2,337 research outputs found

    The Bayesian Formulation of EIT: Analysis and Algorithms

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    We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models - log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.Comment: 30 pages, 10 figure

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

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    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

    Bayesian approach to inverse scattering with topological priors

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    We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution. Assuming the number of objects known, comparison of the results obtained by Markov Chain Monte Carlo sampling and by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate show reasonable agreement. The latter procedure has low computational cost, which makes it an interesting tool for uncertainty studies in 3D. However, MCMC sampling provides a more complete picture of the posterior distribution and yields multi-modal posterior distributions for problems with larger measurement noise. When the number of objects is unknown, we devise a stochastic model selection framework.FEDER/MICINN - AEI grant MTM2017-84446-C2-1-
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