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In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs. • FBSDE deep neural network (DNN) with loss comparing two stochastic processes for the PDEs from Pardoux-Peng theory. • Strong 1/2 Convergence as the Euler-Maruyama method in solving a 100-dimensional Black-Scholes-Barenblatt equation. • Use of Richardson extrapolation for accuracy enhancement. • Multiscale DNN captures temporal high frequency content of the PDEs more accurately. [ABSTRACT FROM AUTHOR]

Abstract: In [J. Halton, Sequential Monte Carlo, Proc. Comb. Phil. Soc. 58 (1962), J. Halton, Sequential Monte Carlo Techniques for the Solution of Linear Systems, J. Sci. Comp. 9 (1994) 213–257] Halton introduced a strategy to be used in Monte Carlo algorithms for the efficient solution of certain matrix problems. We showed in [R. Kong, J. Spanier, Sequential correlated sampling methods for some transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000, R. Kong, J. Spanier, Error analysis of sequential Monte Carlo methods for transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000] how Halton’s method based on correlated sampling can be extended to continuous transport problems and established their geometric convergence for a family of transport problems in slab geometry. In our algorithm, random walks are processed in batches, called stages, each stage producing a small correction that is added to the approximate solution developed from the previous stages. In this paper, we demonstrate that strict error reduction from stage to stage can be assured under rather general conditions and we illustrate this rapid convergence numerically for a simple family of two dimensional transport problems. [Copyright &y& Elsevier]

Abstract: At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed finite state projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon this approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in Escherichia coli. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a multiscale stochastic simulation algorithm (MSSA) which makes use of Gillespie’s stochastic simulation algorithm (SSA) together with a new stochastic formulation of the partial equilibrium assumption (PEA). This method is much more efficient than SSA alone. It works even with a very small population of fast species. Implementation details are discussed, and an application to the modeling of the heat shock response of E. Coli is presented which demonstrates the excellent efficiency and accuracy obtained with the new method. [Copyright &y& Elsevier]

In recent years the kinetic Fokker-Planck approach for modeling gas flows has become increasingly popular. In the Fokker-Planck ansatz the collision integral of the Boltzmann equation is approximated by a Fokker-Planck operator in velocity space. Instead of solving the resulting Fokker-Planck equation directly, the underlying random process is modeled, which leads to an efficient stochastic solution algorithm. Despite the attention to the Fokker-Planck ansatz, the modeling of polyatomic gases has been addressed only in a few works. In this paper a scheme is presented to extend arbitrary monatomic Fokker-Planck models to model polyatomic species. A master equation approach is used to model internal energy relaxation, but instead of solving the master equation directly, the underlying random process is simulated. Three different models are suggested to describe internal particle energies as continuous scalars or as a set of discrete energy levels. The proposed models are applied on different test cases to demonstrate their accuracy. Within the bounds of expectations, a very good agreement with reference DSMC simulations is achieved. • A Master equation approach is applied to model internal energy relaxation. • Three models of varying fidelity are constructed to describe internal energy states. • Prediction of vibrational energy levels consistent with the Larsen-Borgnakke model. • Test cases show very good agreement with reference DSMC simulations. [ABSTRACT FROM AUTHOR]