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Kolmogorov widths and low-rank approximations of parametric elliptic PDEs

Authors: Markus Bachmayr and Albert Cohen
Journal: Math. Comp. 86 (2017), 701-724
MSC (2010): Primary 35C10, 41A46; Secondary 41A63, 65Nxx
Published electronically: July 20, 2016
MathSciNet review: 3584545
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Abstract: Kolmogorov $ n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $ n$-widths can be controlled by that of the error achieved by best $ n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $ n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $ n$-widths.

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Additional Information

Markus Bachmayr
Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

Albert Cohen
Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

Received by editor(s): February 10, 2015
Received by editor(s) in revised form: September 22, 2015
Published electronically: July 20, 2016
Additional Notes: This research was supported by the European Research Council under grant ERC AdG BREAD
Article copyright: © Copyright 2016 American Mathematical Society

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