[Paper Review] A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise

Arnulf Jentzen, Michael Roeckner|arXiv (Cornell University)|Jan 15, 2010

Stochastic processes and financial applications50 references8 citations

TL;DR

This paper introduces a novel infinite-dimensional Milstein-type algorithm for simulating nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise, which simplifies the simulation of iterated stochastic integrals compared to prior methods. The approach achieves a significant break in computational complexity, demonstrating substantial efficiency gains in numerical experiments on stochastic heat, reaction-diffusion, and Burgers equations.

ABSTRACT

A new algorithm for simulating stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense an infinite dimensional analog of Milstein’s scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation showing significant computational savings.

Motivation & Objective

• To address the high computational cost of simulating nonlinear SPDEs with multiplicative trace class noise.
• To extend the efficiency of Milstein’s scheme—known for SODEs—into the infinite-dimensional setting of SPDEs.
• To overcome the difficulty of simulating iterated stochastic integrals in general multidimensional SODEs by exploiting structural advantages in the infinite-dimensional case.
• To develop a numerically feasible and computationally efficient algorithm for evolutionary-type SPDEs.
• To demonstrate significant reductions in computational complexity compared to existing methods through numerical validation.

Proposed method

• Proposes an infinite-dimensional analog of Milstein’s scheme for SPDEs, leveraging the structure of trace class noise.
• Utilizes a spectral or Galerkin-type spatial approximation to reduce the SPDE to a system of SDEs, enabling application of Milstein-type time discretization.
• Simplifies the simulation of iterated stochastic integrals in the infinite-dimensional setting, where such terms become tractable due to the noise structure.
• Employs a stochastic Taylor expansion up to second order to derive the scheme, ensuring weak order of convergence.
• Applies the scheme to various SPDEs, including stochastic heat, reaction-diffusion, and Burgers equations, using Galerkin projections.
• Employs numerical experiments to validate the theoretical complexity reduction and assess computational savings.

Experimental results

Research questions

• RQ1 Can an infinite-dimensional extension of Milstein’s scheme be constructed that avoids the computational burden of iterated stochastic integrals in general SPDEs?
• RQ2 Does the proposed algorithm achieve a significant reduction in computational complexity compared to existing schemes for nonlinear SPDEs with multiplicative noise?
• RQ3 Is the scheme numerically stable and efficient across different types of nonlinear SPDEs, such as stochastic heat and Burgers equations?
• RQ4 How does the complexity of the new algorithm compare quantitatively to standard Euler-type or other higher-order schemes in terms of operations and random variables?
• RQ5 Can the structural properties of trace class noise in infinite dimensions be exploited to simplify the simulation of stochastic integrals?

Key findings

• The proposed infinite-dimensional Milstein-type scheme achieves a notable reduction in computational complexity compared to previously used algorithms for nonlinear SPDEs with multiplicative trace class noise.
• The simulation of iterated stochastic integrals, which is computationally prohibitive in finite-dimensional SODEs, becomes feasible and efficient in the infinite-dimensional setting due to the noise structure.
• Numerical experiments on the stochastic heat equation, stochastic reaction-diffusion equation, and stochastic Burgers equation show significant computational savings.
• The algorithm maintains high accuracy while drastically reducing the number of required random variables and computational operations.
• The method outperforms standard Euler-type schemes in terms of efficiency, particularly for problems requiring high accuracy.
• The key insight is that the infinite-dimensional setting allows for simplifications not present in finite-dimensional cases, enabling practical implementation of higher-order schemes.