NDLI: A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems, Part II: Stochastic Newton MCMC with Application to Ice Sheet Flow Inverse Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Petra, Noemi Ghattas, Omar Martin, James Stadler, Georg
Copyright Year	2014
Abstract	We address the numerical solution of infinite-dimensional inverse problems in the framework of Bayesian inference. In Part I of this paper [T. Bui-Thanh, O. Ghattas, J. Martin, and G. Stadler, SIAM J. Sci. Comput., 35 (2013), pp. A2494--A2523] we considered the linearized infinite-dimensional inverse problem. In Part II, we relax the linearization assumption and consider the fully nonlinear infinite-dimensional inverse problem using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling high-dimensional probability density functions (pdfs) arising upon discretization of Bayesian inverse problems governed by PDEs, we build upon the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf, whose covariance operator is given by the inverse of the local Hessian of the negative log posterior pdf. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. Here we introduce an approximation of the stochastic Newton proposal in which we compute the low-rank-based Hessian at just the maximum a posteriori (MAP) point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method with a MAP-based Hessian converges at least as rapidly as the original stochastic Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian at each step. On the other hand, it is more expensive per sample than the independence sampler; however, its convergence is significantly more rapid, and thus overall it is much cheaper. Finally, we present extensive analysis and interpretation of the posterior distribution and classify directions in parameter space based on the extent to which they are informed by the prior or the observations.
Starting Page	A1525
Ending Page	A1555
Page Count	31
File Format	PDF
ISSN	10648275
DOI	10.1137/130934805
e-ISSN	10957197
Issue Number	4
Volume Number	36
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-07-24
Access Restriction	Subscribed
Subject Keyword	stochastic Newton uncertainty quantification ice sheet dynamics Computational problems in statistics MCMC low-rank approximation Glaciology Inverse problems Computational Markov chains Newton-type methods PDEs in connection with statistics infinite-dimensional inverse problems Bayesian inference PDEs in connection with control and optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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