Adaptive integration of nonlinear evolution equations on tensor manifolds

Abstract

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.

Low-rank Parareal: a low-rank parallel-in-time integrator

“…Other methods that require integrating parts of the vector field can be found in [4,22]. Another approach, proposed in [9,24,35], is based on projecting standard Runge-Kutta methods (sometimes including their intermediate stages). Most of these methods are formulated for constant rank r .…”

“…Most of these methods are formulated for constant rank r . Rank adaptivity can be incorporated without much difficulty for splitting and for projected schemes; see [3,6,9,35]. Finally, given the importance of DLRA in problems from physics (like the Schrödinger and Vlasov equation), the integrators in [7,29] also preserve certain invariants, like energy.…”

A low-rank power iteration scheme for neutron transport criticality problems

“…Problems in which dynamical low-rank is successfully applied to reduce memory and computational costs are, e.g., kinetic theory [7,8,31,30,9,5,6,13,20] as well as uncertainty quantification [10,28,29,33,18]. Furthermore, DLRA allows for adaptive model refinement [4,2,32], where the main idea is to pick the rank of the solution representation adaptively. Recently, an approach to employ DLRA for computing rightmost eigenpairs has been proposed in [12].…”