Title: A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations
Authors: Vandereycken, Bart
Vandewalle, Stefan
Issue Date: Jul-2009
Publisher: Department of Computer Science, K.U.Leuven
Series Title: TW Reports vol:TW544
Abstract: We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a low-rank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semi-definite matrices of fixed rank. We detail how objects from differential geometry, like the Riemannian gradient and Hessian, can be efficiently computed for this manifold. As minimization algorithm we use the Riemannian Trust-Region method of [Found. Comput. Math., 7 (2007), pp. 303--330] based on a second-order model of the objective function on the manifold. Together with an efficient preconditioner this method can find low-rank solutions with very little memory. We illustrate our results with numerical examples.
Publication status: published
KU Leuven publication type: IR
Appears in Collections:Numerical Analysis and Applied Mathematics Section

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