From low-rank retractions to dynamical low-rank approximation and back

Area of Science:

• Numerical analysis
• Differential geometry
• Matrix computations

Background:

• Retractions are essential for algorithms solving optimization problems on smooth manifolds.
• Retractions are increasingly used for manifold-based computational tasks like interpolation.
• Numerical integration of differential equations on fixed-rank matrix manifolds is crucial for dynamical low-rank approximation (DLRA).

Purpose of the Study:

• To explore the application of retractions in numerical integration of differential equations on fixed-rank matrix manifolds.
• To introduce novel numerical integration schemes based on retractions.
• To analyze the properties and performance of these new schemes in the context of DLRA.

Main Methods:

• Utilizing retractions as a core component for numerical integration on manifolds.
• Introducing the KLS retraction derived from an unconventional integrator for DLRA.
• Developing two new integration schemes: Accelerated Forward Euler (AFE) and Projected Ralston-Hermite (PRH).
• Proving local truncation error of order three for AFE and PRH methods.

Main Results:

• Demonstrated that retractions naturally lead to numerical integrators for manifold problems.
• Introduced the KLS retraction as a novel tool for DLRA.
• Showcased how retractions can unify existing DLRA techniques and inspire new ones.
• Established third-order local truncation error for the novel AFE and PRH methods.

Conclusions:

• Retractions are versatile tools applicable to numerical integration on fixed-rank matrix manifolds, closely linked to DLRA.
• The proposed AFE and PRH methods provide accurate (third-order) numerical integration schemes for differential equations on general manifolds.
• Numerical experiments confirm the utility and highlight the trade-offs of these new retraction-based integration methods for DLRA.