Multillevel Weiner-Hopf Monte Carlo and Euler-Poisson schemes for L\'evy processes
Abstract
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).
Numerical Methods for PDEs with Random Coefficients
Abstract
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.
Energy minimising properties of regular and singular equilibria in nonlinear elasticity
15:45
Near-critical survival probability of branching Brownian motion with an absorbing barrier"
Abstract
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)
Iterative methods for PDE eigenvalue problems
Abstract
Preconditioning for 3D sedimentary basin simulations
Abstract
The simulation of sedimentary basins aims at reconstructing its historical
evolution in order to provide quantitative predictions about phenomena
leading to hydrocarbon accumulations. The kernel of this simulation is the
numerical solution of a complex system of time dependent, three
dimensional partial differential equations of mixed parabolic-hyperbolic
type in highly heterogeneous media. A discretisation and linearisation of
this system leads to large ill-conditioned non-symmetric linear systems
with three unknowns per mesh element.
\\
\\
In the seminar I will look at different preconditioning approaches for
these systems and at their parallelisation. The most effective
preconditioner which we developed so far consists in three stages: (i) a
local decoupling of the equations which (in addition) aims at
concentrating the elliptic part of the system in the "pressure block'';
(ii) an efficient preconditioning of the pressure block using AMG; (iii)
the "recoupling'' of the equations. Numerical results on real case
studies, exhibit (i) a significant reduction of sequential CPU times, up
to a factor 5 with respect to the current ILU(0) preconditioner, (ii)
robustness with respect to physical and numerical parameters, and (iii) a
speedup of up to 4 on 8 processors.