Approximation of SPDE covariance operators by finite elements: A semigroup approach
Preprint, 2021

The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition formula is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert--Schmidt norms with numerical simulations illustrating the results.

stochastic partial differential equations

finite element method

stochastic wave equations

covariance operators

stochastic advection-diffusion equations

integral equations


Mihaly Kovacs

Chalmers, Mathematical Sciences

Annika Lang

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Efficient approximation methods for random fields on manifolds

Swedish Research Council (VR) (2020-04170), 2021-01-01 -- 2024-12-31.

Stochastic Continuous-Depth Neural Networks

Chalmers AI Research Centre (CHAIR), 2020-08-15 -- .

Nonlocal deterministic and stochastic differential equations: analysis and numerics

Swedish Research Council (VR) (2017-04274), 2019-01-01 -- 2021-12-31.

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis


Basic sciences

Related datasets

arXiv:2107.10109 [math.NA] [dataset]


More information