Parametric PDEs: sparse or low-rank approximations?. (16th September 2017)

• Record Type: Journal Article
• Authors: Bachmayr, Markus; Cohen, Albert; Dahmen, Wolfgang
• Is Part Of: IMA journal of numerical analysis. Volume 38:Number 4(2018)
• Page Start: 1661
• Page End: 1708
• Publication Date: 2017-09-16
• Subjects: parameter-dependent partial differential equations -- low-rank approximations -- sparse polynomial expansions -- a posteriori error estimates -- adaptive methods -- complexity bounds; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drx052
• Languages: English
• ISSNs: 0272-4979

Abstract

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.