Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs
Preprint, 2019

The computation of quadratic functionals of the solution to a linear stochastic partial differential equation with multiplicative noise is considered. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. Weak error rates are also derived for a fully discrete approximation of the stochastic partial differential equation, using the results obtained from the approximation of the Lyapunov equation. In the setting of finite element approximations, a computational complexity comparison reveals that approximating the Lyapunov equation allows for cheaper computation of quadratic functionals compared to applying Monte Carlo or covariance-based methods directly to the discretized stochastic partial differential equation. Numerical simulations illustrates the theoretical results.

white noise

stochastic heat equation

stochastic partial differential equations

weak convergence

parabolic Anderson model

Lyapunov equations

multiplicative noise

numerical approximation

finite element method

Author

Adam Andersson

Smarter AI Sweden

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Annika Lang

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Leander Schroer

Sopra Steria SE

Stochastics for big data and big systems - bridging local and global

Knut and Alice Wallenberg Foundation, 2013-01-01 -- 2018-09-01.

Approximation and simulation of Lévy-driven SPDE

Swedish Research Council (VR), 2015-01-01 -- 2018-12-31.

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

Roots

Basic sciences

More information

Created

12/20/2019