Principal investigators: U. Langer and B. Buchberger

This coordination and service project is designed to provide the coherence of the SFB, give organizational support, and provide necessary computing power. The tasks of integrating graphical and parallel computing are currently assigned to this project.

Title: **Solving and Proving in General Domains**

Principal investigator: B. Buchberger

The goal of this project is to design and implement an automatic theorem prover in natural deduction style, for the use of the working mathematician. This prover will be an interactive tool both for presentation of mathematical knowledge and deleloping mathematical knowledge in the form of theorems and their (interactive) proofs and algorithms and their behavior.

Title: **Proving and Solving over the Reals**

Principal investigators: B. Buchberger and J. Schicho

The objective is to develop mathematical theories, algorithms, and software libraries/packages for efficiently proving/solving real algebraic constraints (quantified boolean expressions of polynomial equations/inequalities over real numbers) of moderate sizes.

Title: **Symbolic-Numerical Computation on Algebraic Curves and Surfaces**

Principal investigator: F. Winkler

Algebraic curves and surfaces play an essential rôle in modeling physical and virtual objects. The main goal of this project is the improvement and/or perfection of existing methods and the derivation of new methods for designing, manipulating, and visualizing algebraic curves and surfaces.

Title: **Symbolic Summation and Combinatorial Identities**

Principal investigator: P. Paule

The algorithmic problem of symbolic summation can be viewed as a discrete analogue of the well-known problem of symbolic integration. Typical applications for symbolic summation algorithms are combinatorial identities. The main objective of this project is research in algorithms design for symbolic summation and holonomic functions.

Title: **Coupled Field Problems: Advanced Numerical Methods and Applications
to Nonlinear Magnetomechanical Systems**

Principal investigators: U. Langer and R. Lerch

The modeling of complex electromechanical systems often leads to coupled field problems involving the interaction of different systems of nonlinear partial differential equations in complicated 3D geometries. The complexity of these systems requires the development of efficient simulation tools working on powerful workstations as well as on workstation clusters and parallel computers.

Title: **Large Scale Inverse Problems**

Principal investigator: H.W. Engl

Inverse problems are concerned with determining causes for a desired or an observed effect. The mathematical formulation of inverse problems usually leads to ill-posed problems which have to be treated by so-called regularization methods. The major aim of this project is to develop theory-based methods for the optimal coupling between regularization methods and direct solvers for (mainly nonlinear) inverse problems.

Title: **Hierarchical Methods for Simulation and Optimal Design with
Applications to Magnetic Field Problems**

Principal investigators: U. Langer and E. Lindner

The subject of this project is the development, analysis, and implementation of numerically efficient algorithms for solving optimal design problems arising, e.g., in magnetostatics. The project aims at a unified treatment of direct simulation and optimization based on hierarchical methods.

Title: **Estimation of Discontinuous Parameters in Differential Equations**

Principal investigators: O. Scherzer and H.W. Engl

In many practical problems governed by differential equations one in principle knows the type of equation but not the explicit values or forms of the parameters. Regularization techniques have to be applied to solve such problems. It is the aim of this project to develop new regularization methods and their efficient numerical realization in the case of parameters with singularities.