Prof. Dr. Kaspar Nipp

Address:
Seminar for Applied Mathematics
ETH Zentrum / HG G57.3
CH-8092 Zürich
Switzerland

Phone: +41 44 632 3407
Fax: +41 44 632 1104
E-mail: nipp@sam.math.ethz.ch

• Vita
• Research
• Selected Publications
• Projects

• Professional Activities

VITA 

Born March 5, 1949 in Vaduz, Liechtenstein. 

Education:

1974

M. S. in Mathematics at ETH Zürich;

1980

PhD. in Mathematics at ETH Zürich in the field of singular perturbations in ordinary differential equations.

Professional Career:

1974 - 1980

Research Associate HILTI AG, Liechtenstein

1980 - 1982

Assistent at the Seminar für Angewandte Mathematik ETH Zürich

1982 - 1983

Research year at Rensselaer Polytechnic Institute, Troy N.Y., supported by a grant of the Swiss NSF

since 1983

Research Associate and Lecturer at the Seminar für Angewandte Mathematik, ETH Zürich

RESEARCH

The fields of research I have been working in are ordinary differential equations and dynamical systems, in particular, singular perturbations, invariant manifolds and applications in numerical analysis. 

Invariant manifold results with conditions that are easy to verify and appropriate for the applications in mind have been derived (with D. Stoffer). Applications are in the numerical integration of differential equations. Under usual assumptions a singularly perturbed system of differential equations has an important geometric property: It admits a highly attractive smooth invariant manifold. From a numerical point of view singularly perturbed systems are a model class for so-called stiff systems which are difficult to treat. The question is which numerical integration methods are suitable for stiff systems and, in particular, which methods preserve the geometric property of the continuous system, i.e., admit an attractive invariant manifold. These results allow, e.g., to derive global error bounds of stiff integration schemes, to describe the behaviour of stiff integration schemes near a hyperbolic equilibrium and to prove the existence of closed hyperbolic invariant curves for the discrete systems. 

Another question of interest in this area is how to approximate an invariant manifold numerically in an efficient way. 

A topic also related to invariant manifolds is the reduction of high dimensional systems of differential equations to lower dimensional ones which still possess the essential properties. This plays an important role in chemical reaction mechanisms. A chemical reaction model usually contains a very large number of species and is characterized by a large number of time scales. The aim is to reduce the large original system to a low-dimensional simplified one by seeking the relevant time scales. 

Most recently, I have been concerned with differential algebraic systems (DAEs). The flow of a DAE lives on a manifold. In numerical analysis one has to ask what happens to this manifold under discretisation. Results are obtained by relating DAEs to singularly perturbed systems. Applying invariant manifold theorems also allows to obtain error estimates. 

Most of my research in the last few years has been done with D. Stoffer of the group of U. Kirchgraber at ETH. 

Outside the ETH I have started a cooperation with the group of G. Söderlind at Lund University, Sweden, in the field of differential algebraic systems. In the field of singular perturbation theory I am in contact with P. Szmolyan of the Technical University of Vienna. 

SELECTED PUBLICATIONS

Books and Monographs: 

with D. Stoffer, Lineare Algebra, eine Einführung für Ingenieure unter besonderer Berücksichtigung numerischer Aspekte, Vdf Hochschulverlag AG an der ETH Zürich; 1st Edition 1992, pp. 175, (vgl. Inhaltsverzeichnis als PostScript).
...; 2nd Edition 1992, pp. 217. 
...; 3rd Edition 1994, pp. 225. 
...; 4th Edition 1998, pp. 251.
...; 5th Edition 2002, pp. 251.

with D. Stoffer, Invariant Manifolds for Discrete and Continuous Dynamical Systems, EMS, 2013.

Refereed Articles: 

with Ch. Lubich and D. Stoffer, Runge-Kutta solutions of stiff differential equations near stationary points, SIA M J. Numer. Anal. 32, 8 (1995). (PostScript) 

with D. Stoffer, Invariant manifolds of numerical integration schemes applied to stiff systems of singular perturbation type - Part I: RK-methods, Numer. Math., 70 (1995), pp. 245-257. (PostScript)

with D. Stoffer, Invariant manifolds of numerical integration schemes applied to stiff systems of singular perturbation type - Part II: Linear multistep methods, Numer. Math., 74 (1996), pp. 305-323. (PostScript) 

Other Publications: 

with D. Stoffer, Attractive invariant manifolds for maps: Existence, smoothness and continuous dependence on the map, Research Report No. 92-11, Seminar für Angewandte Mathematik, ETH-Zürich (1992). (PostScript) 

Smooth attractive invariant manifolds of singularly perturbed ODE's, Research Report No. 92-13, Seminar für Angewandte Mathematik, ETH-Zürich (1992). (PostScript) 

PROJECTS 

• Geometric Singular Perturbation Theory for Discrete Dynamical Systems
• Invariant Manifolds for Maps and Flows
• Numerical Integration of Differential Algebraic Systems
• Phase Space Reduction in Chemical Kinetics

PROFESSIONAL ACTIVITIES 

Managing Editor of the Journal of Applied Mathematics and Physics (ZAMP) 

Member of the Committee for Computational Science and Engineering at ETH Zürich (Ausschuss Rechnergestützte Wissenschaften) 

Reviewing for journals (ZAMP, BIT, Computing, ...)