Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics

The main objective of this project is to develop and analyse specially tailored numerical methods for approximating the solution to differential equations. Such equations are by far the most used mathematical tools for describing dynamical processes in science and engineering. Any numerical solution obtained is just an approximation to the exact solution of the differential equation. The project focuses on numerical methods which retain selected qualitative properties of the solution. Examples of such properties are preservation of energy or momentum, preservation of volume or some other measure, or preservation of a dissipation property. It is well-known that such structure preserving numerical methods often yield solutions with excellent qualitative behaviour for instance when the simulation is run over a very long time. Examples in which such behaviour is of high significance are the simulation of the solar system and in climate modelling. A new aspect of this project is the use of tools and results from two other areas of mathematics, called discrete integrable systems and algebraic combinatorics. The project will develope open source software. We will aim at establishing a network of young researchers having some connection to Norway, ha ving in mind the future recruitment to positions in the university sector or other research institutions. In order to build an international network, common meeting places are of great importance. Every winter, the project organises a winter workshop with social and scientific activities. This is called MaGIC and always includes students and researchers in the project as well as som guests from abroad. In addition the project participants contribute with talks at many international conferences and meetings, the most important ones being the biennual SciCADE meetings and the triennial FoCM meetings. In the wake of SPIRIT some investigators in the project have collaborated on parallel activities in two EC projects, in the Seventh Framework Programme and Horizon 2020. Two proposals have been funded, they have acronyms CRISP and CHiPS and are of type IRSES (FP7) and RISE (H2020). There have been several important results in the project activities on algebraic combinatorics. For instance, there is now a complete characterisation of expansions of numerical methods in B-series, in the sense that it is known precisely which numerical methods that have such B-series. There is a subset of all B-series which form a group known as the Butcher group, and a result of the project is that this group is in fact a Lie group. In an M.Sc. thesis, computer software for making calculations with these algebraic structures has been developed. In the last year these new theories have also found their application to a kind stochastic processes called rough paths. On the preservation of first integrals such as energy and momentum, there have also been a number of new results. Several new properties have been obtained for the Kahan method for quadratic vector fields, and in particular the known set of discrete integrable maps have een significantly extended thanks to this project. The method of Kahan was recently extended to polynomial vector fields of higher degree. This revealed even more integrable maps in the disguise of a numerical integrator. Finally, integral preserving numerical schemes for PDEs on moving grids have been developed. Shape analysis methods have in the past few years become very popular. Originally developed for planar curves, these methods have been extended to higher dimensional curves, surfaces, activities, character motions and many other objects. We have developed a framework for shape analysis of curves in Lie groups for problems of computer animations. In particular, we use these methods to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones. An activity in the project is image processing by means of energy minimising models where numerically dissipative methods have been developed to search for minima. Recently, there have been some important developments in applying methods and techniques developed in the project within image processing. In the last year, structure preserving model reduction has been developed. It is well-known and of great significance that large and complex processes can be simulated at low computational cost because the dynamics of the problem is evolving approximately on some low-dimensional subspace. In the project this reduction is done in a structure preserving way.

Prosjektets effekt og virkning kan i hovedsak knyttes til to hovedkategorier. Det ene er utvikling av unge forskertalenter der en doktorstudent har avlagt sin doktorgrad og er nå forsker i SINTEF. Postdocstipendiaten er fast ansatt som førsteamanuensis ved NTNU Gjøvik. Det andre og mest omfattende er nettverksbygging. Dette har ikke minst vært rettet mot unge forskere som prosjektet på ulike måter har støttet i å skape et internasjonalt kontaktnett, dette går langt utover de som har hatt rekrutteringsstillinger i prosjektet. Konkrete oppnådde mål er: 1 forsker kom til andre runde i ERC Advanced Grant. NTNU ble partner i det store ETN-prosjektet THREAD. En prosjektdeltaker ble arrangør av et spesialsemester innen prosjektets område ved Isaac Newton instituttet i Cambridge, UK, som hadde en betydelig deltakelse fra prosjektets medarbeidere. Prosjektets deltakere har produsert et høyt antall artikler og gitt foredrag ved viktige internasjonale konferanser.

SPIRIT is a basic research project aimed at developing new numerical methods for solving differential equations, a particular focus will be on structure preserving, geometric integration methods. The Norwegian partners of this project are The Norwegian Un iversity of Science and Technology (NTNU) and the University of Bergen (UiB). The project will uncover new links between numerical analysis, discrete integrable systems and algebraic combinatorics. The research will be of mutual benefit and contribute to discover new directions of research in all three of these fields of mathematics. This project should be seen in the context of an emerging trend where new branches of mathematics enter applications. We see a potential for developing brand new mathematica l tools to meet the upcoming challenges of science and engineering. Validation of theoretical results in applications will be an important activity in the project, some selected application areas, like structure preserving image processing, will be given a stronger focus. Symbolic and numerical software will be developed. Under this overarching scientific aim the project will facilitate the recruitment of the next generation mathematicians to Norwegian universities by developing a network of young mathema ticians in pursuit of an academic career in computational mathematics.