Related papers: Error Estimates for Arnoldo-Tikhonov Regularizatio…
We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields…
A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we…
We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using…
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 approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic…
We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the…
We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence…
In this work, we propose a high-order regularization method to solve the ill-conditioned problems in robot localization. Numerical solutions to robot localization problems are often unstable when the problems are ill-conditioned. A typical…
Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function $u(x)$ is \begin{eqnarray*}…
The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm…
GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as Dirichlet boundary value problems for elliptic partial…
In this paper, we apply randomized algorithms to approximate the total least squares (TLS) solution of the problem $Ax\approx b$ in the large-scale discrete ill-posed problems. A regularization technique, based on the multiplicative…
Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers…
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator…
Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed…
Primal-dual splitting involving proximity operators in order to be able to find some approximation to the minimizer for a general form of Tikhonov type functional is in the focus of this work. This approximation is produced by a pair of…
Many statistical problems can be reduced to a linear inverse problem in which only a noisy version of the operator is available. Particular examples include random design regression, deconvolution problem, instrumental variable regression,…
In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under…
In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a…
We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the…