University of Bath

Let $K\subset {\mathbb R}$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g. a semi-algebraic or a real analytic set. A definable family $\{S_\delta\ |  0<\delta\in{\mathbb R}\}$ of compact subsets of $K$, is called a monotone family if $S_\delta\subset S_\eta$ for all sufficiently small $\delta>\eta>0$. The main result in the talk is that when $\dim K=2$ or $\dim K=n=3$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta>0$, the intersections $S_\delta\cap\Lambda$ is equivalent to one of five (respectively, nine) standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). As a consequence, we prove the two-dimensional case of the topological conjecture on approximation of definable sets by compact families.

This is joint work with Andrei Gabrielov (Purdue).

The usual variational formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive). In this talk, I will argue that this indefiniteness is not an inherent feature of the Helmholtz equation itself, only of its standard formulations. I will do this by presenting new sign-definite formulations of several Helmholtz boundary value problems.

This is joint work with Andrea Moiola (Reading).
 

The coefficients in mathematical models of physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. It is of great importance to quantify the uncertainty in the model outputs based on the (uncertain) information that is available on the model inputs. This invariably leads to very high dimensional quadrature problems associated with the computation of statistics of quantities of interest, such as the time it takes a pollutant plume in an uncertain subsurface flow problem to reach the boundary of a safety region or the buckling load of an airplane wing. Higher order methods, such as stochastic Galerkin or polynomial chaos methods, suffer from the curse of dimensionality and when the physical models themselves are complex and computationally costly, they become prohibitively expensive in higher dimensions. Instead, some of the most promising approaches to quantify uncertainties in continuum models are based on Monte Carlo sampling and the “multigrid philosophy”. Multilevel Monte Carlo (MLMC) Methods have been introduced recently and successfully applied to many model problems, producing significant gains. In this talk I want to recall the classical MLMC method and then show how the gains can be improved further (significantly) by using quasi-Monte Carlo (QMC) sampling rules. More importantly the dimension independence and the improved gains can be justified rigorously for an important model problem in subsurface flow. To achieve uniform bounds, independent of the dimension, it is necessary to work in infinite dimensions and to study quadrature in sequence spaces. I will present the elements of this new theory for the case of lognormal random coefficients in a diffusion problem and support the theory with numerical experiments.

Biharmonic maps are the solutions of a variational problem for maps

between Riemannian manifolds. But since the underlying functional

contains nonlinear differential operators that behave badly on the usual

Sobolev spaces, it is difficult to study it with variational methods. If

the target manifold has enough symmetry, however, then we can combine

analytic tools with geometric observations and make some statements

about existence and regularity.

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of L\'evy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. We also provide here a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant. We find that the rate of convergence is uniformly with respect to the ``jump activity'' (e.g. characterised by the Blumenthal-Getoor index).