Related papers: Approximation of Bayesian Inverse Problems for PDE…

Bayesian inverse problems use observed data to update a prior probability distribution for an unknown state or parameter of a scientific system to a posterior distribution conditioned on the data. In many applications, the unknown parameter…

Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their…

We construct optimal low-rank approximations for the Gaussian posterior distribution in linear Gaussian inverse problems with possibly infinite-dimensional separable Hilbert parameter spaces and finite-dimensional data spaces. We first…

The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only…

We show logarithmic stability for the point source inverse backscattering problem under the assumption of angularly controlled potentials. Radial symmetry implies H\"older stability. Importantly, we also show that the point source equation…

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a…

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution…

Iron loss determination in the magnetic core of an electrical machine, such as a motor or a transformer, is formulated as an inverse heat source problem. The sensor positions inside the object are optimized in order to minimize the…

Diffusion models have emerged as powerful learned priors for Bayesian inverse problems (BIPs). Diffusion-based solvers rely on a presumed likelihood for the observations in BIPs to guide the generation process. Likelihood misspecification…

Time-distance helioseismology is a set of powerful tools to study features below the Sun's surface. Inverse methods are needed to interpret time-distance measurements, with many examples in the literature. However, techniques that utilize a…

We consider the problem of determining the initial heat distribution in the heat equation from a point measurement. We show that this inverse problem is naturally related to the one of recovering the coefficients of Dirichlet series from…

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method…

We propose a control-oriented optimal experimental design (cOED) approach for linear PDE-constrained Bayesian inverse problems. In particular, we consider optimal control problems with uncertain parameters that need to be estimated by…

We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing from exact Bayesian inference, variational inference…

We consider the statistical linear inverse problem of recovering the unknown initial heat state from noisy interior measurements over an inhomogeneous domain of the solution to the heat equation at a fixed time instant. We employ…

In geophysics, inverse modelling can be applied to a wide range of goals, including, for instance, mapping the distribution of rock physical parameters in applied geophysics and calibrating models to forecast the behaviour of natural…

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise…

Many physical problems can be formulated as operator equations of the form Au = f. If these operator equations are ill-posed, we then resort to finding the approximate solutions numerically. Ill-posed problems can be found in the fields of…

The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows the formulation of expert knowledge or physical constraints in a probabilistic fashion and plays an important role for the…