A computational framework for infinite-dimensional bayesian inverse problems part I

The linearized case, with application to global seismic inversion

Tan Bui-Thanh, Omar Ghattas, James Martin, Georg Stadler

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalSIAM Journal on Scientific Computing
Volume35
Issue number6
DOIs
StatePublished - 2013

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Inverse problems
Inversion
Inverse Problem
Uncertainty
Forward Problem
Probability distributions
Probability Distribution
Discretization
Covariance Operator
Seismic Waves
Low-rank Approximation
Prior Probability
Seismic waves
Formulation
Posterior Probability
Number of Components
Bayesian inference
Covariance matrix
Prior distribution
Posterior distribution

Keywords

  • Bayesian inference
  • Infinite-dimensional inverse problems
  • Low rank approximation
  • Scalable algorithms
  • Seismic wave propagation
  • Uncertainty quantification

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

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title = "A computational framework for infinite-dimensional bayesian inverse problems part I: The linearized case, with application to global seismic inversion",
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.",
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author = "Tan Bui-Thanh and Omar Ghattas and James Martin and Georg Stadler",
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AU - Martin, James

AU - Stadler, Georg

PY - 2013

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AB - 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.

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