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High-throughput characterization often requires estimating parameters and model dimension from experimental data of limited quantity and quality. Such data may result in an ill-posed inverse problem, where multiple sets of parameters and…

We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates -- the score functions -- are learned with a neural network, we transform the score matching…

This paper presents an improved implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is based on Markov chain Monte Carlo (MCMC). However, the samples generated by…

Markov chain Monte Carlo samplers based on discretizations of (overdamped) Langevin dynamics are commonly used in the Bayesian inference and computational statistical physics literature to estimate high-dimensional integrals. One can…

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm,…

Langevin Monte Carlo (LMC) is a popular Bayesian sampling method. For the log-concave distribution function, the method converges exponentially fast, up to a controllable discretization error. However, the method requires the evaluation of…

We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte…

One of the major challenges in the Bayesian solution of inverse problems governed by partial differential equations (PDEs) is the computational cost of repeatedly evaluating numerical PDE models, as required by Markov chain Monte Carlo…

Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give…

In this paper we consider a new probability sampling methods based on Langevin diffusion dynamics to resolve the problem of existing Monte Carlo algorithms when draw samples from high dimensional target densities. We extent…

The methodology developed in this article is motivated by a wide range of prediction and uncertainty quantification problems that arise in Statistics, Machine Learning and Applied Mathematics, such as non-parametric regression, multi-class…

The Hamiltonian Monte Carlo (HMC) sampling algorithm exploits Hamiltonian dynamics to construct efficient Markov Chain Monte Carlo (MCMC), which has become increasingly popular in machine learning and statistics. Since HMC uses the gradient…

Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem, both from a practical and theoretical perspective. Compared with samplers with unbiased stationary…

In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian statistical framework with a preconditioned Markov…

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…

We propose a very fast approximate Markov Chain Monte Carlo (MCMC) sampling framework that is applicable to a large class of sparse Bayesian inference problems, where the computational cost per iteration in several models is of order…

The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior…

Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem.…

Performing stochastic inversion on a computationally expensive forward simulation model with a high-dimensional uncertain parameter space (e.g. a spatial random field) is computationally prohibitive even with gradient information provided.…

In Bayesian inverse problems sampling the posterior distribution is often a challenging task when the underlying models are computationally intensive. To this end, surrogates or reduced models are often used to accelerate the computation.…