The aim of the research is to apply Monte Carlo methods to matrix computations, partial differential equations and integral equations.

The research on matrix computations is concentrating on developing algorithms to solve systems of linear algebraic equations, invert matrices, finding bilinear forms of a solution, and finding eigenvalues. These are problems of unquestionable importance in many scientific and engineering applications: e.g. real-time speech coding, digital signal processing, communications, stochastic modelling, and many physical problems involving partial differential equations.

The research considers:

• hybrid methods, which use Monte Carlo to quickly generate a rough inverse and improving this with deterministic refinement;
• parallel algorithms, which harness multiple processing elements to efficiently and effectively solve the problems;
• block methods;
• quasi Monte Carlo, using Sobol and Halton sequences for the Markov Chains;
• sparse matrices, since sparse problems are common in many of the applications of these algorithms;
• resolvent methods, to produce an unbiased estimate of the inverse;
• combining the different methods into a code library.

The research on partial differential equations and integral equations is concentrated on using adaptive techniques for applications in environmental modelling, gas discharge plasma, and semi-conductor physics.