Dynamical low-rank approximation

Abstract: This talk reviews differential equations and their numerical solution on manifolds of low-rank matrices or of tensors with a rank structure such as tensor trains or general tree tensor networks. These low-rank differential equations serve to approximate, in a data-compressed format, large time-dependent matrices and tensors or multivariate functions that are either given explicitly via their increments or are unknown solutions to high-dimensional evolutionary differential equations, with multi-particle time-dependent Schrödinger equations and kinetic equations such as Vlasov equations as noteworthy examples of applications.

Recently developed numerical time integrators are based on splitting the projection onto the tangent space of the low-rank manifold at the current approximation. In contrast to all standard integrators, these projector-splitting methods are robust to the unavoidable presence of small singular values in the low-rank approximation. This robustness relies on exploiting geometric properties of the manifold of low-rank matrices or tensors: in each substep of the projector-splitting algorithm, the approximation moves along a flat subspace of the low-rank manifold. In this way, high curvature due to small singular values does no harm.

This talk is based on work done intermittently over the last decade with Othmar Koch, Bart Vandereycken, Ivan Oseledets, Emil Kieri, Hanna Walach and Gianluca Ceruti.

computational engineering, finance, and sciencenumerical analysis

Audience: researchers in the topic

E-NLA - Online seminar series on numerical linear algebra

Series comments: E-NLA is an online seminar series dedicated to topics in Numerical Linear Algebra. Talks take place on Wednesdays at 4pm (Central European Time) via Zoom and are initially scheduled on a weekly basis.

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