Rapid Covariance-Based Sampling of Linear SPDE Approximations in the Multilevel Monte Carlo Method
Paper in proceedings, 2020

The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along with an implicit Euler scheme to arrive at a fully discrete approximation of the mild solution to the equation. A scheme is presented to compute the covariance of this approximation, which allows for rapid sampling in a Monte Carlo method. This is then extended to a multilevel Monte Carlo method, for which a scheme to compute the cross-covariance between the approximations at different levels is presented. In contrast to traditional path-based methods it is not assumed that the Galerkin subspaces at these levels are nested. The computational complexities of the presented schemes are compared to traditional methods and simulations confirm that, under suitable assumptions, the costs of the new schemes are significantly lower.

Finite element method

Covariance operators

Multilevel Monte Carlo

Monte Carlo

Stochastic partial differential equations

Author

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Springer Proceedings in Mathematics and Statistics

21941009 (ISSN) 21941017 (eISSN)

Vol. 324 423-443

13th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2018
Rennes, France,

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1007/978-3-030-43465-6_21

More information

Latest update

12/2/2020