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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > High-Dimensional Collocation for Lognormal Diffusion Problems

## High-Dimensional Collocation for Lognormal Diffusion ProblemsAdd to your list(s) Download to your calendar using vCal - Oliver Ernst (Technische Universität Chemnitz)
- Friday 09 February 2018, 11:30-12:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact INI IT. UNQW02 - Surrogate models for UQ in complex systems 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. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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