Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case

By Thilo Penzl. 2000

197 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/39d42113505c9d8588c6c6f6f154142a37284f58

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In Computational Theory and Mathematics

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Citations in the corpus

Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey (2013, background): “In several contributions [3, 25, 46, 51, 59], it is shown that under certain assumptions, the eigenvalues of the solution to Lyapunov equations decay fast.”
BDF Methods for Large-Scale Differential Riccati Equations (2004, methodology): “Stopping criteria for the modified ADI iteration can be based either on the fact that ‖Vj‖ → 0 very rapidly or on the residual norm ‖FYjY T j + YjY T j F T + WW T‖; see [31] for an efficient way to compute the Frobenius norms of the residuals.”
Introduction to Hierarchical Matrices with Applications (2003, background): “If the matrix G in (29) is of (global) low rank and the spectra of A and −A are disjoint, then the singular values of the solution X decay exponentially ([ 22 ], [9]).”
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations (2010, background): “This has been studied in [38], [43], [5], and [18] for the case of M = I and a low-rank matrix C in (1.”
Low-Rank Solution of Lyapunov Equations (2004, background): “Frequently the exact solution X itself has low numerical rank [1, 28].”
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations (2007, background): “The possibility of obtaining a good low rank approximate solution is supported by recent theoretical results showing that a rapid decay of the eigenvalues of the symmetric and positive semidefinite exact solution may be observed under certain conditions on A; see [16], [29], [37].”
Analysis and Numerical Solution of Generalized Lyapunov Equations (2002, background): “It was observed that the eigenvalues of the symmetric solutions of large scale Lyapunov equations with low-rank right-hand side generally decay very rapidly, see [5, 128].”
Balanced truncation model reduction for semidiscretized Stokes equation (2006, background): “14) with a low rank right-hand side decay very fast, see [2, 27, 33].”
On critical points of quadratic low-rank matrix optimization problems (2020, background): “For instance, when B in the Lyapunov equation (1.3) itself is low rank, it can be proven (Penzl, 2000, Thm. 1) that X ∗ has exponentially decaying singular values and, hence, can be approximated well by a low-rank matrix.”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, background): “This was motivated by the observation that in the large-scale setting often the Grami-ans have a low numerical rank r P , r Q n , [7,99,155,183].”
A Survey of Model Reduction Methods for Large-Scale Systems (2000, background): “Other related approaches [9, 17, 21] work directly with projected forms of the two Lyapunov equations (2.”
Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form (2005, background): “It was observed that the eigenvalues of the symmetric solutions of Lyapunov equations with low rank right-hand side generally decay very rapidly, and such solutions may be well approximated by low rank matrices, see [5, 57 , 67].”
Computational Methods for Linear Matrix Equations (2016, methodology): “9) are compared in [221] with earlier estimates in [204].”