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Stochastic Galerkin-collocation splitting for PDEs with random parameters

Jahnke, Tobias; Stein, Benny

We propose a numerical method for time-dependent, semilinear partial differential equations (PDEs) with random parameters and random initial data. The method is based on an operator splitting approach. The linear part of the right-hand side is discretized by a stochastic Galerkin method in the stochastic variables and a pseudospectral method in the physical space, whereas the nonlinear part is approximated by a stochastic collocation method in the stochastic variables. In this setting both parts of the random PDE can be propagated very efficiently. The Galerkin method and the collocation method are combined with sparse grids in order to reduce the computational costs. This approach is discussed in detail for the Lugiato-Lefever equation, which serves as a motivating example throughout, but also applies to a much larger class of random PDEs. For such problems our method is computationally much cheaper than standard stochastic Galerkin methods, and numerical tests show that it outperforms standard stochastic collocation methods, too.

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DOI: 10.5445/IR/1000086905
Veröffentlicht am 24.10.2018
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000086905
Verlag KIT, Karlsruhe
Umfang 32 S.
Serie CRC 1173 ; 2018/28
Schlagworte Uncertainty quantification, splitting methods, spectral methods, (generalized) polynomial chaos, Lugiato-Lefever equation, nonlinear Schrödinger equation, sparse grids, stochastic Galerkin method, stochastic collocation
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