Statistical inverse problems: Discretization, model reduction and inverse crimes
The article discusses the discretization of linear inverse problems. When an inverse problem is formulated in terms of infinite-dimensional function spaces and then discretized for computational purposes, a discretization error appears. Since inverse problems are typically ill-posed, neglecting this...
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| Vydáno v: | Journal of computational and applied mathematics Ročník 198; číslo 2; s. 493 - 504 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article Konferenční příspěvek |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier B.V
15.01.2007
Elsevier |
| Témata: | |
| ISSN: | 0377-0427, 1879-1778 |
| On-line přístup: | Získat plný text |
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| Abstract | The article discusses the discretization of linear inverse problems. When an inverse problem is formulated in terms of infinite-dimensional function spaces and then discretized for computational purposes, a discretization error appears. Since inverse problems are typically ill-posed, neglecting this error may have serious consequences to the quality of the reconstruction. The Bayesian paradigm provides tools to estimate the statistics of the discretization error that is made part of the measurement and modelling errors of the estimation problem. This approach also provides tools to reduce the dimensionality of inverse problems in a controlled manner. The ideas are demonstrated with a computed example. |
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| AbstractList | The article discusses the discretization of linear inverse problems. When an inverse problem is formulated in terms of infinite-dimensional function spaces and then discretized for computational purposes, a discretization error appears. Since inverse problems are typically ill-posed, neglecting this error may have serious consequences to the quality of the reconstruction. The Bayesian paradigm provides tools to estimate the statistics of the discretization error that is made part of the measurement and modelling errors of the estimation problem. This approach also provides tools to reduce the dimensionality of inverse problems in a controlled manner. The ideas are demonstrated with a computed example. |
| Author | Kaipio, Jari Somersalo, Erkki |
| Author_xml | – sequence: 1 givenname: Jari surname: Kaipio fullname: Kaipio, Jari email: kaipio@venda.uku.fi organization: Department of Applied Physics, University of Kuopio, P.O. Box 1627, FIN–70211 Kuopio, Finland – sequence: 2 givenname: Erkki surname: Somersalo fullname: Somersalo, Erkki email: erkki.somersalo@hut.fi organization: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN–02015 TKK, Finland |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=18411584$$DView record in Pascal Francis |
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| CODEN | JCAMDI |
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| Keywords | Inverse problems Discretization Bayesian statistics Modelling error Bayes estimation Numerical analysis Function space Error estimation Applied mathematics Statistical model Ill posed problem Inverse problem |
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| References | H. Pikkarainen, A mathematical model for electrical impedance process tomography, Doctoral dissertation, Helsinki University of Technology, Espoo, Finland, 2005. ISBN 951-22-7651-8. Lassas, Siltanen (bib5) 2004; 20 Gelfand, Vilenkin (bib2) 1964; vol. 4 Mandelbaum (bib7) 1884; 65 D. Calvetti, J. Kaipio, E. Somersalo, Aristotelian prior boundary conditions, Int. J. Math. Comp. Sci. 1 (2006) in press. S. Lasanen, Discretizations of generalized random variables with applications to inverse problems, Ann. Acad. Sci. Fenn. Dissertationes 2002. Ju.A. Rozanov, Infinite-dimensional Gaussian Distributions, Proceedings of the Steklov Institute of Math 108 (1968) (English translation: AMS 1971). Kaipio, Somersalo (bib3) 2004; vol. 160 Lehtinen, Päivärinta, Somersalo (bib6) 1989; 5 Nguen, Strang (bib8) 1996 10.1016/j.cam.2005.09.027_bib1 10.1016/j.cam.2005.09.027_bib10 10.1016/j.cam.2005.09.027_bib4 Lehtinen (10.1016/j.cam.2005.09.027_bib6) 1989; 5 Lassas (10.1016/j.cam.2005.09.027_bib5) 2004; 20 10.1016/j.cam.2005.09.027_bib9 Gelfand (10.1016/j.cam.2005.09.027_bib2) 1964; vol. 4 Kaipio (10.1016/j.cam.2005.09.027_bib3) 2004; vol. 160 Mandelbaum (10.1016/j.cam.2005.09.027_bib7) 1884; 65 Nguen (10.1016/j.cam.2005.09.027_bib8) 1996 |
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| Title | Statistical inverse problems: Discretization, model reduction and inverse crimes |
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