Numerical approximation of PDEs on tensor manifolds

Recently, there has been a growing interest in approximating nonlinear functions and PDEs on tensor manifolds. The reason is simple: tensors can drastically reduce the computational cost of high-dimensional problems when the solution has a low-rank structure. In this talk, I will review recent developments on rank-adaptive algorithms for temporal integration of PDEs on tensor manifolds. Such algorithms combine functional tensor train (FTT) series expansions, operator splitting time integration, and an appropriate criterion to adaptively add or remove tensor modes from the FTT representation of the PDE solution as time integration proceeds. I will also present a new tensor rank reduction method that leverages coordinate flows. The idea is very simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has a smaller tensor rank. I will restrict the analysis to linear coordinate transformations, which give rise to a new class of functions that we refer to as tensor ridge functions. Numerical applications are presented and discussed for linear and nonlinear advection equations, and for the Fokker-Planck equation.