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A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems


Authors: Simone Brugiapaglia, Fabio Nobile, Stefano Micheletti and Simona Perotto
Journal: Math. Comp. 87 (2018), 1-38
MSC (2010): Primary 65N30, 65Y20, 94A20; Secondary 65T40, 65K10, 42A61
DOI: https://doi.org/10.1090/mcom/3209
Published electronically: May 11, 2017
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Abstract: We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best $ s$-term approximation of the weak solution of a PDE with respect to a system of $ N$ trial functions, can be recovered via a Petrov-Galerkin approach using $ m \ll N$ test functions. This recovery is guaranteed if the local $ a$-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.


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Additional Information

Simone Brugiapaglia
Affiliation: MOX, Dipartimento di Matematica, Politecnico di Milano, 20133, Milano, Italy
Email: simone.brugiapaglia@polimi.it

Fabio Nobile
Affiliation: MATHICSE-CSQI, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
Email: fabio.nobile@epfl.ch

Stefano Micheletti
Affiliation: MOX, Dipartimento di Matematica, Politecnico di Milano, 20133, Milano, Italy
Email: stefano.micheletti@polimi.it

Simona Perotto
Affiliation: MOX, Dipartimento di Matematica, Politecnico di Milano, 20133, Milano, Italy
Email: simona.perotto@polimi.it

DOI: https://doi.org/10.1090/mcom/3209
Keywords: Compressed sensing, Petrov-Galerkin formulation, advection-diffusion-reaction equation, inf-sup property, local coherence
Received by editor(s): September 8, 2015
Received by editor(s) in revised form: August 3, 2016
Published electronically: May 11, 2017
Additional Notes: The first author acknowledges the INdAM research group “Gruppo Nazionale per il Calcolo Scientifico” for the economic support.
The second author acknowledges the support from the Center for ADvanced MOdeling Science (CADMOS)
The work of the third author was supported by the Project MIUR-PRIN 2010/2011 “Data-Centric Genomic Computing” (GenData 2020)
The financial support of MIUR (Project “Innovative Methods for Water Resources under Hydro-Climatic Uncertainty Scenarios”, PRIN 2010/2011) is gratefully acknowledged by the last author
Article copyright: © Copyright 2017 American Mathematical Society