Section: New Results

Application of greedy algorithms

Participants : Eric Cancès, Claude Le Bris, Tony Lelièvre, David Pommier, Atsushi Suzuki.

Following the work [35] where C. Le Bris and T. Lelièvre, in collaboration with Y. Maday have studied a numerical method based on greedy algorithms to solve high-dimensional partial differential equations, we are now working in various directions:

• with D. Pommier (within the PhD thesis of Jose Infante Acevedo), we propose a pricing method for pricing multi-asset options. The option pricing problem can be reduced to a computation of multi-dimensional integrals. Deterministic methods (as opposed to Monte Carlo methods) suffer from the so-called curse of dimensionality , i.e. , the exponential increase in the number of unknowns as the dimension d increase. In order to reduce the complexity, we consider a type of nonlinear approximation approach, presented in [35] , to compute numerical integrations on the Fourier domain. This method can readily be applied to solving the problems under various asset price dynamics, for which the characteristic function (i.e. the Fourier transform of the probability density function) is available. This is the case for models from the class of regular affine processes, which also includes some jump-diffusion models. We compare our method to the sparse combination technics proposed by Leentvaar and Oosterlee.

• with A. Suzuki, we are working on the convergence of the method, and on its application to molecular dynamics, more precisely, to the evaluation of the so-called commitor function which is very important to understand various properties of "reactive trajectories", namely trajectories which start in a metastable state and ends in another one.

• with E. Cancès, within the PhD thesis of V. Ehrlacher, we are working on extensions of the method to nonlinear problem, and in particular to contact problems, with applications to uncertainty propagation in solid mechanics problems.