Systematic matrix formulation for efficient computational path integration

In this work we introduce a novel methodological treatment of the numerical path integration method, used for computing the response probability density function of stochastic dynamical systems. The method is greatly accelerated by transforming the corresponding Chapman-Kolmogorov equation to a matrix multiplication. With a systematic formulation we split the numerical solution of the Chapman-Kolmogorov equation into three separate parts: we interpolate the probability density function, we approximate the transitional probability density function of the process and evaluate the integral in the Chapman-Kolmogorov equation. We provide a thorough error and efficiency analysis through numerical experiments on a one, two, three and four dimensional problem. By comparing the results obtained through the Path Integration method with analytical solutions and with previous formulations of the path integration method, we demonstrate the superior ability of this formulation to provide accurate results. Potential bottlenecks are identified and a discussion is provided on how to address them.