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Parametric PDEs: Sparse Polynomial or Low-Rank Approximation?

Presented by: 
Wolfgang Dahmen RWTH Aachen University
Monday 5th March 2018 - 09:45 to 10:30
INI Seminar Room 1
We discuss some recent results obtained jointly with Markus Bachmayr and Albert Cohen on the adaptive approximation of parametric solutions for a class of uniformly elliptic parametric PDEs. We briefly review first essential approximability properties of the solution manifold with respect to several approximation types (sparse polynomial expansions, low-rank approximations, hierarchical tensor formats) which then serve as benchmarks for numerical algorithms. We then discuss a fully adaptive algorithmic template with respect to both spatial and parametric variables which can be specified to any of the above approximation types. It completely avoids the inversion of large linear systems and can be shown to achieve any given target accuracy with a certified bound without any a priori assumptions on the solutions. Moreover, the computational complexity (in terms of operation count) can be proven to be optimal (up to uniformly constant factors) for these benchmark classes. That is, it achieves a given target accuracy by keeping the number of adaptively generated degrees of freedom near-minimal at linear computatiuonal cost. We discuss these findings from several perspectives such as: which approximation type is best suited for which problem specification, the role of parametric expansion types, or intrusive versus non-intrusive schemes.
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    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons