Related papers: Regularization Methods for Ill-Posed Problems in M…

We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is…

A new class of control problems is discussed - homeostasis control. Homeostasis control problems can be considered as control problems with a given target set, in particular, as a problem of stabilizing the values of some target function,…

Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation space. When solving the inverse…

We provide a statistical analysis of regularization-based continual learning on a sequence of linear regression tasks, with emphasis on how different regularization terms affect the model performance. We first derive the convergence rate…

We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the…

We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a…

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and…

This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…

In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic…

This paper proposes an algorithm for computing regularized solutions to linear rational expectations models. The algorithm allows for regularization cross-sectionally as well as across frequencies. A variety of numerical examples illustrate…

Regularization methods allow one to handle a variety of inferential problems where there are more covariates than cases. This allows one to consider a potentially enormous number of covariates for a problem. We exploit the power of these…

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which…

We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization…

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the…

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical…

In this article we show the crucial role of elliptic regularity theory for the development of efficient numerical methods for the solution of some variational problems. Here we focus to a class of elliptic multiobjective optimal control…

In this paper, we propose a new Fully Composite Formulation of convex optimization problems. It includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems of Composite…

For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the…

Point processes are stochastic models generating interacting points or events in time, space, etc. Among characteristics of these models, first-order intensity and conditional intensity functions are often considered. We focus on…