1,423 research outputs found
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems
We address the numerical solution of infinite-dimensional inverse problems in
the framework of Bayesian inference. In the Part I companion to this paper
(arXiv.org:1308.1313), we considered the linearized infinite-dimensional
inverse problem. Here 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 pdfs arising from Bayesian inverse problems governed
by PDEs, we build on 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 construction is made tractable by
invoking a low-rank approximation of its 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 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.Comment: 31 page
Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty
We consider optimal experimental design (OED) for Bayesian nonlinear inverse
problems governed by partial differential equations (PDEs) under model
uncertainty. Specifically, we consider inverse problems in which, in addition
to the inversion parameters, the governing PDEs include secondary uncertain
parameters. We focus on problems with infinite-dimensional inversion and
secondary parameters and present a scalable computational framework for optimal
design of such problems. The proposed approach enables Bayesian inversion and
OED under uncertainty within a unfied framework. We build on the Bayesian
approximation error (BAE) framework, to incorporate modeling uncertainties in
the Bayesian inverse problem, and methods for A-optimal design of
infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a
Gaussian approximation to the posterior at the maximum a posteriori probability
point is used to define an uncertainty aware OED objective that is tractable to
evaluate and optimize. In particular, the OED objective can be computed at a
cost, in the number of PDE solves, that does not grow with the dimension of the
discretized inversion and secondary parameters. The OED problem is formulated
as a binary bilevel PDE constrained optimization problem and a greedy
algorithm, which provides a pragmatic approach, is used to find optimal
designs. We demonstrate the effectiveness of the proposed approach for a model
inverse problem governed by an elliptic PDE on a three-dimensional domain. Our
computational results also highlight the pitfalls of ignoring modeling
uncertainties in the OED and/or inference stages.Comment: 26 Page
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