Towards High-order Methods for Stochastic Differential Equations with White Noise: A Spectral Approach

zhang, zhongqiang

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Year:: 2014
Contributor:: zhang, zhongqiang (creator); Karniadakis, George (Director); Rozovskii, Boris (Director); Tretyakov, Michael (Reader); Sarkis, Marcus (Reader); Brown University. Applied Mathematics (sponsor)
Genre:: theses
Subject:: Wiener chaos expansion; stochastic collocation method; stochastic differential equation; Brownian motion; spectral approximation; time integration; strong and weak convergence; numerical schemes; Wong-Zakai approximation; Stochastic differential equations; Brownian motion processes
Extent:: xvi, 285 p.
DOI: https://doi.org/10.7301/Z06Q1VKH

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Abstract:: We develop a recursive multistage Wiener chaos expansion method (WCE) and a recursive multi-stage stochastic collocation method (SCM) for numerical integration of linear stochastic advection-diffusion-reaction equations with multiplicative white noise. We show that both methods are comparably efficient in computing the first two moments of solutions for long time intervals, compared to a direct application of WCE and SCM while both methods are more efficient than the standard Monte Carlo method if high accuracy is required. Both methods belong to Wong-Zakai approximation, where Brownian motion is truncated using its spectral expansion before any discretization in time and space. For computational convenience, WCE is associated with the Ito formulation of underlying equations and SCM is associated with the Stratonovich formulation. We apply SCM using Smolyak’s sparse grid construction to obtain the shock location of the one-dimensional piston problem, which is modeled by stochastic Euler equations with multiplicative white noise. We show numerically that SCM is efficient for short time simulations and for small magnitudes of noises and quasi-Monte Carlo methods are efficient for moderate large-time simulations. We also illustrate the efficiency of SCM through error estimates for a linear model problem. We further investigate the effect of a spectral approximation of Brownian motion, rather than a piecewise linear approximation, for both spatial and temporal noise. For spatial noise, we consider semilinear elliptic equations with additive noise and show that when the solution is smooth enough, the spectral approximation is superior to the piecewise linear approximation while both approximations are comparable when the solution is not smooth. For temporal noise, we use this spectral approach to design numerical schemes for stochastic delay differential equations under the Stratonovich formulation. We show that the spectral approach admits higher-order accuracy only for higher-order schemes. Besides equations with coefficients of linear growth, we also consider stochastic ordinary differential equations with coefficients of polynomial growth. We formulate a basic relationship between local truncation error and global error of numerical methods for these equations, and apply this relationship for our explicit balanced scheme to obtain the convergence order.
Notes:: Thesis (Ph.D. -- Brown University (2014)

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zhang, zhongqiang, "Towards High-order Methods for Stochastic Differential Equations with White Noise: A Spectral Approach" (2014). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.7301/Z06Q1VKH

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Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
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