A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

Yang, Minglei; Zhang, Guannan; Del-Castillo-Negrete, Diego; Stoyanov, Miroslav

doi:10.1016/j.jcp.2021.110564

Title: A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

Journal Article · Fri Jul 16 00:00:00 EDT 2021 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2021.110564· OSTI ID:1817572

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Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. In this work, we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

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Research Organization:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1817572

Journal Information:: Journal of Computational Physics, Vol. 444, Issue NA; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (30)

Simulation of stopped diffusions Buchmann, F. M. Journal of Computational Physics, Vol. 202, Issue 2 https://doi.org/10.1016/j.jcp.2004.07.009	journal	January 2005
A generalized $\theta$-scheme for solving backward stochastic differential equations Zhang, Guannan; Li, Yang; Zhao, Weidong Discrete and Continuous Dynamical Systems - Series B, Vol. 17, Issue 5 https://doi.org/10.3934/dcdsb.2012.17.1585	journal	March 2012
Multi-scale full-orbit analysis on phase-space behavior of runaway electrons in tokamak fields with synchrotron radiation Wang, Yulei; Qin, Hong; Liu, Jian Physics of Plasmas, Vol. 23, Issue 6 https://doi.org/10.1063/1.4953608	journal	June 2016
First-passage-time problem for simulated stochastic diffusion processes Lánský, Petr; Lánská, Věra Computers in Biology and Medicine, Vol. 24, Issue 2 https://doi.org/10.1016/0010-4825(94)90068-X	journal	March 1994
An asymptotic expansion for one-sided Brownian exit densities Ferebee, Brooks Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 63, Issue 1 https://doi.org/10.1007/BF00534172	journal	March 1983
Fokker-Planck Equations of Stochastic Acceleration: A Study of Numerical Methods Park, Brian T.; Petrosian, Vahe The Astrophysical Journal Supplement Series, Vol. 103 https://doi.org/10.1086/192278	journal	March 1996
Anomalous diffusion of tracer in convection rolls Young, W.; Pumir, A.; Pomeau, Y. Physics of Fluids A: Fluid Dynamics, Vol. 1, Issue 3 https://doi.org/10.1063/1.857415	journal	March 1989
New Kinds of High-Order Multistep Schemes for Coupled Forward Backward Stochastic Differential Equations Zhao, Weidong; Fu, Yu; Zhou, Tao SIAM Journal on Scientific Computing, Vol. 36, Issue 4 https://doi.org/10.1137/130941274	journal	January 2014
Numerical simulation of runaway electrons: 3-D effects on synchrotron radiation and impurity-based runaway current dissipation del-Castillo-Negrete, D.; Carbajal, L.; Spong, D. Physics of Plasmas, Vol. 25, Issue 5 https://doi.org/10.1063/1.5018747	journal	May 2018
Chaotic particle transport in time-dependent Rayleigh-Bénard convection Solomon, T. H.; Gollub, J. P. Physical Review A, Vol. 38, Issue 12 https://doi.org/10.1103/PhysRevA.38.6280	journal	December 1988
A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains Yang, Jie Journal of Computational Mathematics, Vol. 36, Issue 2 https://doi.org/10.4208/jcm.1612-m2016-0582	journal	June 2018
Monotone Piecewise Cubic Interpolation Fritsch, F. N.; Carlson, R. E. SIAM Journal on Numerical Analysis, Vol. 17, Issue 2 https://doi.org/10.1137/0717021	journal	April 1980
Evaluations of barrier-crossing probabilities of Wiener paths Park, C.; Schuurmann, F. J. Journal of Applied Probability, Vol. 13, Issue 2 https://doi.org/10.2307/3212830	journal	June 1976
Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions Rubin, Jacob Water Resources Research, Vol. 19, Issue 5 https://doi.org/10.1029/WR019i005p01231	journal	October 1983
The first-passage density of a continuous gaussian process to a general boundary Durbin, J. Journal of Applied Probability, Vol. 22, Issue 01 https://doi.org/10.1017/S0021900200029041	journal	March 1985
Exact Asymptotics for the Probability of Exit from a Domain and Applications to Simulation Baldi, Paolo The Annals of Probability, Vol. 23, Issue 4 https://doi.org/10.1214/aop/1176987797	journal	October 1995
Escape probability and mean residence time in random flows with unsteady drift Brannan, James R.; Duan, Jinqiao; Ervin, Vincent J. Mathematical Problems in Engineering, Vol. 7, Issue 1 https://doi.org/10.1155/S1024123X01001521	journal	January 2001
On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary Salminen, Paavo Advances in Applied Probability, Vol. 20, Issue 02 https://doi.org/10.1017/S0001867800017043	journal	June 1988
Brownian first exit from and sojourn over one sided moving boundary and application Uchiyama, K?hei Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 54, Issue 1 https://doi.org/10.1007/BF00535355	journal	January 1980
The tangent approximation to one-sided Brownian exit densities Ferebee, Brooks Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 61, Issue 3 https://doi.org/10.1007/BF00539832	journal	January 1982
A backward Monte-Carlo method for time-dependent runaway electron simulations Zhang, Guannan; del-Castillo-Negrete, Diego Physics of Plasmas, Vol. 24, Issue 9 https://doi.org/10.1063/1.4986019	journal	September 2017
Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme Gobet, Emmanuel; Menozzi, Stéphane Stochastic Processes and their Applications, Vol. 112, Issue 2 https://doi.org/10.1016/j.spa.2004.03.002	journal	August 2004
Numerical solutions for solving time fractional Fokker–Planck equations based on spectral collocation methods Yang, Yin; Huang, Yunqing; Zhou, Yong Journal of Computational and Applied Mathematics, Vol. 339 https://doi.org/10.1016/j.cam.2017.04.003	journal	September 2018
Weak approximation of killed diffusion using Euler schemes Gobet, Emmanuel Stochastic Processes and their Applications, Vol. 87, Issue 2 https://doi.org/10.1016/S0304-4149(99)00109-X	journal	June 2000
The development of chaotic advection Aref, Hassan Physics of Fluids, Vol. 14, Issue 4 https://doi.org/10.1063/1.1458932	journal	April 2002
First exit time probability for multidimensional diffusions: A PDE-based approach Patie, P.; Winter, C. Journal of Computational and Applied Mathematics, Vol. 222, Issue 1 https://doi.org/10.1016/j.cam.2007.10.043	journal	December 2008
On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems Spencer, B. F.; Bergman, L. A. Nonlinear Dynamics, Vol. 4, Issue 4 https://doi.org/10.1007/BF00120671	journal	August 1993
Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs Zhang, Guannan; Zhao, Weidong; Webster, Clayton Computers & Mathematics with Applications, Vol. 71, Issue 11 https://doi.org/10.1016/j.camwa.2015.11.002	journal	June 2016
Space dependent, full orbit effects on runaway electron dynamics in tokamak plasmas Carbajal, L.; del-Castillo-Negrete, D.; Spong, D. Physics of Plasmas, Vol. 24, Issue 4 https://doi.org/10.1063/1.4981209	journal	April 2017
Transport phenomena Bird, R. Byron Applied Mechanics Reviews, Vol. 55, Issue 1 https://doi.org/10.1115/1.1424298	journal	January 2002

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Title: A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

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