The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

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dc.citation.epage536
dc.citation.issue3
dc.citation.spage526
dc.contributor.affiliationЧуайб Дуккаі університет
dc.contributor.affiliationChouaib Doukkali University
dc.contributor.authorСадек, Л.
dc.contributor.authorТалібі Алауї, Г.
dc.contributor.authorSadek, L.
dc.contributor.authorTalibi Alaoui, H.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2023-10-25T07:19:11Z
dc.date.available2023-10-25T07:19:11Z
dc.date.created2021-03-01
dc.date.issued2021-03-01
dc.description.abstractУ статті представено новий підхід до розв’язання великомасштабних диференціальних рівнянь Ляпунова. Запропонований підхід базується на проектуванні початкової задачі на розширеному блоці підпростору Крилова, використовуючи розширений несиметричний алгоритм Ланцоша. У результаті отримується низькорозмірне диференціальне матричне рівняння Ляпунова. Це диференціальне матричне рівняння розв’язується методом диференціаціювання назад або методом Розенброка. Отриманий розв’язок дозволяє створювати наближений розв’язок початкової задачі. Крім того, дано деякі теоретичні результати. Чисельні результати демонструють продуктивність запропонованого підходу.
dc.description.abstractIn this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.
dc.format.extent526-536
dc.format.pages11
dc.identifier.citationSadek L. The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations / L. Sadek, H. Talibi Alaoui // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 3. — P. 526–536.
dc.identifier.citationenSadek L. The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations / L. Sadek, H. Talibi Alaoui // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 3. — P. 526–536.
dc.identifier.doidoi.org/10.23939/mmc2021.03.526
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/60406
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 3 (8), 2021
dc.relation.references[1] Abou-Kandil H., Freiling G., Ionescu V., Jank G. Matrix Riccati equations in control and systems theory. Birkh¨auser (2012). .
dc.relation.references[2] Datta B. Numerical methods for linear control systems (Vol. 1). Academic Press (2004).
dc.relation.references[3] Sadek L., Talibi Alaoui H. The extended block Arnoldi method for solving generalized differential Sylvester equations. Journal of Mathematical Modeling. 8 (2), 189–206 (2020).
dc.relation.references[4] Sadek E. M., Bentbib A. H., Sadek L., Talibi Alaoui H. Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations. J. Appl. Math. Comput. 62, 157–177 (2020).
dc.relation.references[5] Agoujil S., Bentbib A. H., Jbilou K., Sadek E. M. A minimal residual norm method for large-scale Sylvester matrix equations. Electronic Transactions on Numerical Analysis. 43, 45–59 (2014).
dc.relation.references[6] Hached M., Jbilou K. Numerical solutions to large-scale differential Lyapunov matrix equations. Numer Algor. 79, 741–757 (2018).
dc.relation.references[7] Sadek L., Talibi Alaoui H. Numerical methods for solving large-scale systems of differential equations. Ricerche di Matematica. (2021).
dc.relation.references[8] Bentbib A., Jbilou K., Sadek E. M. On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems. Computers & Mathematics with Applications. 70 (10), 2555–2565 (2015).
dc.relation.references[9] Mena H., Ostermann A., Pfurtscheller L. M., Piazzola C. Numerical low-rank approximation of matrix differential equations. Journal of Computational and Applied Mathematics. 340, 602–614 (2018).
dc.relation.references[10] Bentbib A. H., Jbilou K., Sadek E. L. On some extended block Krylov based methods for large scale nonsymmetric Stein matrix equations. Mathematics. 5 (2), 21 (2017).
dc.relation.references[11] Behr M., Benner P., Heiland J. Solution formulas for differential Sylvester and Lyapunov equations. Calcolo. 56, 51 (2019).
dc.relation.references[12] Lang N., Mena H., Saak J. On the benefits of the LDLT factorization for large-scale differential matrix equation solvers. Linear Algebra and its Applications. 480, 44–71 (2015).
dc.relation.references[13] Druskin V., Knizhnerman L. Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM Journal on Matrix Analysis and Applications. 19 (3), 755–771 (1998).
dc.relation.references[14] Barkouki H., Bentbib A. H., Heyouni M., Jbilou K. An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo. 55, 13 (2018).
