The Numerical Approximation of Stochastic Partial Differential Equations

Jentzen, Arnulf; Kloeden, Peter E.

doi:10.1007/s00032-009-0100-0

Cited by 120 publications

(91 citation statements)

References 97 publications

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“…Applications of such numerical schemes to the deterministic (nonlinear) Schrödinger equation can be found in, for example, [4][5][6][7][8][9][10]17,21] and references therein. Furthermore, these numerical methods were investigated for stochastic parabolic partial differential equations in, for example, [23][24][25], more recently for the stochastic wave equations in [2,11,12,27], where they are termed stochastic trigonometric methods, and lately to stochastic Schrödinger equations driven by Ito noise in [1].…”

Section: Introductionmentioning

confidence: 99%

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Cohen

Dujardin

2017

Stoch PDE: Anal Comp

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This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L 2 -norm of the solution.

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Section: Introductionmentioning

confidence: 99%

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Cohen

Dujardin

2017

Stoch PDE: Anal Comp

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“…Numerical simulations of SPDEs are typically based on the Monte Carlo (MC) or the Polynomial Chaos (PC) methods [2,3,4,5]. In both cases, the long-time simulation of SPDEs proves to be quite expensive [6,7,8].…”

Section: Introductionmentioning

confidence: 99%

A dynamical polynomial chaos approach for long-time evolution of SPDEs

Ozen

Bal

2017

Journal of Computational Physics

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We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen-Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

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“…Jentzen and Kloeden [17] give an overview of strong and pathwise schemes. Weak approximation schemes are more difficult.…”

Section: Introductionmentioning

confidence: 99%

Cubature methods for stochastic (partial) differential equations in weighted spaces

Dörsek

Teichmann

Velušček

2013

Stoch PDE: Anal Comp

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The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da PratoZabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory.

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The Numerical Approximation of Stochastic Partial Differential Equations

Cited by 120 publications

References 97 publications

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

A dynamical polynomial chaos approach for long-time evolution of SPDEs

Cubature methods for stochastic (partial) differential equations in weighted spaces

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