Parallel numerical tensor methods for high-dimensional PDEs
Provides parallel tensor-based solvers for high-dimensional PDEs, which are computationally challenging in fields like physics and engineering.
The paper develops parallel algorithms for high-dimensional PDEs using canonical and hierarchical tensor methods, demonstrating accuracy on a 6D advection equation and linearized Boltzmann equation.
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 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.