We develop a new class of algorithms, which we call step-truncation methods, to integrate in time an initial value problem for an ODE or a PDE on a low-rank tensor manifold. The new methods are based on performing a time step with a conventional time-stepping scheme  followed by a truncation operation into a tensor manifold with prescribed rank. By considering such truncation operation as a nonlinear operator in the space of tensors , we prove various consistency results and errors estimates for a wide range of step-truncation algorithms. In particular, we establish consistency between the best step-truncation method and the best tangent space projection integrator via perturbation analysis . Numerical applications are presented and discussed for a Fokker-Planck equation on a torus of dimension two and four.
We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme , followed by a truncation operation onto a tensor manifold . By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements , we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control . Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.
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