# General Overview

### Research Context

The interdisciplinary PhD research program (DK) involves the Institutes of Applied Geometry (Geo), of Computational Mathematics (NuMa), of Industrial Mathematics (IndMath), and of Symbolic Computation (RISC) of the Johannes Kepler University, and the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences. The DK builds 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.

Historical note: The Special Research Program SFB F013 on "Numerical and Symbolic Scientific Computing" (1998–2008)

The overall and long-term scientific goal of the SFB was the design, verification, implementation, and analysis of numerical, symbolic, and geometrical methods for solving large-scale direct and inverse problems with constraints and their synergetical use in scientific computing for real life problems of high complexity. This included so-called field problems, usually described by partial differential equations (PDEs), and algebraic problems, e.g., involving constraints in algebraic formulation.

### Overall scientific DK concept

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:

The overall goal of the DK program is 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 is accomplished by course work that involves lectures from both areas and by interdisciplinary seminars. --- The long-term goal is to establish a corresponding distinguished PhD Program at JKU which will attract young researchers from all over the world. We expect that 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 Gröbner bases; DK graduates will design new procedures for geometrical scientific computing by utilizing implementations of computer algebra algorithms for problems in algebraic geometry; DK graduates will use symbolic special function algorithms to speed-up hp finite element methods; DK graduates will use symbolic optimization techniques for constructing new algebraic multilevel preconditioners, DK graduates will be able to develop formal mathematical theories by using automated reasoning tools established within the DK, etc. --- It should be noted that first results in each of these directions have been already achieved within the SFB; see the book Numerical and Symbolic Scientific Computing - Progress and Prospects.