Computational Mathematics: Numerical Analysis and Symbolic Computation

The proposed DK involves the Institutes of Applied Geometry, of Computational Mathematics, of Industrial Mathematics, and of Symbolic Computation (RISC) of the Johannes Kepler University (JKU), and the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences. The proposed DK will build on the expertise accumulated within the framework of the FWF Special Research Program SFB F013 ``Numerical and Symbolic Scientific Computing``, a research network consisting of exactly the same participating institutes. The SFB started its work in April 1998. Its overall and long-term scientific goals concerned a non-trival mathematical combination of numerical and symbolic methods for solving large-scale direct and inverse problems with constraints. As a result of the hearing from December 2003, the SFB was granted significantly increased funding for a third and last period of four years, from April 2004 until March 2008. The reviewers (from Canada, France, Germany, Holland, and Norway) also evaluated the educational aspects of combining symbolic and numerical computing; from the FWF-review-protocol: ``The reviewers especially appreciated the activities and the success of the SFB in educating young scientists: good training is provided for young people of different ages.`` As a consequence of the success of the SFB in this regard, the SFB participants unanimously decided to make an effort to establish a focused continuation of this kind of graduate education in the form of an FWF Doktoratskolleg. In the initial phase of the DK, SFB infrastructure will be available. In order to utilize the expertise accumulated during the period of the SFB, the long-term educational goal of the proposed DK is based on similar scientific grounds but stated slightly more generally as follows: Already for the first period, the overall goal of the DK program will be to provide intensive PhD training in two fundamental areas of computational mathematics: numerical analysis (in particular, of direct and inverse field problems) and symbolic computation. This will be accomplished by course work hat involves lectures from both areas and by interdisciplinary seminars. - The long-term goal is to establish a corresponding distinguished PhD Program at the JKU which will attract young researchers world-wide. Such kind of DK education will stimulate numerous new avenues of research. For example, DK graduates will be able to develop new solvers (to direct or inverse problems) by combining numerical approaches with symbolic methods like Groebner bases, or, DK graduates will use symbolic special function algorithms to speed-up finite element methods. First results in each of these directions have been already achieved within the SFB. To achieve such results on a larger scale, special PhD training and interdisciplinary interaction are necessary ingredients. The proposed DK should provide a suitable frame to make it happen.