A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations

By Valeria Simoncini. SIAM Journal on Scientific Computing, 2007

386 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/beb6eb5a98f9776f0f8e6d687b8d672f4a7a66ff

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Citations in the corpus

Krylov Subspace Methods for Linear Systems with Tensor Product Structure (2010, background): “In particular, several variants of Krylov subspace algorithms for solving (1.7) have been developed and analysed, see [16] [17] [20] [22] [23].”
Computational Methods for Linear Matrix Equations (2016, methodology): “In [232] the method was experimentally shown to be highly superior to the Arnoldi method for sparse and large coefficient matrices, allowing the computation of an equally accurate solution with a significantly smaller dimensional subspace, at lower computation costs.”
Power System Dynamic Model Reduction Based on Extended Krylov Subspace Method (2016, methodology): “In comparison, the extended Krylov subspace is reported to be more efficient, and also possesses a better convergence property than the Krylov subspace method [22].”
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].”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, neutral): “, [46, 47, 152, 219] for ADI-type methods, [117, 191, 199, 206] for the Smith method and its variants, [78, 135, 203, 210] for Krylov-based methods, or the recent survey [48] and the references therein.”
On two numerical methods for the solution of large-scale algebraic Riccati equations (2014, methodology): “In the context of GP methods, EKSM was implemented as in [46], where only the inner call to the Matlab care function substituted that to the lyap function; the computation of the residual norm remained unchanged.”
Krylov subspace methods for projected Lyapunov equations (2012, methodology): “In [32], it was proposed to use the extended block Krylov subspace”
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations (2010, methodology): “Next, we performed the same comparison with the KPIK algorithm from [42].”
AN EXTENDED BLOCK ARNOLDI ALGORITHM FOR LARGE-SCALE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATI ON ∗ (2008, methodology): “The extended block Krylov subspace is generated by means of the extended block Arnoldi process first introduced in [15] and used for solving Lyapunov equations in [ 30 ].”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, methodology): “, • Krylov subspace methods [188, 67, 68, 111, 113, 114, 116, 118, 136, 171, 179, 180], and tensor Krylov subspace methods [31, 124], • the factorized sign function iteration [34, 42, 43], and data-sparse implementations of the method [22, 24, 99], • multi-grid methods [152, 167], based on low-rank ”
On the Solution of Large-Scale Algebraic Riccati Equations by Using Low-Dimensional Invariant Subspaces (2016, methodology): “Indeed, the literature describes methods using the ordinary Krylov subspaces generated by the matrix A∗ [13], or the extended ones generated by both A∗ and A−∗ [22, 11], or the rational Krylov subspaces generated with A∗ [8, 23].”
The extended Krylov subspace method and orthogonal Laurent polynomials (2009, methodology): “Simoncini [24] investigated the structure of H2m−1 by other means.”
An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure (2013, methodology): “This includes methods based on Krylov subspaces [17, 24], extended Krylov subspaces [26], and rational Krylov subspaces [6, 18].”