In this work, we are concerned with the high-order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the backward SDE, which contain the conditional expectations and their derivatives. Then, our high-order multistep schemes are obtained by carefully approximating the conditional expectations and the derivatives, in the reference ODEs. Motivated by the local property of the generator of diffusion processes, the Euler method is used to solve the forward SDE; however, it is noticed that the numerical solution of the backward SDE is still of high-order accuracy. Such results are obviously promising: on one hand, the use of the Euler method (for the forward SDE) can dramatically simplify the entire computational scheme, and on the other hand, one might be only interested in the solution of the backward SDE in many real applications such as option pricing. Several numerical experiments are presented to demonstrate the effectiveness of the numerical method.

This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. We use the sparse grid Gaussian-Hermite quadrature rule to approximate the conditional expectations. And for the associated high dimensional interpolations, we adopt an spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and high accurate approximations in high-dimensions, provided that the solutions are sufficiently smooth. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.

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Electron−phonon coupling emerges as a growing frontier in the heart of condensed matter from physical symmetry to the electronic quantum state, but its quantitative strength dependence on the chemical structure has not been assessed. Here, we originally proposed the anion-centered polyhedron (ACP) strategy for elaborating the electron−phonon coupling interaction in rare-earth (RE) materials comprising three chemical factors, RE−O bond length, the effective charge of the coordinated atom, and structural dimensionality. Using Gd 3+ cation with 4f 7 configuration as a fluorescence probe, we found that the "free-O"-centered polyhedron is the most crucial motif in strengthening the phonon-assisted energy transfer and photon emission. The temperature-dependent Huang−Rhys S factors were calculated to identify the electron−phonon coupling intensity based on the fluorescence spectrum quantitatively. Finally, beyond conventional wisdom, a series of structural criteria were presented, serving as useful guidelines for discovering strongly coupled rare-earth optical materials. Our study breaks the long-time "blind"-searching diagram and provides reliable principles for many functional materials associated with electron−phonon coupling, such as superconductors, multiferroics, and phosphors.

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