Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Sadek, El Mostafa; Bentbib, Abdeslem Hafid; Sadek, Lakhlifa; Alaoui, Hamad Talibi

doi:10.1007/s12190-019-01278-7

Cited by 14 publications

(12 citation statements)

References 28 publications

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“…In contrast, equations (3) are solved in an alternative way in IMM by introducing the Lanczos method. It is based on Krylov subspace method [28], [29] to find basis vectors to approximate the original high dimensional matrix. For the original matrix L e , the Lanczos method generates basis vectors by making an initial guess on the first basis vector and following the iteration procedure:…”

Section: Governing Equations and Hybrid Eigenmode Restoration A Probl...mentioning

confidence: 99%

A Hybrid Eigenmode Restoration Algorithm for Computational Lithography Problems Based on Mode Matching Principle

Liu

et al. 2022

IEEE Access

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The mode matching method is an efficient solution to conventional microwave waveguide problems for its dimensionality advantages. It is theoretically extendable to computational lithography problems in nano scales with periodic boundary condition. The high computational complexity of conventional mode matching limits its application significance in highly complicated nano scale problems. This paper introduces an optimized efficient mode matching method for computational lithography problems whose complexity is O(N 1.5 ). This bottleneck is breached by transforming governing equations with Lanczos algorithm. A novel hybrid eigenmode restoration algorithm is proposed to solve the eigenmode accuracy loss by involving the Arnoldi method. Benchmark cases verify the accuracy of the restored eigenmode and mode matching method with from microwave to nano scale problems. The efficiency is further optimized by exploring the relevance between iteration step and structure complexity. The non-uniform iteration step is introduced for efficiency optimization. A real mask component is modelled as a multi-junctional waveguide structure with periodic boundary condition. Simulation results validate the effectiveness of the proposed method and efficiency benefits are discussed through comparison with other solvers including HFSS and RCWA. Results indicate that the proposed method possesses good flexibility and application significance for computational lithography problems with high complexity.

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Section: Governing Equations and Hybrid Eigenmode Restoration A Probl...mentioning

confidence: 99%

A Hybrid Eigenmode Restoration Algorithm for Computational Lithography Problems Based on Mode Matching Principle

Liu

et al. 2022

IEEE Access

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“…For more details on extended block Krylov projection method for solving large matrix equations see [3][4][5]10]. When we apply the ENBL algorithm 2 on the triple A, B, B B F , we get two biorthonormal matrices…”

Section: Low-rank Approximate Solutionmentioning

confidence: 99%

“…Now we have to solve the last differential Lyapunov equation ( 11) by ROS method or BDF method, see [3,4,6].…”

Section: Low-rank Approximate Solutionmentioning

confidence: 99%

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

Sadek¹,

Alaoui²

2021

Math. Model. Comput.

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In 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.

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“…To remedy this problem, combining Krylov subspaces techniques with BDF methods or with Taylor series expansions have recently been proposed [5,19]. Other existing methods described in the recent literature for solving large-scale differential Sylvester matrix equation rely on using the integral formula or some numerical ODE solver [20,37]. The strategy we pursue in this manuscript is different in the sense that our approach for solving differential Sylvester (or Lyapunov) matrix equations is based on the use of the constant solution to the differential equation.…”

Section: Introductionmentioning

confidence: 99%

The constant solution method for solving large scale differential Sylvester matrix equations with time invariant coefficients

Bouhamidi

Elbouyahyaoui

Heyouni

2022

Preprint

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This paper is mainly focused on the solution of Sylvester matrix differential equations with time independent coefficients. We propose a new approach based on the construction of a particular constant solution which allows to construct an approximate solution of the differential equation from that of the corresponding algebraic equation. Moreover, when the matrix coefficients of the differential equation are large, we combine the constant solution approach with Krylov subspace methods for obtaining an approximate solution of the Sylvester algebraic equation, and thus form an approximate solution of the large-scale Sylvester matrix differential equation. We establish some theoretical results including error estimates and convergence as well as relations between the residuals of the differential and its corresponding algebraic Sylvester matrix equation. We also give explicit benchmark formulas for the solution of the differential equation.To illustrate the efficiency of the proposed approach, we perform numerous numerical tests and make various comparisons with other methods for solving Sylvester matrix differential equations.

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Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Cited by 14 publications

References 28 publications

A Hybrid Eigenmode Restoration Algorithm for Computational Lithography Problems Based on Mode Matching Principle

A Hybrid Eigenmode Restoration Algorithm for Computational Lithography Problems Based on Mode Matching Principle

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

The constant solution method for solving large scale differential Sylvester matrix equations with time invariant coefficients

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