  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.
