University of Bath

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).

Partial differential equations (PDEs) with random coefficients are used in computer simulations of physical processes in science, engineering and industry applications with uncertain data. The goal is to obtain quantitative statements on the effect of input data uncertainties for a comprehensive evaluation of simulation results. However, these equations are formulated in a physical domain coupled with a sample space generated by random parameters and are thus very computing-intensive.

We outline the key computational challenges by discussing a model elliptic PDE of single phase subsurface flow in random media. In this application the coefficients are often rough, highly variable and require a large number of random parameters which puts a limit on all existing discretisation methods. To overcome these limits we employ multilevel Monte Carlo (MLMC), a novel variance reduction technique which uses samples computed on a hierarchy of physical grids. In particular, we combine MLMC with mixed finite element discretisations to calculate travel times of particles in groundwater flows.

For coefficients which can be parameterised by a small number of random variables we employ spectral stochastic Galerkin (SG) methods which give rise to a coupled system of deterministic PDEs. Since the standard SG formulation of the model elliptic PDE requires expensive matrix-vector products we reformulate it as a convection-diffusion problem with random convective velocity. We construct and analyse block-diagonal preconditioners for the nonsymmetric Galerkin matrix for use with Krylov subspace methods such as GMRES.

We will consider a branching Brownian motion where particles have a drift $-\rho$, binary branch at rate $\beta$ and are killed if they hit the origin. This process is supercritical  if $\beta>\rho^2/2$ and we will discuss the survival probability in the regime as criticality is approached. (Joint work with Elie Aidekon)