MSI Weekly Bulletin - Week starting Monday 24 April, 2006

In this talk we will discuss new results on domain decomposition preconditioning for linear systems arising from finite element approximation of symmetric elliptic problems of the form - Div. alpha Grad u = f, with highly variable coefficient alpha. The work is motivated by problems arising in the computation of flow through complex heterogeneous media. We will describe condition number bounds for classical two-level additive Schwarz preconditioners which are built from (i) local solves on overlapping subdomains (obtained, for example, by graph partitioning software) and (ii) a global solve on a suitable space of functions defined on a coarse mesh. Letting A denote the finite element stiffness matrix and M^{-1} the preconditioner, our condition number bounds take the form: kappa(M^{-1}A) <= pi(alpha) gamma(alpha) (1 + beta) . Here beta is coefficient-independent and is bounded in terms of the ratio of the coarse mesh diameter and the overlap and pi(alpha) and gamma(alpha) are explicit coefficient-dependent ``robustness indicators'' which depend respectively on the choice of overlapping subdomains and on the choice of coarse space. While this estimate shows that it is the coarse mesh diameter (and not the subdomain diameter) which determines beta, we focus more in this talk on ``coefficient robustness'', i.e. on ensuring that the indicators pi(alpha) and gamma(alpha) are well-behaved. We show that this is possible even when the coefficient alpha varies rapidly on a finer scale than the coarse mesh (a scenario highly likely in real applications, but not covered by classical estimates). In particular, coarse space robustness can be achieved by choosing coarse space basis functions with moderate energy in the H1- inner product weighted by alpha. We shall give extensive numerical experiments which show that ideas from subgrid modelling (or ``multiscale finite elements'') may be used to choose good coarse spaces for a variety of situations. The methods which can be analysed have close connections to some algebraic coarsening procedures which we will discuss briefly at the end of the talk.

GENERALIZATIONS OF THE BOREL-CANTELLI LEMMA Valentin Petrov StPetersburg University and University of Sydney A generalization of the second part of the Borel-Cantelli lemma is obtained. For a sequence of arbitrary events A^1, A^2, ... on a probability space with the divergent series of probabilities of these events a lower bound for P(lim sup A^n) is found. This lower bound is sharp. Several earlier generalizations due to Erdos and Renyi, Spitzer, Kochen and Stone and some others are special cases.

Recently there has been a growing interest in volatility models in finance. In Thavaneswaran et. al. (Math and Comp. Modeling, 2005 and Stats. and Prob. Letters, 2005) new volatility models have been introduced and volatility forecasts are studied. In this talk, financial application of the ARMA processes with GARCH errors in option pricing is discussed. Application to continuous time volatility models is given. Fuzzy forecasts of volatility including parameter variability is also discussed in some detail.