Stabilities of shape identification inverse problems in a Bayesian framework

A general shape identification inverse problem is studied in a Bayesian framework. This problem requires the determination of the unknown shape of a domain in the Euclidean space from finite-dimensional observation data with some Gaussian random noise. Then, the stability of posterior is studied for...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 486; no. 2; p. 123903
Main Author Kawakami, Hajime
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.06.2020
Subjects
Bayesian
Heat equation
Inverse problem
Shape identification
Stability
Shape identification
Stability
Heat equation
Bayesian
Inverse problem
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Summary:A general shape identification inverse problem is studied in a Bayesian framework. This problem requires the determination of the unknown shape of a domain in the Euclidean space from finite-dimensional observation data with some Gaussian random noise. Then, the stability of posterior is studied for observation data. For each point of the space, the conditional probability that the point is included in the unknown domain given the observation data is considered. The stability is also studied for this probability distribution. As a model problem for our inverse problem, a heat inverse problem is considered. This problem requires the determination of the unknown shape of cavities in a heat conductor from temperature data of some portion of the surface of the heat conductor. To apply the above stability results to this model problem, one needs the measurability and some boundedness of the forward operator. These properties are shown.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2020.123903