Fri, 30 Nov 2012

16:00 - 17:00
Gibson Grd floor SR

Multillevel Weiner-Hopf Monte Carlo and Euler-Poisson schemes for L\'evy processes

Albert Ferreiro-Castilla
(University of Bath)
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).

Thu, 25 Oct 2012

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Numerical Methods for PDEs with Random Coefficients

Dr Elisabeth Ullmann
(University of Bath)
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.

Mon, 07 Nov 2011
15:45
Oxford-Man Institute

Near-critical survival probability of branching Brownian motion with an absorbing barrier"

Simon Harris
(University of Bath)
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)

Thu, 31 Jan 2002

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Iterative methods for PDE eigenvalue problems

Prof Ivan Graham
(University of Bath)
Abstract
When steady solutions of complex physical problems are computed numerically it is often crucial to compute their stability in order to, for example, check that the computed solution is "physical", or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an eigenvalue problem which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: \\ $Ax=\lambda Mx$ (1) \\ with $A$ and $M$ large and sparse. In general $A$ is unsymmetric and $M$ is positive semi-definite. Only a small number of "dangerous" eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform "shift-invert" iterations, which require repeated solution of systems of the form \\ $(A - \sigma M)y = Mx$, (2) \\ for some shift $\sigma$ (which may be near a spectral point) and for various right-hand sides $x$. In large applications systems (2) have to be solved iteratively, requiring "inner iterations". \\ \\ In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising algebraic domain decomposition techniques for the inner iterations. \\ \\ In the first part we will describe an analysis of inverse iteration techniques for (1) for a model problem in the presence of errors arising from inexact solves of (2). The delicate interplay between the convergence of the (outer) inverse iteration and the choice of tolerance for the inner solves can be used to determine an efficient iterative method provided a good preconditioner for $A$ is available. \\ \\ In the second part we describe an application to the computation of bifurcations in Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation. We describe the construction of appropriate preconditioners for the corresponding (3 x 3 block) version of (2). These use additive Schwarz methods and can be applied to any unstructured mesh in 2D or 3D and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko. \\ \\ An application to the computation of a Hopf bifurcation of planar flow around a cylinder will be given. \\ \\ This is joint work with Jörg Berns-Müller, Andrew Cliffe, Alastair Spence and Eero Vainikko and is supported by EPSRC Grant GR/M59075.
Thu, 30 Oct 2003

14:00 - 15:00
Comlab

Preconditioning for 3D sedimentary basin simulations

Dr Robert Scheichl
(University of Bath)
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.

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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.

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