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23.01.2019 Prof. Dr. Martin Gander "One Class of Iterative Solvers for Helmholtz Problems..."

Vortrag im Oberseminar Numerik und Optimierung am 23. Januar 2019

Im Rahmen des Oberseminares "Numerik und Optimierung" hält

Herr Prof. Dr. Martin Gander (Universität Genf)

am Mittwoch, 23. Januar 2019, einen Vortrag mit dem Titel

"One Class of Iterative Solvers for Helmholtz Problems: AILU Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimal and Optimized Schwarz Methods"

Beginn: 16:15 Uhr, Raum

Alle Interessierten sind herzlich willkommen!


Solving time-harmonic wave propagation problems by iterative methods
is a difficult task, and over the last two decades, an important
research effort has gone into developing preconditioners for the
simplest representative of such wave propagation problems, the
Helmholtz equation. We consider here a specific class of these new
preconditioners, which can be found in the literature under various
names like Sweeping Preconditioners, Source Transfer Domain
Decomposition, the Method With Single Layer Potentials, and the Method
of Polarized Traces. These methods were developed by researchers with
various backgrounds using formulations and notations that are very
different, and all are among the most promising preconditioners for
the Helmholtz equation and more general time harmonic wave propagation
problems like Maxwell's equations.

We will show that all preconditioners in this class are based on a
common mathematical principle, and they can all be formulated in the
context of a type of domain decomposition methods called optimized
Schwarz methods. This common formulation allows us to explain in
detail how and why these methods work.  All these methods are based on
one-dimensional decompositions of the problem in space into a sequence
of subproblems, and they have in their optimal form the property to
lead to nilpotent iterations, like an exact block LU
factorization. They thus represent at the discrete level approximate
block LU decomposition preconditioners. While we focus in the
presentation on the specific case of the Helmholtz equation, our
formulations are completely general and hold for other partial
differential equations as well. In the case of time harmonic wave
propagation, we will also show important limitations of these new,
promising methods.