Related papers: The Bayesian Approach To Inverse Problems
The Bayesian approach to inverse problems provides a practical way to solve ill-posed problems by augmenting the observation model with prior information. Due to the measure-theoretic underpinnings, the approach has raised theoretical…
A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are considered, such as…
The classical approach to inverse problems is based on the optimization of a misfit function. Despite its computational appeal, such an approach suffers from many shortcomings, e.g., non-uniqueness of solutions, modeling prior knowledge,…
Traditionally, the MaxEnt workshops start by a tutorial day. This paper summarizes my talk during 2001'th workshop at John Hopkins University. The main idea in this talk is to show how the Bayesian inference can naturally give us all the…
We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with…
Deep neural networks have achieved impressive results on a wide variety of tasks. However, quantifying uncertainty in the network's output is a challenging task. Bayesian models offer a mathematical framework to reason about model…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of uncertainty through a computational (forward) model - are strongly connected. In the form of conditional expectation the Bayesian update…
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…
Using a new Bayesian method for solving inverse quantum problems, potentials of quantum systems are reconstructed from coordinate measurements in non-stationary states. The approach is based on two basic inputs: 1. a likelihood model,…
The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP. The proposed methodology extends PINN to account for prior knowledge on the nature of the…
In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be…
The Bayesian approach to data analysis provides a powerful way to handle uncertainty in all observations, model parameters, and model structure using probability theory. Probabilistic programming languages make it easier to specify and fit…
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) --- the propagation of uncertainty through a computational (forward) model --- are strongly connected. In the form of conditional expectation the Bayesian update…
In this paper, first a great number of inverse problems which arise in instrumentation, in computer imaging systems and in computer vision are presented. Then a common general forward modeling for them is given and the corresponding…
Bayesian approach, as a useful tool for quantifying uncertainties, has been widely used for solving inverse problems of partial differential equations (PDEs). One of the key difficulties for employing Bayesian approach for the issue is how…
A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and "a priori" information, provide flexible tools for such empirical learning problems. The…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
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…
Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do…