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We address the numerical solution of infinite-dimensional inverse problems inthe framework of Bayesian inference. In the Part I companion to this paper(arXiv.org:1308.1313), we considered the linearized infinite-dimensionalinverse problem. Here in Part II, we relax the linearization assumption andconsider the fully nonlinear infinite-dimensional inverse problem using aMarkov chain Monte Carlo (MCMC) sampling method. To address the challenges ofsampling high-dimensional pdfs arising from Bayesian inverse problems governedby PDEs, we build on the stochastic Newton MCMC method. This method exploitsproblem structure by taking as a proposal density a local Gaussianapproximation of the posterior pdf, whose construction is made tractable byinvoking a low-rank approximation of its data misfit component of the Hessian.Here we introduce an approximation of the stochastic Newton proposal in whichwe compute the low-rank-based Hessian at just the MAP point, and then reusethis Hessian at each MCMC step. We compare the performance of the proposedmethod to the original stochastic Newton MCMC method and to an independencesampler. The comparison of the three methods is conducted on a synthetic icesheet inverse problem. For this problem, the stochastic Newton MCMC method witha MAP-based Hessian converges at least as rapidly as the original stochasticNewton MCMC method, but is far cheaper since it avoids recomputing the Hessianat each step. On the other hand, it is more expensive per sample than theindependence sampler; however, its convergence is significantly more rapid, andthus overall it is much cheaper. Finally, we present extensive analysis andinterpretation of the posterior distribution, and classify directions inparameter space based on the extent to which they are informed by the prior orthe observations.

A computational framework for infinite-dimensional Bayesian inverse  problems: Part II. Stochastic Newton MCMC with application to ice sheet flow  inverse problems