A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion
We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertaint...
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| Veröffentlicht in: | SIAM journal on scientific computing Jg. 35; H. 6; S. A2494 - A2523 |
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| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
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Philadelphia
Society for Industrial and Applied Mathematics
01.01.2013
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| ISSN: | 1064-8275, 1095-7197 |
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| Abstract | We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertainty, and a prior probability distribution describing uncertainty in the parameter field, find the posterior probability distribution over the parameter field. The prior must be chosen appropriately in order to guarantee well-posedness of the infinite-dimensional inverse problem and facilitate computation of the posterior. Furthermore, straightforward discretizations may not lead to convergent approximations of the infinite-dimensional problem. And finally, solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters. Our computational framework builds on the infinite-dimensional formulation proposed by Stuart [Acta Numer., 19 (2010), pp. 451--559] and incorporates a number of components aimed at ensuring a convergent discretization of the underlying infinite-dimensional inverse problem. The framework additionally incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions. We demonstrate this computational framework on the Bayesian solution of an inverse problem in three-dimensional global seismic wave propagation with hundreds of thousands of parameters. [PUBLICATION ABSTRACT] |
|---|---|
| AbstractList | We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertainty, and a prior probability distribution describing uncertainty in the parameter field, find the posterior probability distribution over the parameter field. The prior must be chosen appropriately in order to guarantee well-posedness of the infinite-dimensional inverse problem and facilitate computation of the posterior. Furthermore, straightforward discretizations may not lead to convergent approximations of the infinite-dimensional problem. And finally, solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters. Our computational framework builds on the infinite-dimensional formulation proposed by Stuart [Acta Numer., 19 (2010), pp. 451--559] and incorporates a number of components aimed at ensuring a convergent discretization of the underlying infinite-dimensional inverse problem. The framework additionally incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions. We demonstrate this computational framework on the Bayesian solution of an inverse problem in three-dimensional global seismic wave propagation with hundreds of thousands of parameters. [PUBLICATION ABSTRACT] |
| Author | Ghattas, Omar Bui-Thanh, Tan Martin, James Stadler, Georg |
| Author_xml | – sequence: 1 givenname: Tan surname: Bui-Thanh fullname: Bui-Thanh, Tan – sequence: 2 givenname: Omar surname: Ghattas fullname: Ghattas, Omar – sequence: 3 givenname: James surname: Martin fullname: Martin, James – sequence: 4 givenname: Georg surname: Stadler fullname: Stadler, Georg |
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| Cites_doi | 10.3934/ipi.2009.3.87 10.1111/j.1365-246X.2009.04368.x 10.1137/100791634 10.1137/090780717 10.1016/0031-9201(81)90046-7 10.1017/S0962492910000061 10.1088/0266-5611/24/3/034015 10.1016/j.jcp.2010.09.008 10.1137/090778250 10.1137/110828010 10.1126/science.1175298 10.1088/0266-5611/28/5/055001 10.4310/CMS.2012.v10.n4.a6 10.3934/ipi.2013.7.397 10.1111/j.1365-246X.2011.04969.x 10.1111/1467-9868.00288 10.1111/j.1365-246X.2011.05044.x 10.1111/j.1467-9868.2011.00777.x 10.1002/nme.394 10.1214/08-AAP563 10.1137/110845598 10.1046/j.1365-246X.2002.01631.x 10.1006/jcph.1996.0225 10.1038/ngeo1501 10.1088/0266-5611/28/5/055002 10.1111/j.1467-9868.2010.00765.x |
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| Snippet | We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems.... |
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| SubjectTerms | Algorithms Approximation Inverse problems Probability distribution Propagation |
| Title | A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion |
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| Volume | 35 |
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