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

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    Type
    Article
    Authors
    Nobile, Fabio
    Tempone, Raul cc
    Wolfers, Sören
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    KAUST Grant Number
    2281
    Date
    2017-11-16
    Online Publication Date
    2017-11-16
    Print Publication Date
    2018-05
    Permanent link to this record
    http://hdl.handle.net/10754/626181
    
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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.
    Citation
    Nobile F, Tempone R, Wolfers S (2017) Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. Numerische Mathematik. Available: http://dx.doi.org/10.1007/s00211-017-0932-4.
    Sponsors
    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.
    Publisher
    Springer Nature
    Journal
    Numerische Mathematik
    DOI
    10.1007/s00211-017-0932-4
    arXiv
    1609.00246
    Additional Links
    http://link.springer.com/article/10.1007/s00211-017-0932-4
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00211-017-0932-4
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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