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Markus Bachmayr, Albert Cohen, Wolfgang Dahmen, Parametric PDEs: sparse or low-rank approximations?, IMA Journal of Numerical Analysis, Volume 38, Issue 4, October 2018, Pages 1661–1708, https://doi.org/10.1093/imanum/drx052
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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.