Michael Hintermüller

The Primal-Dual Active Set Strategy as a Semismooth Newton Method

This paper addresses complementarity problems motivated by constrained optimal control problems. It is shown that the primal-dual active set strategy, which is known to be extremely efficient for this class of problems, and a specific semismooth Newton method lead to identical algorithms. The notion of slant differentiability is recalled and it is argued that the

A Comparison of a Moreau--Yosida-Based Active Set Strategy and Interior Point Methods for Constrained Optimal Control Problems

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A Second Order Shape Optimization Approach for Image Segmentation

-function is slantly differentiable in Lp-spaces when appropriately combined with a two-norm concept. This leads to new local convergence results of the primal-dual active set strategy. Global unconditional convergence results are obtained by means of appropriate merit functions.

Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration

This research is devoted to the numerical solution of constrained optimal control problems governed by elliptic partial differential equations. The main purpose is a comparison between a recently developed Moreau--Yosida-based active set strategy involving primal and dual variables and two implementations of interior point algorithms.

Path-following Methods for a Class of Constrained Minimization Problems in Function Space

The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newtons method are presented.

A mesh-independence result for semismooth Newton methods

Multi-scale total variation models for image restoration are introduced. The models utilize a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to improve the performance of the initial algorithm a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

Topography and isotherms revisited: the influence of laterally migrating drainage divides

Path-following methods for primal-dual active set strategies requiring a regularization parameter are introduced. Existence of a primal-dual path and its differentiability properties are analyzed. Monotonicity and convexity of the primal-dual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path-following strategies are developed and their efficiency is demonstrated by means of examples.

An Efficient Primal-Dual Method for

Abstract.For a class of semismooth operator equations a mesh independence result for generalized Newton methods is established. The main result states that the continuous and the discrete Newton process, when initialized properly, converge q-linearly with the same rate. The problem class considered in the paper includes MCP-function based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given.

L^1

Abstract Drainage divides of mountain ranges will shift laterally during asymmetric erosion of the range. Such asymmetric erosion may occur due to differential precipitation on different sides of a range, or because of different topographic gradients, for example at passive margins. Here, a semi-analytical solution is presented that can be used to evaluate the shape of isotherms underneath an asymmetrically eroding, randomly shaped topography. It is shown that, because of this asymmetric erosion, cooling curves of rocks from the slowly denuding side of the range (in shift direction of divide) are characterized by a decrease in cooling rate with decreasing temperature, while cooling curves of rocks from the rapidly denuding side (in lee of the shift direction) are characterized by increasing cooling rate with decreasing temperature. Moreover, in rapidly denuding terrains, cooling through the 110°C and 75°C isotherms (approximating the retention temperatures of the apatite fission track and (U–Th)/He systems, respectively) may occur on the slowly denuding side of a range at twice the age of samples from the rapidly denuding side. In general, differences in cooling curves and cooling ages below 110°C should be recognizable in terrains where the erosion and lateral migration rates are of the order of 2 mm per year or more. In view of the recent developments of low temperature geochronological methods, our model provides an important tool to estimate the rate of lateral migration of drainage divides in regions where this is not constrained by the geomorphological record.

TV Image Restoration

Image restoration based on an