Related papers: Non-Stationary Multi-layered Gaussian Priors for B…
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is…
The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
We propose to use L\'evy {\alpha}-stable distributions for constructing priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian…
Bayesian Last Layer (BLL) models focus solely on uncertainty in the output layer of neural networks, demonstrating comparable performance to more complex Bayesian models. However, the use of Gaussian priors for last layer weights in…
This paper deals with Gibbs samplers that include high dimensional conditional Gaussian distributions. It proposes an efficient algorithm that avoids the high dimensional Gaussian sampling and relies on a random excursion along a small set…
Bayesian inference is used extensively to quantify the uncertainty in an inferred field given the measurement of a related field when the two are linked by a mathematical model. Despite its many applications, Bayesian inference faces…
Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is…
We present a new technique, based on semivariogram methodology, for obtaining point estimates for use in prior modeling for solving Bayesian inverse problems. This method requires a connection between Gaussian processes with covariance…
Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended…
In this paper, the problem of state estimation, in the context of both filtering and smoothing, for nonlinear state-space models is considered. Due to the nonlinear nature of the models, the state estimation problem is generally intractable…
This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is…
This article revisits the problem of Bayesian shape-restricted inference in the light of a recently developed approximate Gaussian process that admits an equivalent formulation of the shape constraints in terms of the basis coefficients. We…
Bayesian methods are developed for the multivariate nonparametric regression problem where the domain is taken to be a compact Riemannian manifold. In terms of the latter, the underlying geometry of the manifold induces certain symmetries…
In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. We reconstruct a high-order piecewise polynomial space that arbitrary order…
It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a…
This paper considers the objective comparison of stochastic models to solve inverse problems, more specifically image restoration. Most often, model comparison is addressed in a supervised manner, that can be time-consuming and partly…
A machine-learnable variational scheme using Gaussian radial basis functions (GRBFs) is presented and used to approximate linear problems on bounded and unbounded domains. In contrast to standard mesh-free methods, which use GRBFs to…
We propose practical deep Gaussian process models on Riemannian manifolds, similar in spirit to residual neural networks. With manifold-to-manifold hidden layers and an arbitrary last layer, they can model manifold- and scalar-valued…
We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian…