### 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 language | English (US) |
---|---|

Journal | SIAM Journal on Scientific Computing |

Volume | 35 |

Issue number | 6 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### 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

**A computational framework for infinite-dimensional bayesian inverse problems part I : The linearized case, with application to global seismic inversion.** / Bui-Thanh, Tan; Ghattas, Omar; Martin, James; Stadler, Georg.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 35, no. 6. https://doi.org/10.1137/12089586X

}

TY - JOUR

T1 - A computational framework for infinite-dimensional bayesian inverse problems part I

T2 - The linearized case, with application to global seismic inversion

AU - Bui-Thanh, Tan

AU - Ghattas, Omar

AU - Martin, James

AU - Stadler, Georg

PY - 2013

Y1 - 2013

N2 - 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.

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.

KW - Bayesian inference

KW - Infinite-dimensional inverse problems

KW - Low rank approximation

KW - Scalable algorithms

KW - Seismic wave propagation

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=84892572639&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892572639&partnerID=8YFLogxK

U2 - 10.1137/12089586X

DO - 10.1137/12089586X

M3 - Article

VL - 35

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 6

ER -