A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations

By Bart Vandereycken, Stefan Vandewalle. 2010

127 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Examples include nonlinear conjugate gradient or Newton-like methods on manifolds of low-rank matrices [35,191,198,257,258] or low-rank tensors [52,53,131,132,255], see also Section 3.4.”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, methodology): “One can regard S+(p, n) as a submanifold embedded in Rn×n and equip it with the usual Euclidean metric, as was done in, e.g., Helmke & Moore (1994), Helmke & Shayman (1995), Orsi et al. (2004), Orsi et al. (2006), Vandereycken et al. (2009), Vandereycken & Vandewalle (2010).”
Computational Methods for Linear Matrix Equations (2016, methodology): “A computational improvement to decrease the number of solves per iteration was suggested in [175] (where the LR-ADI method was called CF-ADI, in which CF stands for Cholesky Factor).”
Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations (2014, methodology): “In [4] it is shown, by drawing connections to a Riemannian optimization framework [39], that IRKA can also be used for the computation of low-rank solutions of large Lyapunov equations.”
On critical points of quadratic low-rank matrix optimization problems (2020, methodology): “It is used in the Riemannian algorithms [38, 27, 37, 39] for low-rank optimization.”
On two numerical methods for the solution of large-scale algebraic Riccati equations (2014, methodology): “Certain solution strategies, such as the ADI method [36] (see, further [7, §4], [8] and references therein), also implicitly build rational Krylov subspaces [7], [34].”
The geometry of algorithms using hierarchical tensors (2013, unknown): “A number of large-scale applications [42] [43] [44] [45] [46] show that exploiting the smooth structure of the manifold indeed accelerates the algorithms compared to other, more ad-hoc ways of dealing with low-rank constraints.”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, methodology): “The dynamical approximation may thus be used as a retraction as required in optimization methods on low-rank manifolds like CG and Newton; see [1, 31].”
Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey (2013, background): “In their SIAM Outstanding Paper Prize awarded contribution [60] Vandereycken and Vandewalle propose an optimization on manifolds based approach to finding a low rank solution of (3).”
On the benefits of the LDLT factorization for large-scale differential matrix equation solvers (2015, methodology): “Many methods for solving largescale ALEs have been proposed [34, 35, 40, 44, 45, 46, 48].”