Carlos N. Rautenberg

Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction

Variable splitting schemes for the function space version of the image reconstruction problem with total variation regularization (TV-problem) in its primal and pre-dual formulations are considered. For the primal splitting formulation, while existence of a solution cannot be guaranteed, it is shown that quasi-minimizers of the penalized problem are asymptotically related to the solution of the original TV-problem. On the other hand, for the pre-dual formulation, a family of parametrized problems is introduced and a parameter dependent contraction of an associated fixed point iteration is established. Moreover, the theory is validated by numerical tests. Additionally, the augmented Lagrangian approach is studied, details on an implementation on a staggered grid are provided and numerical tests are shown.

A Sequential Minimization Technique for Elliptic Quasi-variational Inequalities with Gradient Constraints

A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradient constraints in function space is considered. Problems of this type arise, for instance, in the mathematical description of the magnetization of superconductors, in problems in elastoplasticity, or in electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs by a sequential minimization technique relying on the repeated solution of variational inequality--type problems. A monotone operator theoretic approach is developed which does not resort to Mosco convergence as is often done in connection with existence analysis for QVIs. For the numerical solution of the QVIs a penalty approach combined with a semismooth Newton iteration is proposed. The paper ends with a report on numerical tests involving the

Parabolic Quasi-variational Inequalities with Gradient-Type Constraints

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Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests

-Laplace operator and various types of nonlinear constraints.

THE INFINITE-DIMENSIONAL OPTIMAL FILTERING PROBLEM WITH MOBILE AND STATIONARY SENSOR NETWORKS

This paper considers a class of nonlinear evolution quasi-variational inequality (QVI) problems with pointwise gradient constraints in vector-valued function spaces. The existence and approximation of solutions is addressed based on a combination of

Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part I: Modelling and Theory

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Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

-semigroup methods, Mosco convergence, and monotone operator techniques developed by Brezis. An algorithm based on semi-discretization in time is proposed and its numerical implementation based on a penalty approach and semismooth Newton methods is studied. This paper ends with a report on numerical tests which involve the

Optimal Sensor Placement: A Robust Approach

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Solutions and Approximations to the Riccati Integral Equation with Values in a Space of Compact Operators

-Laplacian and several types of nonlinear constraints.

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

Based on the weighted total variation model and its analysis pursued in Hintermüller and Rautenberg 2016, in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.