# Efficient Solution of Fokker-Planck Equations in Two Dimensions
###### Abstract
[PERSON] and [PERSON] [26] and [PERSON] and [PERSON] [27] obtained the generalized probability density evolution equation by using decoupled physical equations. [PERSON] [28] studied the stationary probability density function of a multi-degree-of-freedom nonlinear system subjected to external, independent Poisson white noise, reducing the high-dimensional generalized FP equation to a low-dimensional equation, and [PERSON] et al. [29] improved on this approach. In contrast to most of these studies, the reduction method used herein does not depend on the physics of the problem considered. Independent of the dimension of the problem, operator splitting provides a numerical method to compute the solution to matrix differential equations that can be more efficient than straightforward integration using the original coefficient matrices. Operator splitting methods have been known for more than a century, starting with the scheme introduced by [PERSON] in the mid-1870s [30]. [PERSON] and [PERSON] [31] developed alternating direction implicit (ADI) methods. In the mid-1970s, close relationships between the augmented Lagrangian methods of [PERSON] [32] and [PERSON] [33; 34] and ADI methods were identified, which inspired the alternating direction method of multipliers (ADMM). Based on a symmetrization principle, [PERSON] [35] introduced a second-order variant of the [PERSON] scheme, which was motivated by a need for the accurate solution of hyperbolic problems. This second-order scheme is unconditionally stable, which makes it popular in computing solutions to PDEs. [PERSON] al. [36] further developed balanced splitting and rebalanced splitting methods; compared with earlier splitting methods, these have higher accuracy and can correctly capture steady-state behavior. [PERSON] and [PERSON] [37] discussed an algorithm that is equivalent to the ADMM, and this method has been used in many image-processing applications. Operator splitting has attracted attention for its computational efficiency in the solution of FP equations. [PERSON] et al. [38] combined an ADI scheme with the finite difference method to solve FP problems in particle accelerators. For studying the influence of noise in the particle dynamics in a storage ring, a method based on operator splitting was developed by [PERSON] and [PERSON] [39], which can also be used in stochastic beam-dynamics problems in accelerators. [PERSON] et al. [40] provided a robust finite difference scheme with operator splitting (ADI) for the FP equation in two spatial variables and time. Because of the rapid growth of the FE equation with the increasing dimension of the underlying dynamical system, it now seems unlikely that this approach to the solution of the FP equation will be extended beyond the three or four states that have represented a practical limit for several years. However, systems of this size do have practical applications, such as design optimization and nonlinear filtering [41], and so there is some value in pursuing their efficient solution. In addition, we are motivated in part by the goal of providing baseline timing and solution results for the FEA of such problems as the field shifts increasingly toward the routine use of alternative formulations, such as physics-informed neural networks (PINNs) [42; 43; 44], which are very promising for problems of high dimension but whose numerical properties have not yet been fully tested. Our interest is thus pragmatic. The finite element analysis of the FP equation is largely a mature subject, with some gaps where known numerical techniques may be applied. We try to provide practical guidance for the use of two of these, model reduction and operator splitting. While we focus on 2-D problems herein, our approach is directly extensible to larger systems, which have structurally similar coefficient matrices; however, both programming complexity and storage requirements grow very quickly, and the efficiency gains we achieve are not enough to offset this.