Karl Kunisch

Bio: Karl Kunisch is an academic researcher from University of Graz. The author has contributed to research in topics: Optimal control & Nonlinear system. The author has an hindex of 59, co-authored 418 publications receiving 15508 citations. Previous affiliations of Karl Kunisch include Brown University & Rice University.

Total Generalized Variation

Abstract: The novel concept of total generalized variation of a function $u$ is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of $u$. 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.

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

Abstract: 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 $\max$-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.

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

Abstract: 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

Galerkin proper orthogonal decomposition methods for parabolic problems

Abstract: 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.

Estimation Techniques for Distributed Parameter Systems

Abstract: I Examples of Inverse Problems Arising in Applications.- I.1. Inverse Problems in Ecology.- I.2. Inverse Problems in Lake and Sea Sedimentation Analysis.- I.3. Inverse Problems in the Study of Flexible Structures.- I.4. Inverse Problems in Physiology.- II Operator Theory Preliminaries.- II.1. Linear Semigroups.- II.2. Galerkin Schemes.- III Parameter Estimation: Basic Concepts and Examples.- III.1. The Parameter Estimation Problem.- III.2. Application of the Theory to Special Schemes for Linear Parabolic Systems.- III.2.1. Modal Approximations.- III.2.2. Cubic Spline Approximations.- III.3. Parameter Dependent Approximation and the Nonlinear Variation of Constants Formula.- IV Identifiability and Stability.- IV.1. Generalities.- IV.2. Examples.- IV.3. Identifiability and Stability Concepts.- IV.4. A Sufficient Condition for Identifiability.- IV.5. Output Least Squares Identifiability.- IV.5.1. Theory.- IV.5.2. Applications.- IV.6. Output Least Squares Stability.- IV.6.1. Theory.- IV.6.2. An Example.- IV.7. Regularization.- IV.7.1. Tikhonov's Lemma and Its Application.- IV.7.2. Regularization Revisited.- IV.8. Concluding Remarks on Stability.- IV.8.1. A Summary of Possible Approaches.- IV.8.2. Remarks on Implementation.- V Parabolic Equations.- V.1. Modal Approximations: Discrete Fit-to-Data Criteria.- V.2. Quasimodal Approximations.- V.3. Operator Factorization: A = -C*C.- V.4. Operator Factorization: A = A1/2A1/2.- V.5. Numerical Considerations.- V.6. Numerical Test Examples.- V.7. Examples with Experimental Data.- VI Approximation of Unknown Coefficients in Linear Elliptic Equations.- VI.1. Parameter Estimation Convergence.- VI.2. Function Space Parameter Estimation Convergence.- VI.3. Rate of Convergence for a Special Case.- VI.4. Methods Other Than Output-Least-Squares.- VI.4.1. Method of Characteristics.- VI.4.2. Equation Error Method.- VI.4.3. A Variational Technique.- VI.4.4. Singular Perturbation Techniques.- VI.4.5. Adaptive Control Methods.- VI.4.6. An Augmented Lagrangian Technique.- VI.5. Numerical Test Examples.- VII An Annotated Bibliography.- Al) Preliminaries.- A2) Linear Splines.- A3) Cubic Hermite Splines.- A5) Polynomial Splines, Quasi- Interpolation.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Anisotropic diffusion in image processing

Abstract: Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

Stochastic Differential Equations

Abstract: We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.