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PBDW method for state estimation: error analysis for noisy data and nonlinear formulation

hal.structure.identifierEDF-R and D, Research Group R16 (Energy Strategy and Economics),
dc.contributor.authorGong, Helin
hal.structure.identifierLaboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
dc.contributor.authorMaday, Yvon
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorMula, Olga
HAL ID: 1531
ORCID: 0000-0002-3017-6598
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorTaddei, Tommaso
dc.date.accessioned2019-12-18T12:51:35Z
dc.date.available2019-12-18T12:51:35Z
dc.date.issued2019
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/20340
dc.language.isoenen
dc.subjectvariational data assimilationen
dc.subjectparameterized partial differentialequationsen
dc.subjectmodel order reductionen
dc.subject.ddc510en
dc.titlePBDW method for state estimation: error analysis for noisy data and nonlinear formulationen
dc.typeDocument de travail / Working paper
dc.description.abstractenWe present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function utrue living in a high-dimensional Hilbert space from M measurement observations given in the form ym=ℓm(utrue),m=1,…,M, where ℓm are linear functionals. The method approximates utrue with u^=z^+η^. The \emph{background} z^ belongs to an N-dimensional linear space ZN built from reduced modelling of a parameterized mathematical model, and the \emph{update} η^ belongs to the space UM spanned by the Riesz representers of (ℓ1,…,ℓM). When the measurements are noisy {--- i.e., ym=ℓm(utrue)+ϵm with ϵm being a noise term --- } the classical PBDW formulation is not robust in the sense that, if N increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background z^ either on the whole ZN in the noise-free case, or on a well-chosen subset KN⊂ZN in presence of noise. The restriction to KN makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an \emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.en
dc.publisher.nameCahier de recherche CEREMADE, Université Paris-Dauphineen
dc.publisher.cityParisen
dc.identifier.citationpages30en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris-Dauphineen
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-02404316en
dc.subject.ddclabelMathématiquesen
dc.identifier.citationdate2019-12
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2019-12-18T12:41:31Z
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