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High-Dimensional Collocation for Lognormal Diffusion Problems

Presented by: 
Oliver Ernst Technische Universität Chemnitz
Friday 9th February 2018 - 11:30 to 12:30
INI Seminar Room 1
Co-authors: Björn Sprungk (Universität Mannheim), Lorenzo Tamellini (IMATI-CNR Pavia) Many UQ models consist of random differential equations in which one or more data components are uncertain and modeled as random variables. When the latter take values in a separable function space, their representation typically requires a large or countably infinite number of random coordinates. Numerical approximation methods for such functions of an infinite number of parameters based on best N-term approximation have recently been proposed and shown to converge at an algebraic rate. Collocation methods have a number of computational advantages over best N-term approximation, and we show how ideas successful there can be used to show a similar convergence rate for sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with a lognormal diffusion coefficient. We outline a general L2-convergence theory based on previous work by Bachmayr et al. and Chen and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons