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dc.contributor.authorCohen, Albert
dc.contributor.authorDahmen, Wolfgang
dc.contributor.authorMula, Olga
dc.contributor.authorNichols, James
dc.date.accessioned2020-10-21T09:11:07Z
dc.date.available2020-10-21T09:11:07Z
dc.date.issued2020
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/21135
dc.language.isoenen
dc.subjectstate estimationen
dc.subjectparameter estimationen
dc.subject.ddc511en
dc.titleNonlinear reduced models for state and parameter estimationen
dc.typeDocument de travail / Working paper
dc.description.abstractenState estimation aims at approximately reconstructing the solution u to a parametrized partial differential equation from m linear measurements, when the parameter vector y is unknown. Fast numerical recovery methods have been proposed based on reduced models which are linear spaces of moderate dimension n which are tailored to approximate the solution manifold M where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches, and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width dm(M) of the solution manifold. In this paper we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces Vk, each having dimension at most m and leading to different estimators u∗k. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator u∗. Our analysis shows that u∗ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator y∗ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation, that potentially improves the quality of both estimators.en
dc.identifier.citationpages35en
dc.relation.ispartofseriestitleCahier de recherche CEREMADEen
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-02932983en
dc.subject.ddclabelPrincipes généraux des mathématiquesen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2020-10-21T09:04:05Z
hal.person.labIds1005052
hal.person.labIds1028935
hal.person.labIds60
hal.person.labIds6348


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