dc.relation.references[15] Parlett B. N., Taylor D. R., Li Z. A. A look-ahead Lanczos algorithm for unsymmetric matrices. Math. Comput. 44 (169), 105–124 (1985).
dc.relation.references[16] Penzl T. L YAPACK A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems. Available online: http://www.tuchemintz.de/sfb393/lyapack.
dc.relation.referencesen[1] Abou-Kandil H., Freiling G., Ionescu V., Jank G. Matrix Riccati equations in control and systems theory. Birkh¨auser (2012). .
dc.relation.referencesen[2] Datta B. Numerical methods for linear control systems (Vol. 1). Academic Press (2004).
dc.relation.referencesen[3] Sadek L., Talibi Alaoui H. The extended block Arnoldi method for solving generalized differential Sylvester equations. Journal of Mathematical Modeling. 8 (2), 189–206 (2020).
dc.relation.referencesen[4] Sadek E. M., Bentbib A. H., Sadek L., Talibi Alaoui H. Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations. J. Appl. Math. Comput. 62, 157–177 (2020).
dc.relation.referencesen[5] Agoujil S., Bentbib A. H., Jbilou K., Sadek E. M. A minimal residual norm method for large-scale Sylvester matrix equations. Electronic Transactions on Numerical Analysis. 43, 45–59 (2014).
dc.relation.referencesen[6] Hached M., Jbilou K. Numerical solutions to large-scale differential Lyapunov matrix equations. Numer Algor. 79, 741–757 (2018).
dc.relation.referencesen[7] Sadek L., Talibi Alaoui H. Numerical methods for solving large-scale systems of differential equations. Ricerche di Matematica. (2021).
dc.relation.referencesen[8] Bentbib A., Jbilou K., Sadek E. M. On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems. Computers & Mathematics with Applications. 70 (10), 2555–2565 (2015).
dc.relation.referencesen[9] Mena H., Ostermann A., Pfurtscheller L. M., Piazzola C. Numerical low-rank approximation of matrix differential equations. Journal of Computational and Applied Mathematics. 340, 602–614 (2018).
dc.relation.referencesen[10] Bentbib A. H., Jbilou K., Sadek E. L. On some extended block Krylov based methods for large scale nonsymmetric Stein matrix equations. Mathematics. 5 (2), 21 (2017).
dc.relation.referencesen[11] Behr M., Benner P., Heiland J. Solution formulas for differential Sylvester and Lyapunov equations. Calcolo. 56, 51 (2019).
dc.relation.referencesen[12] Lang N., Mena H., Saak J. On the benefits of the LDLT factorization for large-scale differential matrix equation solvers. Linear Algebra and its Applications. 480, 44–71 (2015).
dc.relation.referencesen[13] Druskin V., Knizhnerman L. Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM Journal on Matrix Analysis and Applications. 19 (3), 755–771 (1998).
dc.relation.referencesen[14] Barkouki H., Bentbib A. H., Heyouni M., Jbilou K. An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo. 55, 13 (2018).
dc.relation.referencesen[15] Parlett B. N., Taylor D. R., Li Z. A. A look-ahead Lanczos algorithm for unsymmetric matrices. Math. Comput. 44 (169), 105–124 (1985).
dc.relation.referencesen[16] Penzl T. L YAPACK A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems. Available online: http://www.tuchemintz.de/sfb393/lyapack.
dc.relation.urihttp://www.tuchemintz.de/sfb393/lyapack
dc.rights.holder© Національний університет “Львівська політехніка”, 2021
dc.subjectрозширений блок підпростору Крилова
dc.subjectрозширений несиметричний блок алгоритма Ланцоша
dc.subjectнаближення низького рангу
dc.subjectдиференціальні рівняння Ляпунова
dc.subjectextended block Krylov subspace
dc.subjectextended nonsymmetric block Lanczos algorithm
dc.subjectlow-rank approximation
dc.subjectdifferential Lyapunov equations
dc.titleThe extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations
dc.title.alternativeРозширений несиметричний блок методів Ланцоша для розв’язування великомасштабних диференціальних рівнянь Ляпунова
dc.typeArticle

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Mathematical Modeling And Computing. – 2021. – Vol. 8, No. 3