Partial differential equations
Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems
Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1174-1187.

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.

Nous examinons le problème inverse de l'identification du coefficient de diffusion comme fonction de la résolvante pour des équations elliptiques scalaires. Nous établissons, pour des topologies appropriées, un résultat de stabilité Lipschitz pour une classe de coefficients de diffusion mesurables, minorés et majorés par des constantes positives fixées a priori. Ce résultat intervient de manière essentielle dans le développement d'algorithmes greedy pour l'approximation d'une famille paramétrée de problèmes elliptiques de manière robuste et uniforme par rapport au terme source. Nous traitons séparément le cas de la dimension un, pour lequel nous disposons de formules explicites de représentation des solutions permettant de comparer la distance entre une résolvante et une combinaison linéaire d'un nombre fini d'autres et des coefficients correspondants, et un développement complet de l'approche greedy. Nous étendons ces résultats au problème de l'identification de la densité à partir de l'opérateur résolvant correspondant. Nous signalons aussi quelques problèmes ouverts, en particulier dans le cas multi-dimensionnel.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.017
Choulli, Mourad 1; Zuazua, Enrique 2, 3, 4

1 IECL, UMR CNRS 7502, Université de Lorraine, boulevard des Aiguillettes, BP 70239, 54506 Vandœuvre-lès-Nancy cedex & Île du Saulcy, 57045 Metz cedex 01, France
2 DeustoTech, Fundación Deusto, Avda Universidades, 24, 48007 Bilbao, Spain
3 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
4 Facultad Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007 Bilbao, Spain
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Choulli, Mourad; Zuazua, Enrique. Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1174-1187. doi : 10.1016/j.crma.2016.10.017. http://www.numdam.org/articles/10.1016/j.crma.2016.10.017/

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Cited by Sources:

The second author was partially supported by the ERC Advanced Grant Agreement: 694126 – DYCON (Dynamic Control), the ANR (France) Project ICON (ANR-2016-ACHN-0014-01), Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, and MTM2014-52347 of the MINECO (Spain).