Karl Kunisch

Total Generalized Variation

The novel concept of total generalized variation of a function

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

u

Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of

Galerkin proper orthogonal decomposition methods for parabolic problems

u

Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition

. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.

Lagrange Multiplier Approach to Variational Problems and Applications

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

Convergence rates for Tikhonov regularisation of non-linear ill-posed problems

\max

Primal-Dual Strategy for Constrained Optimal Control Problems

-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.

Level-set function approach to an inverse interface problem

Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved. For the time integration the backward Euler scheme is considered. The asymptotic estimates involve the singular values of the POD snapshot set and the grid-structure of the time discretization as well as the snapshot locations.

The Linear Regulator Problem for Parabolic Systems

Summary. In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.