Past

  The "Cubature on Wiener space" algorithm can be regarded as a general approach to high order weak approximations. Based on this observation we will derive many well known weak discretisation schemes and optimise the computational effort required for a given accuracy of the approximation. We show that cubature can also help to overcome some stability difficulties. The cubature on Wiener space algorithm is frequently combined with partial sampling techniques and we outline an extension to these methods to reduce the variance of the samples. We apply the extended method to examples arising in mathematical finance. Joint work of G. Gyurko, C. Litterer and T. Lyons  

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.