|Research Interests & Project Suggestions|
My research interests cover a wide spectrum of issues in Scientific Computing, including
|Applications in Science and Technology|
The interdisciplinary approach and collaboration with scientists from other fields are essential for my work. I am particularly interested in finding partners for cooperation on the following research topics.
|1. Data local iterative methods|
Risc architectures and thus most high performance computers are generally based on a fast but small cache memory while the access to the global main memory is currently typically one order of magnitude slower than the peak computing speed. This is even more drastic in virtual shared memory architectures. Many algorithms of scientific computing, e.g. matrix-vector operations or relaxation methods do not exploit the hierarchical memory structure so that they run with only a fraction of the CPU peak speed.However, the data access structures of many algorithms can be restructured.
In a project we study the data access patterns of efficient iterative methods (primarily of multilevel and conjugate gradient type) and to investigate modifcations that will improve the performance. However, I expect that for obtaining full efficiency, new algorithmic components must be developed, e.g. non-standard cycling strategies and new smoothers for multigrid. This project would ideally be associated with a larger computer architecture or benchmarking effort. Further aspects to extend the scope of this project are virtual shared memory performance and domain decomposition parallelization.
Presently this project is being funded through the DFG.
|2. Object oriented numerics|
Object oriented techniques can be used for numerical simulations as a systematic software development approach. However, it is not trivial to represent the richness of mathematical structures efficiently with object oriented languages like C++. Based on our previous experience, I propose to focus on realistic prototype applications that can benefit from flexible and dynamic data structures in an object oriented approach. Possible applications are adaptive finite element calculations, sparse matrix algorithms, or particle systems in computational physics. This project could be extended naturally to parallel processing by including concepts like distributed objects. It would also be interesting to study parallel object oriented programs on virtual shared memory systems, because the object oriented programming paradigm can naturally provide the data locality that is necessary on these systems.
|3. Simulation of Reaction Diffusion Systems|
Stationary and instationary reaction diffusion equations in two or three space dimensions consist of systems of coupled nonlinear elliptic or parabolic equations, respectively. Typical applications are the simulation of environmental pollution spreading or the multigroup neutron diffusion equations. Based on experience with multigrid algorithms for these problems, including their parallelization and adaptive techniques, I am interested in further applications to realistic problems.
|4. Multilevel Adaptive Finite Element Methods|
The combination of adaptive hierarchical finite elements with fast multilevel methods provides an elegant and efficient approach to many problems in continuum mechanics. Two recent interesting aspects of these methods are:
|the possibility to combine the multilevel structure efficiently with an (implicit) extrapolation which does not require uniform meshes or global regularity.|
|low communication parallelization techniques, most remarkably those which only require coarse grid communication (and some domain overlap)|
I am interested in applications of these techniques to real-life problems, including (but not limited to) problems in (nonlinear) elasticity and porous media flow. Presently we are working on the simulation of a plasma immersion ion implantation (PII) simulation jointly with the Lehrstuhl für Experimentalphysik IV of the Augsburg University.
|5. Algebraic multigrid for sparse matrix equations|
Algebraic Multigrid (as proposed by Brandt, McCormick, Ruge, and Stüben) is a general large linear system solver based on an analogy to partial differential equations. In a setup phase the coarse levels are constructed recursively from the coefficient matrix alone without using any notion of an underlying discretization. This part of the algorithm consists of a heuristic backtracking and is (remotely) related to the symbolic factorization used in direct methods. The AMG method has been successfully applied to systems, as they arise in the layout optimization of electric circuits. By its generality the AMG method can be applied almost everywhere, where large sparse systems occur. Further interesting research problems in AMG are:
|Additive AMG, and its use as a preconditioner in CG-like algorithms.|
|Parallelization. This is not trivial, especially for the setup phase which is recursive and would have to be modified substantially.|
|Non-standard smoothers (in particular ILU) to further improve the robustness.|
For further information please contact me by e-mail or check these references.