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 an...
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ftdatacite:10.48550/arxiv.1308.6221 2023-05-15T16:40:47+02:00 A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems Petra, Noemi Martin, James Stadler, Georg Ghattas, Omar 2013 https://dx.doi.org/10.48550/arxiv.1308.6221 https://arxiv.org/abs/1308.6221 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Methodology stat.ME Numerical Analysis math.NA Optimization and Control math.OC Statistics Theory math.ST Computation stat.CO FOS Computer and information sciences FOS Mathematics 35Q62, 62F15, 35R30, 35Q93, 65C40, 65C60, 49M15, 86A40 Preprint Article article CreativeWork 2013 ftdatacite https://doi.org/10.48550/arxiv.1308.6221 2022-04-01T13:18:05Z 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. : 31 pages Report Ice Sheet DataCite Metadata Store (German National Library of Science and Technology) |
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DataCite Metadata Store (German National Library of Science and Technology) |
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topic |
Methodology stat.ME Numerical Analysis math.NA Optimization and Control math.OC Statistics Theory math.ST Computation stat.CO FOS Computer and information sciences FOS Mathematics 35Q62, 62F15, 35R30, 35Q93, 65C40, 65C60, 49M15, 86A40 |
spellingShingle |
Methodology stat.ME Numerical Analysis math.NA Optimization and Control math.OC Statistics Theory math.ST Computation stat.CO FOS Computer and information sciences FOS Mathematics 35Q62, 62F15, 35R30, 35Q93, 65C40, 65C60, 49M15, 86A40 Petra, Noemi Martin, James Stadler, Georg Ghattas, Omar A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
topic_facet |
Methodology stat.ME Numerical Analysis math.NA Optimization and Control math.OC Statistics Theory math.ST Computation stat.CO FOS Computer and information sciences FOS Mathematics 35Q62, 62F15, 35R30, 35Q93, 65C40, 65C60, 49M15, 86A40 |
description |
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. : 31 pages |
format |
Report |
author |
Petra, Noemi Martin, James Stadler, Georg Ghattas, Omar |
author_facet |
Petra, Noemi Martin, James Stadler, Georg Ghattas, Omar |
author_sort |
Petra, Noemi |
title |
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
title_short |
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
title_full |
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
title_fullStr |
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
title_full_unstemmed |
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems |
title_sort |
computational framework for infinite-dimensional bayesian inverse problems: part ii. stochastic newton mcmc with application to ice sheet flow inverse problems |
publisher |
arXiv |
publishDate |
2013 |
url |
https://dx.doi.org/10.48550/arxiv.1308.6221 https://arxiv.org/abs/1308.6221 |
genre |
Ice Sheet |
genre_facet |
Ice Sheet |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1308.6221 |
_version_ |
1766031199963185152 |