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Numerical Analysis of Multiscale Problems pp 203–241Cite as

Sparse Tensor Approximation of Parametric Eigenvalue Problems

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 83))

Abstract

We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain. The stable numerical implementation of these methods is discussed and their error analysis is given. Applications to parametric elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients are presented.

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Acknowledgements

Supported by SNF grant PDFMP2-127034/1 and by ERC AdG grant STAHDPDE 247277.

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Authors and Affiliations

  1. Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland

    Roman Andreev & Christoph Schwab

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  1. Roman Andreev
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  2. Christoph Schwab
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Correspondence to Christoph Schwab .

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Editors and Affiliations

  1. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Ivan G. Graham

  2. , Applied & Computational Mathematics, California Institute of Technology, Pasadena, 91125, California, USA

    Thomas Y. Hou

  3. , Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom

    Omar Lakkis

  4. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Robert Scheichl

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Andreev, R., Schwab, C. (2012). Sparse Tensor Approximation of Parametric Eigenvalue Problems. In: Graham, I., Hou, T., Lakkis, O., Scheichl, R. (eds) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22061-6_7

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