A Milstein scheme for SPDEs

Abstract

This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite dimensional commutativity condition. In particular, a suitable infinite dimensional analog of Milstein's algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation and stochastic reaction diffusion equations showing signifficant computational savings.

The article is slightly revised and shortened. In particular, some numerical simulations are removed

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• Stochastic integration in UMD Banach spaces
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• On implicit and explicit discretization schemes for parabolic SPDEs in any dimension
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Cited by in corpus (27)

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• A mild Ito formula for SPDEs
• Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise
• Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise
• Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
• Stochastic delay fractional evolution equations driven by fractional Brownian motion
• An exponential Wagner-Platen type scheme for SPDEs
• Weak error analysis for semilinear stochastic Volterra equations with additive noise
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• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values
• Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises
• Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise
• Numerical Schemes for Rough Parabolic Equations
• Optimal Regularity of Stochastic Evolution Equations in M-type 2 Banach Spaces
• Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations
• Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise
• Strong Convergence of the Linear Implicit Euler Method for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise
• Maximal gamma-regularity
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• Strong convergence of the backward Euler approximation for the finite element discretization of semilinear parabolic SPDEs with non-global Lipschitz drift driven by additive noise
• Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise