UC Santa Cruz

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. In particular, we develop a mathematical framework for the analysis of these algorithms which encompasses both explicit and implicit time stepping. With this framework we prove convergence of a wide range of step-truncation methods, including one-step and multi-step methods. These methods 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 linear advection problem, a clasas of Fokker-Planck equations, the Allen-Cahn equation, the nonlinear Schrodinger, and a Burgers' equation with uncertain initial condition.