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Mathematics of Computation

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Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs


Author: Kristin Kirchner
Journal: Math. Comp. 89 (2020), 2801-2845
MSC (2010): Primary 35R60, 60H15, 65C30, 65M12, 65M60
DOI: https://doi.org/10.1090/mcom/3524
Published electronically: May 26, 2020
MathSciNet review: 4136548
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Abstract:

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process.

In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective–injective tensor product spaces as trial–test spaces. We then propose numerical approximations based on Petrov–Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor norms.

In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov–Galerkin discretizations, which we show to be stable and quasi-optimal.

In both parts, the outcomes are validated by numerical examples.


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Additional Information

Kristin Kirchner
Affiliation: Seminar for Applied Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland
Address at time of publication: Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands
MR Author ID: 1188840
Email: k.kirchner@tudelft.nl

Keywords: Stochastic differential equations, moments, covariance, variational problem, tensor product spaces, stability and convergence, Petrov–Galerkin method
Received by editor(s): August 24, 2018
Received by editor(s) in revised form: July 8, 2019
Published electronically: May 26, 2020
Article copyright: © Copyright 2020 American Mathematical Society