Convergent Snapshot Algorithms for Infinite-Dimensional Lyapunov EquationsIMA Journal of Numerical Analysis
AbstractWe consider two algorithms to approximate the solution Z of a class of stable operator Lyapunov equations of the form AZ + ZA* + BB* = 0. The algorithms utilize time snapshots of solutions of certain linear infinite-dimensional differential equations to construct the approximations. Matrix approximations of the operators a and B are not required and the algorithms are applicable as long as the rank of B is relatively small. The first algorithm produces an optimal low-rank approximate solution using proper orthogonal decomposition. The second algorithm approximates the product of the solution with a few vectors and can be implemented with a minimal amount of storage. Both algorithms are known for the matrix case. However, the extension of the algorithms to infinite dimensions appears to be new. We establish easily verifiable convergence theory and a priori error bounds for both algorithms and present numerical results for two model problems.
Department(s)Mathematics and Statistics
Document TypeArticle - Journal
Document VersionFinal Version
Rights© 2011 Oxford University Press, All rights reserved.
Citation InformationJohn R. Singler. "Convergent Snapshot Algorithms for Infinite-Dimensional Lyapunov Equations" IMA Journal of Numerical Analysis (2011)
Available at: http://works.bepress.com/john-singler/19/