Algorithms for the CMRH method for dense linear systems

Duminil, Sébastien; Heyouni, Mohammed; Marion, P; Sadok, H.

doi:10.1007/s11075-015-9997-2

Cited by 5 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Comparison studies between the CMRH and GMRES methods in terms of floating point operations and memory requirements for dense matrices can be found in the works of Sadok et al among others. At this stage, we give a brief description for the CMRH method, and for more details, see other works . To formulate the CMRH solver for the linear system , we define the maximum and Euclidean norms of a vector

boldv \in R^{N}

{‖boldv‖}_{\infty} = \max_{1 \leq i \leq N} | v_{i} |, {‖ v ‖}_{2} = \sqrt{\sum_{i = 1}^{N} {| v_{i} |}^{2}},

where v i is the i th component of the vector v .…”

Section: Changing Minimal Residual Solvermentioning

confidence: 99%

“…Note that we have used the symbol ↔ to express swapping contents in the CMRH algorithm. For more details on overstorage and the pivoting strategy in the CMRH solver, see other works …”

Section: Changing Minimal Residual Solvermentioning

confidence: 99%

“…At this stage, we give a brief description for the CMRH method, and for more details, see other works. [29][30][31][32] To formulate the CMRH solver for the linear system (10), we define the maximum and Euclidean norms of a vector v ∈ R N as…”

Section: Changing Minimal Residual Solvermentioning

confidence: 99%

See 2 more Smart Citations

Iterative solvers for generalized finite element solution of boundary‐value problems

Mohamed

Seaı̈d

Bouhamidi

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill‐conditioned linear systems that are associated with these methods. In this study, we investigate the performance of a class of iterative solvers for the generalized finite element solution of time‐dependent boundary‐value problems. A fully implicit time‐stepping scheme is used for the time integration in the finite element framework. As enrichment, we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers, we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared with dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix–vector products are involved in their implementation. Two test examples for boundary‐value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill‐conditioned linear systems resulting from the generalized finite element methods for time domain problems.

show abstract

boldv \in R^{N}

{‖boldv‖}_{\infty} = \max_{1 \leq i \leq N} | v_{i} |, {‖ v ‖}_{2} = \sqrt{\sum_{i = 1}^{N} {| v_{i} |}^{2}},

where v i is the i th component of the vector v .…”

Section: Changing Minimal Residual Solvermentioning

confidence: 99%

“…Note that we have used the symbol ↔ to express swapping contents in the CMRH algorithm. For more details on overstorage and the pivoting strategy in the CMRH solver, see other works …”

Section: Changing Minimal Residual Solvermentioning

confidence: 99%

See 1 more Smart Citation

Iterative solvers for generalized finite element solution of boundary‐value problems

Mohamed

Seaı̈d

Bouhamidi

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…The corresponding convergence analysis of the CMRH method and its relation to the GMRES method can be found in [5,6]. There are a great deal of past and recent works and interests in developing the Hessenberg process and CMRH method for linear systems [7][8][9][10][11][12][13]. Specifically, in [7,8], Duminil presents an implementation for parallel architectures and an implementation of the left preconditioned CMRH method.…”

Section: Introductionmentioning

confidence: 99%

“…There are a great deal of past and recent works and interests in developing the Hessenberg process and CMRH method for linear systems [7][8][9][10][11][12][13]. Specifically, in [7,8], Duminil presents an implementation for parallel architectures and an implementation of the left preconditioned CMRH method. A polynomial preconditioner and flexible right preconditioner for CMRH methods are considered in [9] and [10], respectively.…”

Section: Introductionmentioning

confidence: 99%

Heavy Ball Restarted CMRH Methods for Linear Systems

Teng

Wang

2018

MCA

View full text Add to dashboard Cite

Abstract:The restarted CMRH method (changing minimal residual method based on the Hessenberg process) using fewer operations and storage is an alternative method to the restarted generalized minimal residual method (GMRES) method for linear systems. However, the traditional restarted CMRH method, which completely ignores the history information in the previous cycles, presents a slow speed of convergence. In this paper, we propose a heavy ball restarted CMRH method to remedy the slow convergence by bringing the previous approximation into the current search subspace. Numerical examples illustrate the effectiveness of the heavy ball restarted CMRH method.

show abstract