Related papers: Regularization of linear ill-posed problems involv…
A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak $L^\infty$-solutions are renormalized. But then, if the noise is nondegenerate, uniqueness…
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems.…
In this paper, we consider a regularization strategy for the factorization method when there is noise added to the data operator. The factorization method is a qualitative method used in shape reconstruction problems. These methods are…
We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of…
In this paper, we address the problem of approximating solutions of ill-posed problems using mollification. We quickly review existing mollification regularization methods and provide two new approximate solutions to a general ill-posed…
This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the…
We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Schmidt operators (and they require a self-similarity condition…
Logarithmically regularized 2D Euler equations are active scalar equations with the non-local velocity $u = \nabla^\perp \Delta^{-1}T_\gamma \omega$ for the scalar $\omega$. Two types of the regularizing operator $T_\gamma$ with a parameter…
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and…
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…
We consider perturbed nonlinear ill-posed equations in Hilbert spaces, with operators that are monotone on a given closed convex subset. A simple stable approach is Lavrentiev regularization, but existence of solutions of the regularized…
In this book, written in Portuguese, we discuss what ill-posed problems are and how the regularization method is used to solve them. In the form of questions and answers, we reflect on the origins and future of regularization, relating the…
Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…
We propose a regularization method to solve a nonlinear ill-posed problem connected to inversion of data gathered by a ground conductivity meter.
Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by…
Our focus is on the stable approximate solution of linear operator equations based on noisy data by using $\ell^1$-regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where…
In this paper, we consider the $\alpha\| \cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $\alpha\geq\beta\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the…
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…
We present a regularization method to approach a solution of the pessimistic formulation of ill -posed bilevel problems . This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and…
In this study, we investigate the $\left\|\cdot\right\|_{\ell_{1}}^{2}-\eta\left\|\cdot\right\|_{\ell_{2}}^{2}$ sparsity regularization with $0< \eta\leq 1$, in the context of nonlinear ill-posed inverse problems. We focus on the…