[Paper Review] Numerical Solution of Stochastic Partial Differential Equations with Correlated Noise

Dirk Blömker, Minoo Kamrani|arXiv (Cornell University)|Nov 9, 2013

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper presents a numerical scheme for stochastic partial differential equations (SPDEs) with correlated, non-diagonal colored noise using spectral Galerkin spatial discretization and a time-stepping method by Jentzen & Kloeden. It establishes path-wise convergence in the uniform topology with convergence rates of order $ N^{-rac{1}{2}} $ in space and $ ( abla t)^{rac{1}{4}} $ in time under uniform bounds on the numerical solution, extending prior results to general nonlinearities and correlated noise structures.

ABSTRACT

In this paper we investigate the numerical solution of stochastic partial differential equations (SPDEs) for a wider class of stochastic equations. We focus on non-diagonal colored noise instead of the usual space-time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen $\&$ Kloeden, we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.

Motivation & Objective

• To extend the numerical analysis of SPDEs with colored noise beyond space-time white noise and uncorrelated Brownian motions.
• To establish path-wise convergence in the uniform topology for SPDEs with non-diagonal, correlated noise.
• To derive convergence rates for combined space-time discretization using spectral Galerkin and time-stepping schemes.
• To validate the theoretical results with numerical examples under local Lipschitz nonlinearities and polynomial growth.

Proposed method

• Spatial discretization via spectral Galerkin method using eigenfunctions of the Laplacian operator.
• Time discretization based on the numerical scheme proposed by Jentzen & Kloeden for stochastic differential equations.
• Use of uniform bounds on the numerical solution or the Galerkin approximation to control error propagation.
• Application of energy-type a-priori estimates to verify uniform bounds for spectral methods.
• Incorporation of correlated noise through a covariance operator that does not commute with the linear operator $ A $.
• Implementation via Cholesky decomposition to generate correlated Brownian motions in finite-dimensional approximations.

Experimental results

Research questions

• RQ1 What is the convergence rate of the numerical solution for SPDEs with correlated, non-diagonal noise in the uniform topology?
• RQ2 How can uniform bounds on the numerical approximation be established for SPDEs with local Lipschitz nonlinearities?
• RQ3 Can the spectral Galerkin method combined with time discretization achieve path-wise convergence for SPDEs with colored noise?
• RQ4 What is the impact of non-commuting noise covariance and linear operators on the convergence behavior?
• RQ5 How do numerical experiments confirm the theoretical convergence rates in practice?

Key findings

• The paper establishes path-wise convergence in the uniform topology for the numerical solution of SPDEs with correlated noise.
• The convergence rate in space is $ O(N^{-rac{1}{2}}) $, and in time it is $ O(( abla t)^{rac{1}{4}}) $, under uniform boundedness assumptions.
• Numerical examples confirm the theoretical convergence rate of $ \frac{1}{2} $ in the spatial discretization parameter $ N $, as seen in Figure 3.
• The uniform bound assumption allows the treatment of nonlinearities with local Lipschitz conditions and polynomial growth.
• The method is applicable to general SPDEs beyond the Burgers equation, including higher-dimensional domains and different boundary conditions.
• The implementation using Cholesky decomposition successfully generates correlated Brownian motions, enabling accurate simulation of the noise structure.