Approximation of stochastic partial differential equations by a kernel-based collocation method

Abstract

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time-stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centred at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.

Keywords:

• kernel-based collocation
• numerical solutions
• stochastic partial differential equation
• reproducing kernel
• Matérn function
• Gaussian process

2010 AMS Subject Classifications::

• 46E22
• 65D05
• 60G15
• 60H15
• 65N35

Acknowledgements

We thank Phaedon-Stelios Koutsourelakis for the inspiration to solve SPDEs with a maximum likelihood-based approach. The work of Igor Cialenco was partially supported by the National Science Foundation (NSF) grant DMS-0908099. Gregory E. Fasshauer and Qi Ye acknowledge support from NSF grants DMS-0713848 and DMS-1115392. The authors also thank the anonymous referee and the editors for their helpful comments and suggestions which improved greatly the final manuscript.