A stability estimate for data assimilation subject to the heat equation with initial datum
This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for t...
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| Published in | Comptes rendus. Mathématique Vol. 361; no. G9; pp. 1521 - 1530 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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Académie des sciences (Paris)
01.01.2023
Académie des sciences |
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| Abstract | This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature. |
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| AbstractList | This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature. |
| Author | Delay, Guillaume Ern, Alexandre Burman, Erik Oksanen, Lauri |
| Author_xml | – sequence: 1 givenname: Erik surname: Burman fullname: Burman, Erik – sequence: 2 givenname: Guillaume surname: Delay fullname: Delay, Guillaume – sequence: 3 givenname: Alexandre surname: Ern fullname: Ern, Alexandre – sequence: 4 givenname: Lauri surname: Oksanen fullname: Oksanen, Lauri |
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| Cites_doi | 10.1006/jdeq.1993.1088 10.1137/060670961 10.1007/s00211-019-01087-x 10.1016/j.matpur.2018.10.003 10.1088/0266-5611/22/2/007 10.1051/cocv/2011168 10.1137/22M1508637 10.3233/ASY-1995-10201 10.1088/0266-5611/25/12/123013 10.1007/s00211-018-0949-3 10.3934/mcrf.2017012 10.1070/sm1995v186n06abeh000047 10.1051/m2an/2018030 10.1080/03605309108820778 |
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| References | Yamamoto, Masahiro (key2023111010591212365_15) 2009; 25 Burman, Erik (key2023111010591212365_6) 2018; 139 Puel, Jean-Pierre (key2023111010591212365_12) 2009; 48 Le Rousseau, Jérôme (key2023111010591212365_11) 2012; 18 Imanuvilov, Oleg Yu. (key2023111010591212365_8) 2001; 218 Burman, Erik (key2023111010591212365_3) 2018; 52 Robbiano, Luc (key2023111010591212365_13) 1991; 16 Èmanuilov, Oleg Yu. (key2023111010591212365_7) 1995; 186 Burman, Erik (key2023111010591212365_4) 2019; 129 Burman, Erik (key2023111010591212365_5) 2020; 144 Isakov, Victor (key2023111010591212365_9) 1993; 105 Klibanov, Michael V. (key2023111010591212365_10) 2006; 22 Bourgeois, Laurent (key2023111010591212365_1) 2017; 7 Burman, Erik (key2023111010591212365_2) 2023; 61 Robbiano, Luc (key2023111010591212365_14) 1995; 10 |
| References_xml | – volume: 105 start-page: 217 year: 1993 ident: key2023111010591212365_9 article-title: Carleman type estimates in an anisotropic case and applications publication-title: J. Differ. Equations doi: 10.1006/jdeq.1993.1088 – volume: 48 start-page: 1089 issue: 2 year: 2009 ident: key2023111010591212365_12 article-title: A nonstandard approach to a data assimilation problem and Tychonov regularization revisited publication-title: SIAM J. Control Optim. doi: 10.1137/060670961 – volume: 144 start-page: 451 year: 2020 ident: key2023111010591212365_5 article-title: A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime publication-title: Numer. Math. doi: 10.1007/s00211-019-01087-x – volume: 129 start-page: 1 year: 2019 ident: key2023111010591212365_4 article-title: Unique continuation for the Helmholtz equation using stabilized finite element methods publication-title: J. Math. Pures Appl. doi: 10.1016/j.matpur.2018.10.003 – volume: 22 start-page: 495 year: 2006 ident: key2023111010591212365_10 article-title: Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data publication-title: Inverse Probl. doi: 10.1088/0266-5611/22/2/007 – volume: 18 start-page: 712 issue: 3 year: 2012 ident: key2023111010591212365_11 article-title: On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations publication-title: ESAIM, Control Optim. Calc. Var. doi: 10.1051/cocv/2011168 – volume: 218 start-page: 113 year: 2001 ident: key2023111010591212365_8 article-title: Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications publication-title: Control of nonlinear distributed parameter systems (College Station, TX, 1999) – volume: 61 start-page: 2534 issue: 5 year: 2023 ident: key2023111010591212365_2 article-title: The unique continuation problem for the heat equation discretized with a high-order space-time nonconforming method publication-title: SIAM J. Numer. Anal. doi: 10.1137/22M1508637 – volume: 10 start-page: 95 issue: 2 year: 1995 ident: key2023111010591212365_14 article-title: Fonction de coût et contrôle des solutions des équations hyperboliques publication-title: Asymptotic Anal. doi: 10.3233/ASY-1995-10201 – volume: 25 issue: 12 year: 2009 ident: key2023111010591212365_15 article-title: Carleman estimates for parabolic equations and applications publication-title: Inverse Probl. doi: 10.1088/0266-5611/25/12/123013 – volume: 139 start-page: 505 year: 2018 ident: key2023111010591212365_6 article-title: Data assimilation for the heat equation using stabilized finite element methods publication-title: Numer. Math. doi: 10.1007/s00211-018-0949-3 – volume: 7 start-page: 347 issue: 3 year: 2017 ident: key2023111010591212365_1 article-title: Quantification of the unique continuation property for the heat equation publication-title: Math. Control Relat. Fields doi: 10.3934/mcrf.2017012 – volume: 186 start-page: 109 issue: 6 year: 1995 ident: key2023111010591212365_7 article-title: Controllability of parabolic equations publication-title: Mat. Sb. doi: 10.1070/sm1995v186n06abeh000047 – volume: 52 start-page: 2065 issue: 5 year: 2018 ident: key2023111010591212365_3 article-title: Fully discrete finite element data assimilation method for the heat equation publication-title: ESAIM, Math. Model. Numer. Anal. doi: 10.1051/m2an/2018030 – volume: 16 start-page: 789 issue: 4-5 year: 1991 ident: key2023111010591212365_13 article-title: Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques publication-title: Commun. Partial Differ. Equations doi: 10.1080/03605309108820778 |
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| Snippet | This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are... |
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| Title | A stability estimate for data assimilation subject to the heat equation with initial datum |
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