Time: 12/3/2014
Program:
9:00 – 10:00: The first lecture
10:00 – 10:30: Coffee break
10:30 – 11:30: The second lecture
11:30 – 14:00: Lunch time
14:00 – 15:00: The third lecture
15:00 – 15:30: Coffee time
Location: VIASM Lecture Hall (C2).
Lecturer: Prof. Ian H. Sloan, University of New South Wales, Australia
Abstract:
In this series of three lectures I will describe 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 an introduction I will outline 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.
References
[1] J. Dick, F.Y. Kuo, and I.H. Sloan. High dimensional integration – the Quasi-Monte Carlo way. Acta Numerica 13, 133-288, 2013.
[2] I.G. Graham, F.Y. Kuo, D. Nuyens, R. Scheichl, and I.H. Sloan. Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. Journal of Computational Physics, 230, 3668–3694, 2011.
[3] F.Y. Kuo, C. Schwab, and I.H. Sloan. Quasi-Monte Carlo for very high dimensional integration: the standard setting and beyond. ANZIAM Journal, to appear.
[4] F.Y. Kuo, C. Schwab, and I.H. Sloan. Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. ANZIAM J 53, 1-37, 2011.
[5] F.Y. Kuo and I.H. Sloan. Lifting the curse of dimensionality. SIAM J Numer Anal 50, 3351-3374, 2012.
Registration:
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