Arnulf Jentzen

Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise

We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

Taylor Approximations for Stochastic Partial Differential Equations

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence. In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hlder continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right. The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix. Audience: Applied and pure mathematicians interested in using and further developing numerical methods for SPDEs will find this book helpful. It may also be used as a source of material for a graduate course. Contents: Preface; List of Figures; Chapter 1: Introduction; Part I: Random and Stochastic Ordinary Partial Differential Equations; Chapter 2: RODEs; Chapter 3: SODEs; Chapter 4: SODEs with Nonstandard Assumptions; Part II: Stochastic Partial Differential Equations; Chapter 5: SPDEs; Chapter 6: Numerical Methods for SPDEs; Chapter 7: Taylor Approximations for SPDEs with Additive Noise; Chapter 8: Taylor Approximations for SPDEs with Multiplicative Noise; Appendix: Regularity Estimates for SPDEs; Bibliography; Index.

GALERKIN APPROXIMATIONS FOR THE STOCHASTIC BURGERS EQUATION

Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite-dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated. These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation, and a stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations. The main novelty in this article is the estimation of the difference of the finite-dimensional Galerkin approximations and of the solution of the infinite-dimensional SPDE uniformly in space, i.e., in the

Taylor expansions of solutions of stochastic partial differential equations with additive noise

L^\infty

Efficient simulation of nonlinear parabolic SPDEs with additive noise

-topology, instead of the usual Hilbert space estimates in the

A Milstein Scheme for SPDEs

L^2

Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise

-topology, that were shown before.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Ito formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Ito formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

An Improved Maximum Allowable Transfer Interval for

Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649-667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement.

L^{p}

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, stochastic reaction diffusion equations and a stochastic Burgers equation, showing significant computational savings.