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We present a computational framework for estimating the uncertainty in thenumerical solution of linearized infinite-dimensional statistical inverseproblems. We adopt the Bayesian inference formulation: given observational dataand their uncertainty, the governing forward problem and its uncertainty, and aprior probability distribution describing uncertainty in the parameter field,find the posterior probability distribution over the parameter field. The priormust be chosen appropriately in order to guarantee well-posedness of theinfinite-dimensional inverse problem and facilitate computation of theposterior. Furthermore, straightforward discretizations may not lead toconvergent approximations of the infinite-dimensional problem. And finally,solution of the discretized inverse problem via explicit construction of thecovariance matrix is prohibitive due to the need to solve the forward problemas many times as there are parameters. Our computational framework builds onthe infinite-dimensional formulation proposed by Stuart (A. M. Stuart, Inverseproblems: A Bayesian perspective, Acta Numerica, 19 (2010), pp. 451-559), andincorporates a number of components aimed at ensuring a convergentdiscretization of the underlying infinite-dimensional inverse problem. Theframework additionally incorporates algorithms for manipulating the prior,constructing a low rank approximation of the data-informed component of theposterior covariance operator, and exploring the posterior that together ensurescalability of the entire framework to very high parameter dimensions. Wedemonstrate this computational framework on the Bayesian solution of an inverseproblem in 3D global seismic wave propagation with hundreds of thousands ofparameters.

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