Symbolic Calculus & Geometric Integration

Members

Mark Sofroniou

Giulia Spaletta

Geometric integration

Geometric integration is a recent branch of numerical analysis and computational mathematics which aims to the development of numerical methods which incorporate qualitative information of the underlying problem into their structure.
In the last years, the numerical analysis of differential equations has changed significantly. Previously, the main efforts of algorithm developers were aimed at developing robust 'black box' integrators, whose main goal was to obtain fast solvers with small global error.
Recently, it has been increasingly clear that for many applications it is crucial to preserve qualitative features of the underlying continuous equations in the numerical solution. Examples are preservation of energy, momentum or symplectic structures, and numerical solutions that evolve on a given manifold.
During the last decade, we have seen the development of several different new algorithms, many inspired by geometrical ideas. In the same period, the commonly used programming languages in computational science have shifted from procedural languages such as FORTRAN and C towards modern object oriented languages such as C++, MATHEMATICA, MATLAB and others.
This development allows us to capture advanced mathematical concepts as objects in a computer program, and might further boost the interest in applying geometrical concepts in computational mathematics.

Examples of Geomteric Integration algorithms for differential equations include:

Geometric ideas are also important in other areas of numerical analysis such as linear algebra and optimization.

Symbolic Computation

The primary goal is the research and development of algorithms for computer algebra, including both symbolic computation and hybrid symbolic-numeric computation. The algorithms developed are incorporated into the Mathematica computer algebra environment Some of the main research interests include: