Since the 1940s much attention has been devoted to the best ways to discretize differential equations so that they can be solved numerically. Initially, the main emphasis was on all-purpose methods (defined for all differential equations), such as Runge-Kutta methods and linear multistep methods, and their quantitative accuracy. During the last 25 years, however, interest has expanded to considering special classes of differential equations and purpose-built algorithms that are tailored to preserve the special features of each class. In addition to be quantitatively accurate, these novel methods have the advantage of also being qualitatively accurate. This has resulted in methods that preserve symmetries, first integrals, symplectic structure, measure, foliations, Lyapunov functions, etc. These methods are called geometric integration methods.

The present project will address fundamental questions in the field of structure preserving algorithms. The research objectives for the CHiPS project include:

• analysis of algebraic, combinatorial and geometric structures underlying geometric numerical integration schemes for differential equations,
• structure preservation in partial differential equations with highly oscillatory solutions,
• new energy-dissipative techniques for image processing,
• geometric integration methods for Bayesian statistics,
• structure preserving methods for applications in mechanics and control.

Partners involved in the program.

The scientific innovations of the project will be incorporated into usable software tools in cooperation with our non-academic partner MathWorks.

CHiPS project reference at the EU.