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      A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion

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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 (A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 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 3D global seismic wave propagation with hundreds of thousands of parameters.

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          Most cited references17

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          Inverse problems: A Bayesian perspective

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            Riemann manifold Langevin and Hamiltonian Monte Carlo methods

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              p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees

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                Author and article information

                Journal
                06 August 2013
                Article
                1308.1313
                8dd54165-f880-426a-8dc0-50c3445ed2d3

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                35Q62, 62F15, 35R30, 35Q93, 65C60, 35L05
                30 pages; to appear in SIAM Journal on Scientific Computing
                math.NA math.OC stat.CO stat.ME

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