Iterative Methods for Sparse Linear Systems

Saad, Yousef

doi:10.1137/1.9780898718003

Cited by 9,752 publications

(9,915 citation statements)

References 140 publications

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“…We refer the reader to e.g. [29] for a general introduction on Krylov subspace methods and to [29,Section 10] and [25,Section 9.4] for a review on flexible methods. The minimum residual norm GMRES method [26] has been extended by Saad [23] to allow variable preconditioning.…”

Section: General Settingmentioning

confidence: 99%

“…When non variable preconditioning is considered, the full GMRES method [26] is often chosen for the solution of non-symmetric or non-Hermitian linear systems because of its robustness and its minimum residual norm property [25]. Nevertheless to control both the memory requirements and the computational cost of the orthogonalization scheme, restarted GMRES is preferred; it corresponds to a scheme where the maximal dimension of the approximation subspace is fixed.…”

Section: Introductionmentioning

confidence: 99%

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A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

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Section: General Settingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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“…For clarity of presentation, we assume that the partitions are of equal size m and that each overlap is of size τ . Thus, we can rewrite (1) as two linear systems (4) (5) where we choose the adjustment vector y such that the solution of the lower part of (4) coincides with the upper part of (5), in other words, (6) Let us assume, for now, that each overlapped partition is nonsingular, and that (7) Thus, using (4), (5) and (6) we obtain the balance system (8) (9) (10) Notice that once the balance system (8) is solved for y, linear systems (4) and (5) can be solved independently in parallel. Since the coefficient matrix M is not available explicitly, we use a modified iterative method to solve the balance system (8), where we compute the residual and matrix-vector products as described below.…”

Section: B Sparse Matrix Solvermentioning

confidence: 99%

“…Pardiso [6], MUMPS [7], and SuperLU [8] belong to the direct methods. Conjugate gradient [9], multigrid method [10], and streaming multigrid solver [11] are the commonly used iterative solvers. However, due to direct solvers' poor scalability, they are not appropriate for large linear systems, while iterative solvers are not so robust and they usually take a great number of iterations to converge.…”

Section: Introductionmentioning

confidence: 99%

Fast Parallel Compositon of Dunhuang Murals based on Matrix Tearing

Chen¹,

Wang²,

Zhao³

et al. 2017

Proceedings of the 3rd International Conference on Wireless Communication and Sensor Networks (WCSN 2016)

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Abstract-In this paper we have developed a robust scalable hybrid scheme for Fast Parallel compositon of Dunhuang murals based on Matrix Tearing. The large cost associated with solution of partitioned linear systems, which is required to perform the matrix-vector multiplication, was avoided by presenting a scheme that requires only the solution of tiny linear systems of the same size as that of the overlap between two partitions. The difficulties associated with the multiple partition approach are resolved by using the reordering.

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“…The diagonal translation operator T L (k, r M ), the weighting factor W (k θ ), and the probe correction coefficientP(k,r M ) are combined to form the coupling matrix C . The given set of linear equations is solved using the Generalized Minimum Residual Solver (GMRES) [27] in a least mean square sense (LMS) [28] as…”

Section: Near-field Error Analysismentioning

confidence: 99%

Near-Field Error Analysis for Arbitrary Scanning Grids Using Fast Irregular Antenna Field Transformation Algorithm

Qureshi¹,

Schmidt²,

Eibert³

2013

PIER B

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Abstract-An extensive error analysis for arbitrary near-field antenna measurements is performed. Expressions are derived to estimate the far-field uncertainty using the available near-field data together with the measurement inaccuracy but, most importantly, without the knowledge of the reference far field. Error analysis techniques presented so far either assume a specific set of antennas or a specific measurement surface and are difficult to generalize. We present a generalized approach providing realistic error estimates using the recently developed Fast Irregular Antenna Field Transformation Algorithm (FIAFTA). FIAFTA utilizes equivalent plane wave sources to represent radiated antenna fields and is able to process nearfield data collected on arbitrary measurement grids. The unknown plane wave coefficients are determined by solving a linear system of equations. The error model is applied to planar, cylindrical, and spherical near-field measurements and is also valid for arbitrary measurement grids. The estimated far-field uncertainties show good agreement with the reference far-field errors.

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Iterative Methods for Sparse Linear Systems

Cited by 9,752 publications

References 140 publications

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Fast Parallel Compositon of Dunhuang Murals based on Matrix Tearing

Near-Field Error Analysis for Arbitrary Scanning Grids Using Fast Irregular Antenna Field Transformation Algorithm

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