Abstract
In this article, a new method is proposed to approximate the rightmost eigenpair of certain matrix-valued linear operators, in a low-rank setting. First, we introduce a suitable ordinary differential equation, whose solution allows us to approximate the rightmost eigenpair of the linear operator. After analyzing the behaviour of its solution on the whole space, we project the ODE on a low-rank manifold of prescribed rank and correspondingly analyze the behaviour of its solutions. For a general linear operator we prove that—under generic assumptions—the solution of the ODE converges globally to its leading eigenmatrix. The analysis of the projected operator is more subtle due to its nonlinearity; when ca is self-adjoint, we are able to prove that the associated low-rank ODE converges (at least locally) to its rightmost eigenmatrix in the low-rank manifold, a property which appears to hold also in the more general case. Two explicit numerical methods are proposed, the second being an adaptation of the projector splitting integrator proposed recently by Lubich and Oseledets. The numerical experiments show that the method is effective and competitive.
Article PDF
Similar content being viewed by others
References
Absil, P.-A.: Continuous-time systems: that solve computational problems. Intern. J. Unconv. Comp. 2, 291–304 (2006)
Chu, M. -T.: On the continuous realization of iterative processes. SIAM Rev. 30(3), 375–387 (1988)
Embree, M., Lehoucq, R. B.: Dynamical systems and non-Hermitian iterative eigensolverss. SIAM J. Numer. Anal. 47(2), 1445–1473 (2009)
Guglielmi, N., Lubich, C.: Matrix stabilization using differential equations. SIAM J. Numer. Anal. 55(6), 3097–3119 (2017)
Hodel, A. S., Tenison, B., Poolla, K.: Numerical solution of the Lyapunov equation by approximate power iteration. Linear Algebra Appl. 236, 205–230 (1996)
Kieri, E., Lubich, C., Walach, H.: Discretized dynamical low-rank approximation in the presence of small singular values. SIAM J. Numer Anal. 54(2), 1020–1038 (2016)
Kressner, D., Uschmajew, A.: On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems. Linear Algebra Appl. 493, 556–572 (2016)
Koch, O., Lubich, C.: Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29(2), 434–454 (2007)
Kressner, D., Steinlechner, M., Uschmajew, A.: Low-rank tensor methods with subspace correction for symmetric eigenvalue problems. SIAM J. Sci. Comput. 36(5), A2346–A2368 (2014)
Lubich, C., Oseledets, I. V.: A projector-splitting integrator for dynamical low-rank approximation. BIT 54(1), 171–188 (2014)
Nanda, T.: Differential equations the QR algorithm. SIAM J. Numer. Anal. 22(2), 310–321 (1985)
Rakhuba, M., Oseledets, I.: Jacobi–Davidson Method on low-rank matrix manifolds. SIAM J. Sci Comp. 40(2), A1149–A1170 (2018)
Rakhuba, M., Novikov, A., Oseledets, I.: Low-rank Riemannian eigensolver for high-dimensional Hamiltonians. J Comput Phys 396, 718–737 (2019)
Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58(3), 377–441 (2016)
Uschmajew, A., Vandereycken, B.: Geometric methods on low-rank matrix and tensor manifolds. In: Grohs, P. et al. (eds.) Handbook of Variational Methods for Nonlinear Geometric Data, pp 261–313. Springer (2020)
Acknowledgements
The authors thank Giorgio Fusco (University of L’Aquila) for stimulating discussions. Part of this work was prepared while D. Kressner was visiting GSSI.
Funding
Open access funding provided by Gran Sasso Science Institute - GSSI within the CRUI-CARE Agreement. The first author thanks the Italian INdAM GNCS for financial support. The first author thanks the Italian MUR (PRIN project 2017, Discontinuous dynamical systems: theory, numerics and applications). N. Guglielmi and C. Scalone thank the Italian INdAM GNCS for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Valeria Simoncini
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guglielmi, N., Kressner, D. & Scalone, C. Computing low-rank rightmost eigenpairs of a class of matrix-valued linear operators. Adv Comput Math 47, 66 (2021). https://doi.org/10.1007/s10444-021-09895-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-021-09895-2