Strobl22

Applied Harmonic Analysis and Friends

June 19th - 25th 2022

Strobl, AUSTRIA

"High dimensional approximation and application to PDEs with random coefficients"

Potts, Daniel

In this talk, we present algorithms for the approximation of multivariate functions. We start with the approximation by trigonometric polynomials based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. The proposed algorithm based mainly on a one-dimensional fast Fourier transform, and the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. After a detailed introduction we will focus on the following questions in more detail. \begin{itemize} \item We discuss methods where the support of the trigonometric polynomial is unknown. \item We describes an extension of approximation methods for nonperiodic functions via a multivariate change of variables. %\item We present a method based on the analysis of variance (ANOVA) decomposition that aims to detect the structure of the function, i.e., find out which dimension and dimension interactions are \textit{important}. This information is then utilized in obtaining an approximation for the function. \item Based on these methods we develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. \end{itemize} \noindent This talk based on joint work with Lutz Kämmerer, Fabian Taubert and Robert Nasdala.
http://univie.ac.at/projektservice-mathematik/e/talks/Potts_2022-02_abstract_potts.pdf

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