Parametric PDEs: sparse or low-rank approximations?

Abstract We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2018-10, Vol.38 (4), p.1661-1708
Main Authors: Bachmayr, Markus, Cohen, Albert, Dahmen, Wolfgang
Format: Article
Language:English
Subjects:
Mathematics
Numerical Analysis
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Summary:Abstract We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations separating spatial and parametric variables, and hierarchical tensor decompositions separating all variables. We describe corresponding adaptive algorithms based on a common generic template and show their near-optimality with respect to natural approximability assumptions for each type of approximation. A central ingredient in the resulting bounds for the total computational complexity is a new operator compression result in the case of infinitely many parameters. We conclude with a comparison of the complexity estimates based on the actual approximability properties of classes of parametric model problems, which shows that the computational costs of optimized low-rank expansions can be significantly lower or higher than those of sparse polynomial expansions, depending on the particular type of parametric problem.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drx052