Research Hans Z. Munthe-Kaas
Page last updated by HZMK: May, 2012
Overview:
My research is in the borderland between pure- and applied mathematics and computer science. A recurring theme is the role of mathematical abstractions in computational science, both as a tool for constructing efficient algorithms and as an organizing principle for computational software. Thus, as an overall label, most of my work is within Foundations of Computational Mathematics.
I am interested in applications of differential geometric techniques, geometric integration and structure preserving algorithms for solution of differential equations. Lie group integrators are numerical integration methods for differential equations built from coordinate independent operations such as Lie group actions on a manifold. Lie group integrators have been developed in tight cooperation between the research groups in Bergen, DAMTP Cambridge and mathematics NTNU.
Analysis of numerical Lie group integrators lead us to the study of new types of formal power series for flows on manifolds. Lie-Butcher theory combines classical B-series for integration schemes with Lie series. This research activity has connections to many areas of mathematical research such as control theory, stochastic differential equations, renormalization, combinatorial Hopf algebra theory and non-commutative symmetric functions, the subject of the 2010 COCO trimester.
In my PhD (1989) I, among other things, investigated FFTs for functions possessing crystallographic symmetries. After 2004, I have again found interest in this topic, due to connections between this theory and multivariate versions of Chebyshev polynomials. This is applied in the construction of spectral element methods based on triangular and simplicial subdivisions of the domain. A related topic is applications of Fourier transforms on groups, with applications in numerical linear algebra and image processing.
My PhD work, for which I was awarded the Esso prize in 1999, was on numerical linear algebra and parallel algorithms. I was interested in applications of group theory in the construction of fast elliptic solvers, as well as an organizing principle for massively parallel computations. In the early 1990s I was writing software and developing algorithms for massively parallel computers, in particular FFTs and various algebraic techniques for routing permutations on massively parallel computers. I was also interested in construction of permutation networks, in particular I studied some generalized shuffle-exchange networks. These are based on interesting connections between permutation networks and the theory of linear shift registers.
Object oriented program design is founded on the distinction between specification and implementation, or ‘what’ and ‘how’. This is a well established principle in core computer science. In computational mathematics, however, software has traditionally been organized around coordinate based formulations and discrete representations. In the SOPHUS project, initiated around 1991, we investigated coordinate free formulations for numerical algorithm design and for numerical software. In particular we developed coordinate free formulations of tensor computations and this was also an initial motivation for the study of Lie group integrators. The DiffMan Matlab toolbox for solving differential equations on manifolds is based on the ideas of the SOPHUS C++ library.