this talk\, we show a new understanding of how the number of GMRes it
erations depends on frequency in this situation. The non-normal nature of
CFIE makes GMRes the iterative method of choice for solving linear syste
ms stemming from its discretisation. GMRes has the advantage of being abl
e to solve any non-singular linear system\, in particular non-normal. But
the convergence analysis becomes less straightforward in this case\, beca
use it requires more information than just the spectrum. Bounds for GMRes
applied to non-normal matrices can be derived using condition number of
eigenvalues\, numerical range or pseudo-spectrum. But in the case of trapp
ing\, an additional difficulty comes from the solution operator whose nor
m grows exponentially through a sequence of frequencies tending to infini
ty\, with the density of these ``bad’’ frequencies increasing as the f
requency increases. In this case\, the spectrum of the associated matrix
has the form of a cluster associated with eigenvalues near the origin. We
provide a new analysis of the GMRes convergence taking into account this
particular distribution\, which allows to show more precisely why the num
ber of iterations grows with the frequency.\n\nhttps://indico.math.cnrs.f
r/event/8871/
LOCATION:IMT - K. Johnson
URL:https://indico.math.cnrs.fr/event/8871/
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