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Tensor methods for higher-dimensional Fokker-Planck equation

Presented by: 
Tomas Vejchodsky Academy of Sciences of the Czech Republic
Date: 
Wednesday 6th April 2016 - 11:45 to 12:30
Venue: 
INI Seminar Room 1
Abstract: 
In order to analyse stochastic chemical systems, we solve the corresponding Fokker-Planck equation numerically. The dimension of this problem corresponds to the number of chemical species and the standard numerical methods fail for systems with already four or more chemical species due to the so called curse of dimensionality. Using tensor methods we succeeded to solve realistic problems in up to seven dimensions and an academic example of a reaction chain of 20 chemical species.

In the talk we will present the Fokker-Planck equation and discuss its well-posedness. We will describe its discretization based on the finite difference method and we will explain the curse of dimensionality. Then we provide the main idea of tensor methods. We will identify several types of errors of the presented numerical scheme, namely the modelling error, the domain truncation error, discretization error, tensor truncation error, and the algebraic error. We will present an idea that equilibration of these errors based on a posteriori error estimates yields considerable savings of the computational time.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons