A Stable Multistep Scheme for Solving Backward Stochastic Differential Equations

Zhang, Weidong; Zhang, Guannan; Ju, Lili

doi:10.1137/09076979x

Cited by 76 publications

(119 citation statements)

References 28 publications

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“…Further details, and rigorous error analysis of the proposed BMC method can be found in Refs. [15][16][17][18].…”

Section: Algorithmmentioning

confidence: 99%

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A backward Monte-Carlo method for time-dependent runaway electron simulations

Zhang

Del-Castillo-Negrete

2017

Physics of Plasmas

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Kinetic descriptions of runaway electrons (RE) are usually based on Fokker-Planck models that determine the probability distribution function (PDF) of RE in 2-dimensional momentum space.Despite of the simplification involved, the Fokker-Planck equation can rarely be solved analytically and direct numerical approaches (e.g., continuum and particle-based Monte Carlo (MC)) can be time consuming, especially in the computation of asymptotic-type observables including the runaway probability, the slowing-down and runaway mean times, and the energy limit probability.Here we present a novel backward MC approach to these problems based on backward stochastic differential equations (BSDEs) that describe the dynamics of the runaway probability by means of the Feynman-Kac theory. The key ingredient of the backward MC algorithm is to place all the particles in a runaway state and simulate them backward from the terminal time to the initial time. As such, our approach can provide much faster convergence than direct MC methods (by significantly reducing the number of particles required to achieve a prescribed accuracy) while at the same time maintaining the advantages of particle-based methods (compared to continuum approaches). The proposed algorithm is unconditionally stable, can be parallelized as easy as the direct MC method, and its extension to dimensions higher than two is straightforward, thus paving the way for conducting large-scale RE simulation.

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“…Further details, and rigorous error analysis of the proposed BMC method can be found in Refs. [15][16][17][18].…”

Section: Algorithmmentioning

confidence: 99%

“…On the other hand, adding degrees of freedom with diffusive dynamics requires computing a high-dimensional version of the quadrature formula in Eq. (15). For example, a more accurate collision operator would introduce energy diffusion in the model in Eq.…”

Section: Algorithmmentioning

confidence: 99%

A backward Monte-Carlo method for time-dependent runaway electron simulations

Zhang

Del-Castillo-Negrete

2017

Physics of Plasmas

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“…Based on the quadrature rules used in (5.5), we observe that it is highly possible the quadrature points do not belong to the spatial grid S. In this case, we follow the same strategy as in [33,35] to resolve this issue, i.e., constructing piecewise Lagrange interpolating polynomials based on S to interpolate the integrands at non-grid quadrature points. Again, taking Y n+1 (x) as an example, it can be approximated by…”

Section: The Fully Discrete Schemementioning

confidence: 99%

Second-order schemes for solving decoupled forward backward stochastic differential equations

Zhang

Yang

2014

Sci. China Math.

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We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are driven by a ddimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17,25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.Mathematics subject classification: 60H35, 60H10, 65C20, 65C30

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“…To our best knowledge in the literature, up to now, one-step second-order numerical schemes for solving BSDE and decoupled FBSDEs were proposed and studied in [5,25,28,29,32]. In 2006, Zhao, Chen and Peng proposed numerical schemes for solving BSDE in [25], in which the Crank-Nicolson (short for C-N) is included.…”

Section: Introductionmentioning

confidence: 99%

“…It was proposed in [25] for solving BSDE and the extension for solving decoupled FBSDEs was introduced in [29]. The second-order convergence rate of the C-N scheme for BSDE was proved in [28], but for decoupled FBSDEs is still open until now.…”

Section: Introductionmentioning

confidence: 99%

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

Yang

Zhao

2017

Sci. China Math.

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The Crank-Nicolson (short for C-N) scheme for solving backward stochastic differential equation (BSDE), driven by Brownian motions, was first developed by the authors W. Zhao, L. Chen and S. Peng [SIAM J. Sci. Comput., 28 (2006), 1563-1581, and numerical experiments showed that the accuracy of this C-N scheme was of second order for solving BSDE. This C-N scheme was extended to solve decoupled forwardbackward stochastic differential equations (FBSDEs) by W. Zhao, Y. Li and Y. Fu [Sci. China. Math., 57 (2014), 665-686], and it was numerically shown that the accuracy of the extended C-N scheme was also of second order. To our best knowledge, among all one-step (two-time level) numerical schemes with second-order accuracy for solving BSDE or FBSDEs, such as the ones in the above two papers and the one developed by the authors D. Crisan and K. Manolarakis [Ann. Appl. Probab., 24, 2 (2014), 652-678], the C-N scheme is the simplest one in applications. The theoretical proofs of second-order error estimates reported in the literature for these schemes for solving decoupled FBSDEs did not include the C-N scheme. The purpose of this work is to theoretically analyze the error estimate of the C-N scheme for solving decoupled FBSDEs. Based on the Taylor and Itô-Taylor expansions, the Malliavin calculus theory (e.g., the multiple Malliavin integration-by-parts formula), and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

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A Stable Multistep Scheme for Solving Backward Stochastic Differential Equations

Cited by 76 publications

References 28 publications

A backward Monte-Carlo method for time-dependent runaway electron simulations

A backward Monte-Carlo method for time-dependent runaway electron simulations

Second-order schemes for solving decoupled forward backward stochastic differential equations

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

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