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Recent studies demonstrate that diffusion models can serve as a strong prior for solving inverse problems. A prominent example is Diffusion Posterior Sampling (DPS), which approximates the posterior distribution of data given the measure…

Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and…

Gravity inversion is the problem of estimating subsurface density distributions from observed gravitational field data. We consider the two-dimensional (2D) case, in which recovering density models from one-dimensional (1D) measurements…

Time-fractional differential equations offer a robust framework for capturing intricate phenomena characterized by memory effects, particularly in fields like biotransport and rheology. However, solving inverse problems involving fractional…

Inverse problems arise across scientific and engineering domains, where the goal is to infer hidden parameters or physical fields from indirect and noisy observations. Classical approaches, such as variational regularization and Bayesian…

A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few…

This paper proposes a novel Bayesian framework for solving Poisson inverse problems by devising a Monte Carlo sampling algorithm which accounts for the underlying non-Euclidean geometry. To address the challenges posed by the Poisson…

We propose to model the image differentials of astrophysical source maps by Student's t-distribution and to use them in the Bayesian source separation method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC) sampling…

Bayesian formulations of inverse problems are attractive for their ability to incorporate prior knowledge and update probabilistic models as new data become available. Markov chain Monte Carlo (MCMC) methods sample posterior probability…

In the field of inverse estimation for systems modeled by partial differential equations (PDEs), challenges arise when estimating high- (or even infinite-) dimensional parameters. Typically, the ill-posed nature of such problems…

Markov chain Monte Carlo (MCMC) methods are fundamental to Bayesian computation, but can be computationally intensive, especially in high-dimensional settings. Push-forward generative models, such as generative adversarial networks (GANs),…

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training…

We introduce a scheme for probabilistic hypocenter inversion with Stein variational inference. Our approach uses a differentiable forward model in the form of a physics informed neural network, which we train to solve the Eikonal equation.…

Machine learning methods for computational imaging require uncertainty estimation to be reliable in real settings. While Bayesian models offer a computationally tractable way of recovering uncertainty, they need large data volumes to be…

This paper investigates the application of Physics-Informed Neural Networks (PINNs) for solving the inverse advection-diffusion problem to localize pollution sources. The study focuses on optimizing neural network architectures to…

Iterative methods are widely used for solving partial differential equations (PDEs). However, the difficulty in eliminating global low-frequency errors significantly limits their convergence speed. In recent years, neural networks have…

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.…

Many machine learning applications require operating on a spatially distributed dataset. Despite technological advances, privacy considerations and communication constraints may prevent gathering the entire dataset in a central unit. In…

We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Bayesian inverse problems with "Besov random tree prior". These priors are given by a wavelet series with stochastic coefficients, and certain terms in the expansion…

Bayesian inference with deep generative prior has received considerable interest for solving imaging inverse problems in many scientific and engineering fields. The selection of the prior distribution is learned from, and therefore an…