Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems Dépendance Lipschitz des coefficients comme fonction de la résolvante et approximation greedy pour les problèmes elliptiques scalaires

Choulli M, Zuazua E (2016)

Publication Type: Journal article

Publication year: 2016

Journal

Comptes Rendus Mathematique Elsevier Masson

Book Volume: 354

Pages Range: 1174-1187

Journal Issue: 12

DOI: 10.1016/j.crma.2016.10.017

Abstract

We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive constants, the resolvent determines the diffusivity in an unique manner. Furthermore, we prove that the inverse mapping from resolvent to the coefficient is Lipschitz in suitable topologies. This result plays a key role when applying greedy algorithms to the approximation of parameter-dependent elliptic problems in an uniform and robust manner, independent of the given source terms. In one space dimension, the results can be improved using the explicit expression of solutions, which allows us to link distances between one resolvent and a linear combination of finitely many others and the corresponding distances on coefficients. These results are also extended to multi-dimensional elliptic equations with variable density coefficients. We also point out some possible extensions and open problems.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Université de Lorraine FR France (FR) University of Deusto / Universidad de Deusto ES Spain (ES)

How to cite

APA:

Choulli, M., & Zuazua, E. (2016). Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems Dépendance Lipschitz des coefficients comme fonction de la résolvante et approximation greedy pour les problèmes elliptiques scalaires. Comptes Rendus Mathematique, 354(12), 1174-1187. https://dx.doi.org/10.1016/j.crma.2016.10.017

MLA:

Choulli, Mourad, and Enrique Zuazua. "Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems Dépendance Lipschitz des coefficients comme fonction de la résolvante et approximation greedy pour les problèmes elliptiques scalaires." Comptes Rendus Mathematique 354.12 (2016): 1174-1187.

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