The Mini-course: “Three Lectures on Quasi Monte Carlo Methods and Applications to PDE with Random Coefficients” by Prof. Ian Sloan (University of New South Wales, Australia) has taken place at C2 Lecture Hall, VIASM on 12 March, 2014. This is also one of the scientific activities of Prof. Dinh Dung’s research group at VIASM.

In this series of three lectures, Prof. Ian Sloan describes the application of modern Quasi Monte Carlo methods (which are just equal weight quadrature rules over a high-dimensional cube) to the computation of expected values associ-ated with a PDE with random coefficients. A key motivating example is the flow of a liquid (oil or water) through a porous material, with the permeability modelled as a random field. In such a problem the expected value of a physical quantity (such as the pressure at a special point, or the net flow across the outflow boundary) involves many random variables. (In principle an infinite number of random variables are needed to describe a random field. In practice the number needed to achieve an adequate approximation is often very large.) And an expected value over many variable is a high-dimensional integral. Such problems are often treated by Monte Carlo methods.

Quasi-Monte Carlo methods, like Monte Carlo, are sampling methods, but with the difference that the points are chosen deterministically rather than randomly. The aim is always to design point distributions that are better than random. There are also other methods for treating such problems, including polynomial chaos, stochastic Galerkin and stochastic collocation, but all methods face difficulties when the dimensionality of the problem (that is, the number of random variables needed to describe the field) is high. In such cases Monte Carlo and Quasi Monte Carlo methods may be the only alternative.

After the introduction Prof. Sloan outlines the modern theory of Quasi Monte Carlo methods, and then discuss the application of such rules to PDE with random coefficients. There are no specific prerequisites, but it would be helpful to have a good knowledge of analysis and elliptic PDE.

The Mini-course has taken place for the whole day. It attracted more than 30 participants from various academic institutions in Vietnam.