Related papers: A regularization approach for an inverse source pr…

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the…

We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem…

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and…

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in…

In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed…

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy…

In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a…

The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The…

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…

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…

The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to…

In this paper we consider the computation of approximate solutions for inverse problems in Hilbert spaces. In order to capture the special feature of solutions, non-smooth convex functions are introduced as penalty terms. By exploiting the…

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of…

Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, which are a necessity due to the ill-posedness of inverse problems. Tikhonov-type regularization methods are very popular in…

We investigate a level-set type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable…

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to…

In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show…

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for…

We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem…

We explore the possibility for using boundary data to identify sources in elliptic PDEs. Even though the associated forward operator has a large null space, it turns out that box constraints, combined with weighted sparsity regularization,…