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Building on the well-known total-variation (TV), this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID) i.e. the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the non-convexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, this latter is validated by simulations perfomed on computerized tomography (CT).
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve its solutio
This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matern-type Gaussian field priors that enable flexible modeling near the bou
In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the so-called continuous-time Infinite Swapping algorithm. Such a method, found its origins in the molecular dynamics community, and can be understood
In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. Complex computational geometries, given only by surface triangulations, are recast into the isogeometric context by transf
In this paper, by employing the asymptotic method, we prove the existence and uniqueness of a smoothing solution for a singularly perturbed Partial Differential Equation (PDE) with a small parameter. As a by-product, we obtain a reduced PDE model wit