Low-Rank Solution of Lyapunov Equations

By Jing-Rebecca Li, Jacob K. White. SIAM Review, 2004

460 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/901f64d9f56c825a3531d5dbe38b7c694c270f47

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

On Submodularity and Controllability in Complex Dynamical Networks (2014, methodology): “One can obtain further computational benefits by computing low rank approximations of these Gramians also using methods from [24].”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, unknown): “, [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.”
A Survey on Model Reduction of Coupled Systems (2008, methodology): “However, recent results on low rank approximations to the solutions of matrix equations [9, 12, 29, 33, 40, 49] make the balanced truncation model reduction approach attractive for large-scale problems.”
On the benefits of the LDLT factorization for large-scale differential matrix equation solvers (2015, methodology): “in complete analogy to the procedure first employed by Li and White for the ZZ case in [35].”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “Methods to compute the basis in the large-scale setting include approximate balan ced truncation (Gugercin and Antoulas 2004; Li and White 2002; Moore 1981; Penzl 2006; Sorensen and Antoulas 2002), Krylov-based methods (Feldmann and Freund 1995; Gallivan et al.”
A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations (2014, methodology): “This includes the low-rank Smith iteration and ADI method [31, 33], subspace methods [9, 25, 36, 39, 40], and low-rank variants of the sign function iteration [5, 11].”
Computational Methods for Linear Matrix Equations (2016, methodology): “It was observed in [174] that the ADI method actually generates a (block) rational Krylov subspace for the given vector of shifts sk = [s1, .”
Existence and Computation of Low Kronecker-Rank Approximations for Large Linear Systems of Tensor Product Structure (2004, methodology): “A low rank approximation X k to X can, e.g., be computed eﬃciently by iterative methods like the ADI or Smith iteration (see [11] and more recently [10]) that perform few iterative steps such that the i th iterate is of rank at most i .”
Hessian‐based model reduction for large‐scale systems with initial‐condition inputs (2008, methodology): “Methods to compute the basis in the largescale setting include Krylov-subspace methods [15, 16, 19], approximate balanced truncation [20,27,28,31], and proper orthogonal decomposition (POD) [13,24,30].”
On the Solution of Large-Scale Algebraic Riccati Equations by Using Low-Dimensional Invariant Subspaces (2016, methodology): “These methods include the Newton-ADI (Alternate Direction Implicit) [14, 16, 5, 21] and the various projection-type methods, usually based on approximations using Krylov [11, 13, 22] or rational Krylov [9, 8] subspaces generated by the matrices A∗ and A−∗ and the initial (block-)vector C∗.”
A Structure Preserving FGMRES Method for Solving Large Lyapunov Equations (2012, background): “The most efficient way, usually referred to as LRCF-ADI [3] uses elegant way of computing L j from L j−1 such that the computational work grows at most linearly with the number of shifts t.”
Low rank iterative methods for projected generalized Lyapunov equations (2005, methodology): “As in the standard case [17, 22], in order to keep the real factors, we can take the complex shift parameters in complex conjugate pairs « Ø ` ¬ with Ø ` and compute the iterates Í as follows”
BDF Methods for Large-Scale Differential Riccati Equations (2004, methodology): “Inserting this factorization into the above iteration, re-arranging terms and combining two iteration steps, we obtain a factored ADI iteration that in each iteration step yields nw new columns of a full-rank factor of Q (see [6, 27, 30] for several variants of this method):”
Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey (2013, background): “After some further manipulation of equation (14) [36], one finds that in fact only the new columns in the low rank factor update have to be processed, instead of all columns as in the naive approach.”