Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs
HTML articles powered by AMS MathViewer

by Kristin Kirchner HTML | PDF
Math. Comp. 89 (2020), 2801-2845 Request permission


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.

Similar Articles
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:
  • Received by editor(s): August 24, 2018
  • Received by editor(s) in revised form: July 8, 2019
  • Published electronically: May 26, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 2801-2845
  • MSC (2010): Primary 35R60, 60H15, 65C30, 65M12, 65M60
  • DOI:
  • MathSciNet review: 4136548