Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 

 

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
DOI: https://doi.org/10.1090/mcom/3132
Published electronically: July 20, 2016
MathSciNet review: 3584545
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35C10, 41A46, 41A63, 65Nxx

Retrieve articles in all journals with MSC (2010): 35C10, 41A46, 41A63, 65Nxx


Additional Information

Markus Bachmayr
Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France
Email: bachmayr@ljll.math.upmc.fr

Albert Cohen
Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France
Email: cohen@ann.jussieu.fr

DOI: https://doi.org/10.1090/mcom/3132
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