Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map.
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Bibliography:
Soeren Wolfers, Fabio Nobile, Raul Tempone, Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. Submitted arXiv 1609.00246, Sept 2016.
Authors:
Soeren Wolfers, Fabio Nobile, Raul Tempone
Keywords:
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
Year:
2016
Abstract:
We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.
