In geostatistics, Gaussian random fields are often used to model heterogeneities of soil or subsurface parameters. To give spatial approximations of these random fields, they are discretized. Then, different techniques of geostatistical inversion are used to condition them on measurement data. Among these techniques, Markov chain Monte Carlo (MCMC) techniques stand out, because they yield asymptotically unbiased conditional realizations. However, standard Markov Chain Monte Carlo (MCMC) methods suffer the curse of dimensionality when refining the discretization. This means that their efficiency decreases rapidly with an increasing number of discretization cells. Several MCMC approaches have been developed such that the MCMC efficiency does not depend on the discretization of the random field. The pre-conditioned Crank Nicolson Markov Chain Monte Carlo (pCN-MCMC) and the sequential Gibbs (or block-Gibbs) sampling are two examples. In this paper, we will present a combination of the pCN-MCMC and the sequential Gibbs sampling. Our algorithm, the sequential pCN-MCMC, will depend on two tuning-parameters: the correlation parameter $β$ of the pCN approach and the block size $κ$ of the sequential Gibbs approach. The original pCN-MCMC and the Gibbs sampling algorithm are special cases of our method. We present an algorithm that automatically finds the best tuning-parameter combination ($κ$ and $β$) during the burn-in-phase of the algorithm, thus choosing the best possible hybrid between the two methods. In our test cases, we achieve a speedup factors of $1-5.5$ over pCN and of $1-6.5$ over Gibbs. Furthermore, we provide the MATLAB implementation of our method as open-source code.
Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the poster…
We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank--Nicolson (pCN), a contribution of this paper in the development of a novel algorithm, termed the `Mixed' pCN (MpCN). RWM and pCN are shown n…
Bayesian inverse problems highly rely on efficient and effective inference methods for uncertainty quantification (UQ). Infinite-dimensional MCMC algorithms, directly defined on function spaces, are robust under refinement of physical models. Recent development of this class of algorithms has started to incorporate the geometry of the posterior informed by data so that they are capable of exploring complex probability structures. However, the required geometric quantities are usually expensive to obtain in high dimensions. On the other hand, most geometric information of the unknown parameter …
We introduce a new family of MCMC samplers that combine auxiliary variables, Gibbs sampling and Taylor expansions of the target density. Our approach permits the marginalisation over the auxiliary variables yielding marginal samplers, or the augmentation of the auxiliary variables, yielding auxiliary samplers. The well-known Metropolis-adjusted Langevin algorithm (MALA) and preconditioned Crank-Nicolson Langevin (pCNL) algorithm are shown to be special cases. We prove that marginal samplers are superior in terms of asymptotic variance and demonstrate cases where they are slower in computing ti…
University of Leeds（利兹大学）；University of Milan-Bicocca（米兰比可卡大学）
Reconstructing images from downsampled and noisy measurements, such as MRI and low dose Computed Tomography (CT), is a mathematically ill-posed inverse problem. We propose an easy-to-use reconstruction method based on Expectation Propagation (EP) techniques. We incorporate the Monte Carlo (MC) method, Markov Chain Monte Carlo (MCMC), and Alternating Direction Method of Multiplier (ADMM) algorithm into EP method to address the intractability issue encountered in EP. We demonstrate the approach on complex Bayesian models for image reconstruction. Our technique is applied to images from Gamma-cam…
Belgian Nuclear Research Centre（比利时核研究中心）；Normandie Univ（诺曼底大学）；UNIROUEN（鲁昂大学）；UNIHAVRE（勒阿弗尔大学）；INSA Rouen（鲁昂国立应用科学学院）；LITIS（鲁昂信息科学与技术实验室）；University of Lausanne（洛桑大学）
Probabilistic inversion within a multiple-point statistics framework is often computationally prohibitive for high-dimensional problems. To partly address this, we introduce and evaluate a new training-image based inversion approach for complex geologic media. Our approach relies on a deep neural network of the generative adversarial network (GAN) type. After training using a training image (TI), our proposed spatial GAN (SGAN) can quickly generate 2D and 3D unconditional realizations. A key characteristic of our SGAN is that it defines a (very) low-dimensional parameterization, thereby allowi…
In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and non-concave structures in the joint posterior posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we aim …
University of Helsinki（赫尔辛基大学）；University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）
The efficiency of Markov Chain Monte Carlo (MCMC) depends on how the underlying geometry of the problem is taken into account. For distributions with strongly varying curvature, Riemannian metrics help in efficient exploration of the target distribution. Unfortunately, they have significant computational overhead due to e.g. repeated inversion of the metric tensor, and current geometric MCMC methods using the Fisher information matrix to induce the manifold are in practice slow. We propose a new alternative Riemannian metric for MCMC, by embedding the target distribution into a higher-dimensio…
Gaussian Process (GP) models are a powerful and flexible tool for non-parametric regression and classification. Computation for GP models is intensive, since computing the posterior density, $π$, for covariance function parameters requires computation of the covariance matrix, C, a $pn^2$ operation, where p is the number of covariates and n is the number of training cases, and then inversion of C, an $n^3$ operation. We introduce MCMC methods based on the "temporary mapping and caching" framework, using a fast approximation, $π^*$, as the distribution needed to construct the temporary space. W…
The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed. In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo (MCMC) methods are usually required. In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison. We firstly propose a one-dimensional second order Cauchy difference prior, and construct new first a…
Variational Autoencoder for Generating Broader-Spectrum prior Proposals in Markov chain Monte Carlo Methods
Department One（部门一）；Department Two（部门二）；National Laboratory for Scientific Computing（国家科学计算实验室）；Department of Mathematical Sciences, The University of Texas at Dallas（数学科学系，德克萨斯大学达拉斯分校）
Sequential Kalman Tuning of the $t$-preconditioned Crank-Nicolson algorithm: efficient, adaptive and gradient-free inference for Bayesian inverse problems
Inverse parameter estimation of process-based models is a long-standing problem in many scientific disciplines. A key question for inverse parameter estimation is how to define the metric that quantifies how well model predictions fit to the data. This metric can be expressed by general cost or objective functions, but statistical inversion methods require a particular metric, the probability of observing the data given the model parameters, known as the likelihood. For technical and computational reasons, likelihoods for process-based stochastic models are usually based on general assumptions…
Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse …
We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: (1) a scale mixture of normals with respect to an alpha-stable random variable; and (2) a mixture of Bartlett--Fejer kernels (or triangle densities) with respect to a two-component mixture of gamma random variables. Both lead to MCMC methods for posterior simulation, and these methods turn out to have complementary domains of maximum efficiency. The first representation is a well known result due to West (1987), and is the better c…
We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYl…
