Parallel tensor methods for high-dimensional linear PDEs

Highlights: • We develop new parallel algorithms to solve high-dimensional PDEs. • Algorithms use canonical and hierarchical tensor methods, alternating least squares, and hierarchical SVD. • Our algorithms are accurate and efficient when computing seven-dimensional PDEs. High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and Fokker–Planck equations. We develop new parallel algorithms to solve such high-dimensional PDEs. The algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition. Both implicit and explicit integration schemes are presented and discussed. We demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the Boltzmann equation.