"High dimensional approximation and applications"Potts, DanielIn 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 or on scattered data. To this end, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. The proposed algorithm based mainly on a fast Fourier transforms, 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. -We discuss methods where the support of the trigonometric polynomial is unknown. -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 important. This information is then utilized in obtaining an approximation for the function. -Based on these methods we develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations. This talk based on joint work with Lutz Kämmerer, Michael Schmischke and Fabian Taubert. |
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