Related papers: Adaptive cross approximation for Tikhonov regulari…
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based…
We study Tikhonov regularization for certain classes of non-linear ill-posed operator equations in Hilbert space. Emphasis is on the case where the solution smoothness fails to have a finite penalty value, as in the preceding study…
The theory of spectral filtering is a remarkable tool to understand the statistical properties of learning with kernels. For least squares, it allows to derive various regularization schemes that yield faster convergence rates of the excess…
This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
Modern applications require methods that are computationally feasible on large datasets but also preserve statistical efficiency. Frequently, these two concerns are seen as contradictory: approximation methods that enable computation are…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine…
We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence…
Despite recent advances in regularisation theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov…
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if…
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence…
We consider iterative methods for solving linear ill-posed problems with compact operator and right-hand side only available via noise-polluted measurements. Conjugate gradients (CG) applied to the normal equations with an appropriate…
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*}…
This paper focuses on the analysis of conforming virtual element methods for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward…
In this article we propose a novel nonstationary iterated Tikhonov (NIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical…
We deal with a boundary detection problem arising in nondestructive testing of materials. The problem consists in recovering an unknown portion of the boundary, where a Robin condition is satisfied, with the use of a Cauchy data pair…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the…