A probabilistic approach for the computation of local and nonlocal transport

Understanding transport is one of the main challenges of the controlled nuclear fusion program. Although significant progress has been made, the complexity of the problem calls for novel methods to overcome limitations of current approaches. The models of interest in this presentation are Fokker-Planck equations in which local (e.g., diffusive) transport is represented by differential operators and nonlocal transport by integro-differential operators (e.g., Landau-fluid and non-diffusive turbulent transport closures). Computational approaches for these problems can be classified as deterministic (e.g., finite-difference) and stochastic (e.g., Monte-Carlo). Although extensively used, continuum deterministic methods face stability and scalability challenges especially in the case of nonlocal operators that result in non-sparse matrices. On the other hand, particle-based stochastic methods face poor convergence due to statistical sampling. Here we present an alternative novel approach based on the Feynman-Kac theory that establishes a link between the Fokker-Planck equation and the stochastic differential equation of the underlying stochastic process. Intuitively, the method is a hybrid approach based on the stochastic representation of the local, or nonlocal, transport process, but with the actual computation reduced to the deterministic evaluation of mathematical expectations (i.e., integrals) bypassing the need of sampling individual stochastic orbits. The resulting algorithm is unconditionally stable, and parallelizable [Yang, et al., J. of Comput. Phys. 444, 110564 (2021) and https://arxiv.org/pdf/2205.00516.pdf]. We present applications to the initial value and the exit time problems for Fokker-Planck equations. Physics problems of interest include local and nonlocal anisotropic transport in toroidal geometry, confinement, and production rate of runaway electrons.