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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.

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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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Authors and Affiliations

  1. MATHICSE-CSQI, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Fabio Nobile

  2. Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

    Raúl Tempone & Sören Wolfers

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  1. Fabio Nobile
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  2. Raúl Tempone
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  3. Sören Wolfers
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Correspondence to Sören Wolfers.

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Nobile, F., Tempone, R. & Wolfers, S. Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. Numer. Math. 139, 247–280 (2018). https://doi.org/10.1007/s00211-017-0932-4

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  • DOI: https://doi.org/10.1007/s00211-017-0932-4

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