Preconditioned Low-Rank Riemannian Optimization for Multiterm Linear Matrix Equations

by Ivan Bioli (Università degli Studi di Pavia)

Thursday 11 Apr 2024, 16:00 → 17:00 Europe/Rome

Aula Seminari (Dipartimento di Matematica)

We focus on the development and analysis of Riemannian optimization methods for solving multiterm linear matrix equations, when the associated linear operator is symmetric positive definite. Assuming low-rank approximability of the solution, we employ first- and second-order Riemannian optimization algorithms on low-rank matrix manifolds.
To enhance competitiveness with existing solvers on realistic problems, significant efforts are dedicated to developing efficient Riemannian preconditioners, which are interpreted as changing the Riemannian metric. By alternating between fixed-rank Riemannian optimization and rank updates, we devise a rank-adaptive algorithm that does not require fixing the rank a priori. In numerical experiments, we illustrate the effectiveness of the developed Riemannian preconditioners, comparing our approach with established solvers
such as Conjugate Gradient with truncation. Results indicate that the Riemannian optimization approach, in both its fixed-rank and rank-adaptive variants, is competitive with other solvers and can avoid excessive growth of ranks in intermediate iterations.

References

[Bou23] Nicolas Boumal. An Introduction to Optimization on Smooth Manifolds. 1st ed. Cambridge University Press, Mar. 16, 2023.
[KSV16] Daniel Kressner, Michael Steinlechner, and Bart Vandereycken. “Preconditioned Low-rank Riemannian Optimization for Linear Systems with Tensor Product Structure”. In: SIAM Journal on Scientific Computing 38.4 (Jan. 2016), A2018–A2044.
[VV10] Bart Vandereycken and Stefan Vandewalle. “A Riemannian Optimization Ap
proach for Computing Low-Rank Solutions of Lyapunov Equations”. In: SIAM
Journal on Matrix Analysis and Applications 31.5 (Jan. 2010), pp. 2553–2579.