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

hal.structure.identifierEDF R&D [EDF R&D]
dc.contributor.authorGONG, Helin
hal.structure.identifierLaboratoire Jacques-Louis Lions [LJLL]
dc.contributor.authorMADAY, Yvon
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorMULA, Olga
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorTADDEI, Tommaso
dc.date.accessioned2024-04-04T02:58:27Z
dc.date.available2024-04-04T02:58:27Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192659
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 $u^{\rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = \ell_m(u^{\rm true}),\, m=1,\dots,M$, where $\ell_m$ are linear functionals. The method approximates $u^{\rm true}$ with $\hat{u} = \hat{z} + \hat{\eta}$. The \emph{background} $\hat{z}$ belongs to an $N$-dimensional linear space $\mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the \emph{update} $\hat{\eta}$ belongs to the space $\mathcal{U}_M$ spanned by the Riesz representers of $(\ell_1,\dots, \ell_M)$. When the measurements are noisy {--- i.e., $y_m = \ell_m(u^{\rm true})+\epsilon_m$ with $\epsilon_m$ 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 $\hat{z}$ either on the whole $\mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $\mathcal{K}_N \subset \mathcal{Z}_N$ in presence of noise. The restriction to $\mathcal{K}_N$ 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.
dc.language.isoen
dc.title.enPBDW method for state estimation: error analysis for noisy data and nonlinear formulation
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Mathématiques générales [math.GM]
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.identifier.arxiv1906.00810
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-02404316
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-02404316v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=GONG,%20Helin&MADAY,%20Yvon&MULA,%20Olga&TADDEI,%20Tommaso&rft.genre=preprint


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