Markov chain Monte Carlo Using an Approximation

Christen, J. Andrés; Fox, C. Fred

doi:10.1198/106186005x76983

Cited by 272 publications

(313 citation statements)

References 16 publications

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“…The validity of the assumptions for our application is discussed in the paper. We would like to note that two-stage MCMC algorithms have been used previously (e.g., [2,16,22,10]) in different situations.…”

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confidence: 99%

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

Efendiev¹,

Hou²,

Luo³

2006

SIAM J. Sci. Comput.

181

205

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Abstract.We study the preconditioning of Markov Chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite-volume method. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application, and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen-Loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than ten times if MCMC simulations are preconditioned using coarse-scale models.

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confidence: 99%

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

Efendiev¹,

Hou²,

Luo³

2006

SIAM J. Sci. Comput.

181

205

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“…Other possible sampling methods to include in a comparison are the various implementations of the Gibbs sampler for EIT [21,20,27,38], the DRAM method [16], the DREAM algorithm [6,24,37], and the delayed acceptance Metropolis algorithm [11,19]. We can support not including a comparison with DRAM here because in both [4,26], MALA is shown to outperform DRAM on large-scale nonlinear inverse problems similar to EIT.…”

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confidence: 99%

Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography

Bardsley¹,

Seppänen²,

Solonen³

et al. 2015

SIAM/ASA J. Uncertainty Quantification

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Abstract. In a typical inverse problem, a spatially distributed parameter in a physical model is estimated from measurements of model output. Since measurements are stochastic in nature, so is any parameter estimate. Moreover, in the Bayesian setting, the choice of regularization corresponds to the definition of the prior probability density function, which in turn is an uncertainty model for the unknown parameters. For both of these reasons, significant uncertainties exist in the solution of an inverse problem. Thus to fully understand the solution, quantifying these uncertainties is important. When the physical model is linear and the error model and prior are Gaussian, the posterior density function is Gaussian with a known mean and covariance matrix. However, the electrical impedance tomography inverse problem is nonlinear, and hence no closed form expression exists for the posterior density. The typical approach for such problems is to sample from the posterior and then use the samples to compute statistics (such as the mean and variance) of the unknown parameters. Sampling methods for electrical impedance tomography have been studied by various authors in the inverse problems community. However, up to this point the focus has been on the development of increasingly sophisticated implementations of the Gibbs sampler, whose samples are known to converge very slowly to the correct density for large-scale problems. In this paper, we implement a recently developed sampling method called randomize-then-optimize (RTO), which provides nearly independent samples for each application of an appropriate numerical optimization algorithm. The sample density for RTO is not the posterior density, but RTO can be used as a very effective proposal within a Metropolis-Hastings algorithm to obtain samples from the posterior. Here our focus is on implementing the method on synthetic examples from electrical impedance tomography, and we show that it is both computationally efficient and provides good results. We also compare RTO performance with the Metropolis adjusted Langevin algorithm and find RTO to be much more efficient. 1. Introduction. Solving an inverse problem requires estimating unknown parameters in a physical model from measurements of model output. The unknown parameter vector typically corresponds to a numerical discretization of a continuously defined, spatially distributed parameter in the physical model and hence is high-dimensional. This leads to an overparameterized statistical model, and hence estimates have large variance or, in other words, are unstable with respect to perturbations in the observations. Regularization methods [36], which have been studied for decades in the inverse problems literature, are the typical fix for such instability.Over the past ten years or so, an increasing number of researchers in inverse problems have drawn the connection between regularized solutions of inverse problems and Bayesian statistics [20]. Since the measured data in an inverse problem will always contain noise, a ...

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“…Geralmente, as simulações diretas do método MCMC apresentam a desvantagem de necessitar de um elevado esforço computacional, uma vez que um grande número de simulaçõesé necessário para que obtenha-se a convergência da cadeia de Markov e uma boa taxa de aceitação. Portanto, a fim de minimizar este problema recorre-se a uma eficiente e rigorosa metodologia, conhecida como método MCMC a dois estágios [6,12,13], que pode ser descrito pelo Algoritmo 1.…”

Section: O Método De Monte Carlo Via Cadeias De Markovunclassified

Utilização de Modelos Autoregressivos na Quantificação de Incertezas em Problemas de Transporte Linear

Pereira

Souto

2016

Tend. Mat. Apl. Comput.

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Recebido em 28 agosto, 2014 / Aceito em 9 fevereiro, 2016 RESUMO. O fenômeno físico do processo de escoamentos de traçadores em um meio poroso heterogêneó e modelado por um sistema de equações diferenciais parciais, sujeitas a certas condições de contorno e inicial. As variações significativas das propriedades do meio poroso (porosidade e permeabilidade) são responsáveis pela introdução das incertezas contidas no modelo matemático. Com intuito de reduzir as incertezas dos modelos geológicos, deferentes metodologias tem sido desenvolvidas e testadas em diversos problemas de escoamentos de fluidos em meios porosos heterogêneo. O objetivo deste trabalhoé o estudo da quantificação de incertezas em problemas de escoamentos de traçadores em meios porosos heterogêneos empregando uma abordagem Bayesiana para a seleção dos campos de permeabilidades, baseada em um conjunto de medições da concentração do traçador em pontos específicos do meio poroso. O método da Soma Sucessiva de Campos Gaussianos Independentes (SSCGI)é utilizado na parametrização das incertezas contidas nos meios porosos heterogêneos. Na resolução do problema inverso, emprega-se o método de Monte Carlo via Cadeias de Markov (MCMC) a dois estágios que utiliza o algoritmo de MetropolisHastings baseado em passeios aleatórios do tipo autoregressivo. Através deste procedimento, gera-se uma cadeia Markov que converge para a distribuição estacionária, que neste casoé a distribuição a posteriori de interesse. Resultados numéricos são apresentados para um conjunto de realizações dos campos de permeabilidades. Palavras-chave:Campos estocásticos Gaussianos, escoamento em meios porosos, método MCMC a dois estágios, traçador. INTRODUÇÃOO estudo dos escoamentos multifásicos em meios porosos heterogêneos representa uma ferramentaútil em diversasáreas importantes para o desenvolvimento científico e tecnológico, dentre as quais pode-se destacar a indústria do petróleo e a contaminação ambiental. As técnicas † Trabalho apresentado no XXXV Congresso Nacional de Matemática Aplicada e Computacional.

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Markov chain Monte Carlo Using an Approximation

Cited by 272 publications

References 16 publications

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography

Utilização de Modelos Autoregressivos na Quantificação de Incertezas em Problemas de Transporte Linear

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