Show simple item record

Nonlinear reduced models for state and parameter estimation

hal.structure.identifierLaboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
dc.contributor.authorCohen, Albert
dc.contributor.authorDahmen, Wolfgang
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorMula, Olga
hal.structure.identifierDepartment of Mathematics [ANU CANBERRA]
dc.contributor.authorNichols, James
dc.date.accessioned2020-10-21T09:11:07Z
dc.date.available2020-10-21T09:11:07Z
dc.date.issued2022
dc.identifier.issn2166-2525
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/21135
dc.language.isoenen
dc.subjectstate estimation
dc.subjectparameter estimation
dc.subjectreduced order modeling
dc.subjectoptimal recovery
dc.subject.ddc511en
dc.titleNonlinear reduced models for state and parameter estimation
dc.typeArticle accepté pour publication ou publié
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.
dc.relation.isversionofjnlnameSIAM/ASA Journal on Uncertainty Quantification
dc.relation.isversionofjnlvol10
dc.relation.isversionofjnlissue1
dc.relation.isversionofjnldate2022
dc.relation.isversionofjnlpages35
dc.relation.isversionofdoi10.1137/20M1380818
dc.relation.isversionofjnlpublisherSIAM - Society for Industrial and Applied Mathematics
dc.subject.ddclabelPrincipes généraux des mathématiquesen
dc.description.ssrncandidatenon
dc.description.halcandidatenon
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2023-02-20T15:13:04Z
hal.author.functionaut
hal.author.functionaut
hal.author.functionaut
hal.author.functionaut


Files in this item

Thumbnail
Name:
2009.02687.pdf
Size:
1.192Mb
Format:
PDF
View/Open

This item appears in the following Collection(s)

Show simple item record