Related papers: Elliptic Bayesian Inverse Problems on Metric Graph…
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…
Denoising diffusion models have driven significant progress in the field of Bayesian inverse problems. Recent approaches use pre-trained diffusion models as priors to solve a wide range of such problems, only leveraging inference-time…
We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This…
We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial…
We consider a linearised inverse conductivity problem for electromagnetic waves in a three dimensional bounded domain at a high time-harmonic frequency. Increasing stability bounds for the conductivity coefficient in the full Maxwell system…
Partial differential equations are central to describing many physical phenomena. In many applications these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain…
Inverse problems have many applications in science and engineering. In Computer vision, several image restoration tasks such as inpainting, deblurring, and super-resolution can be formally modeled as inverse problems. Recently, methods have…
The present paper is motivated by one of the most fundamental challenges in inverse problems, that of quantifying model discrepancies and errors. While significant strides have been made in calibrating model parameters, the overwhelming…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions. Recently, there has been a growing interest in understanding the computational hardness of these optimization problems,…
We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices…
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…
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.…
These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental…
Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's…
We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a…
A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the…
A general setup for deterministic system identification problems on graphs with Dirichlet and Neumann boundary conditions is introduced. When control nodes are available along the boundary, we apply a discretize-then-optimize method to…
Ill-posed linear inverse problems arise frequently in various applications, from computational photography to medical imaging. A recent line of research exploits Bayesian inference with informative priors to handle the ill-posedness of such…
A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as…