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Fisher information for inverse problems and trace class operators
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.ORCID iD: 0000-0002-7018-6248
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.ORCID iD: 0000-0002-9857-0938
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
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2012 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, no 12, Article ID: 123503- p.Article in journal (Refereed) Published
Abstract [en]

This paper provides a mathematical framework for Fisher information analysis forinverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, andthe Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriatespace for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictlysmaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information tobe trace class, a sufficient condition is formulated based on the singular values ofthe Jacobian as well as of the eigenvalues of the covariance operator, together withsome regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with “external”spherically isotropic noise, as well as “internal” additive uncorrelated noise.

Place, publisher, year, edition, pages
2012. Vol. 53, no 12, Article ID: 123503- p.
National Category
Mathematics Physical Sciences
Research subject
Physics, Waves and Signals
Identifiers
URN: urn:nbn:se:lnu:diva-23304DOI: 10.1063/1.4763470OAI: oai:DiVA.org:lnu-23304DiVA: diva2:582741
Available from: 2013-01-06 Created: 2013-01-06 Last updated: 2017-01-11Bibliographically approved

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Nordebo, SvenKhrennikov, AndreiNilsson, BörjeToft, Joachim
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CiteExportLink to record
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Cite
Citation style
  • apa
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  • de-DE
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  • Other locale
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