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Sparse Tensor Approximation of Parametric Eigenvalue Problems

by R. Andreev and Ch. Schwab

(Report number 2010-40)

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.

Keywords: Stochastic partial differential equations, uncertainty quantification, finite element method, parametric eigenvalue problem, anisotropic Smolyak collocation, multilevel approximation, sparse tensor product

@Techreport{AS10_51,
  author = {R. Andreev and Ch. Schwab},
  title = {Sparse Tensor Approximation of Parametric Eigenvalue Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-40.pdf },
  year = {2010}
}

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