Numerische Mathematik

, Volume 139, Issue 1, pp 247–280 | Cite as

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

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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.

Mathematics Subject Classification

41A05 41A25 41A63 65B99 65C05 65D10 65D32 

Notes

Acknowledgements

S. Wolfers and R. Tempone are members of the KAUST Strategic Research Initiative, Center for Uncertainty Quantification in Computational Sciences and Engineering. R. Tempone received support from the KAUST CRG3 Award Ref: 2281. F. Nobile received support from the Center for ADvanced MOdeling Science (CADMOS). We thank Abdul-Lateef Haji-Ali for many helpful discussions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.MATHICSE-CSQIÉcole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  2. 2.Applied Mathematics and Computational ScienceKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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