The solution of linear systems by using the Sherman–Morrison formula

Maponi, Pierluigi

doi:10.1016/j.laa.2006.07.007

Cited by 21 publications

(18 citation statements)

References 10 publications

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“…Maponi [20] proposed a general approach based on the Sherman Morrison formula to solve linear systems. The application of this generic algorithm to (16) leads to an increased computational cost as the special structure of the system (and special structure of R) are not exploited.…”

Section: Analysis Methodsmentioning

confidence: 99%

“…, w n obs ) ∈ R n obs ×n obs is a diagonal matrix holding the diagonal entries of W (nens) , u i is the i-th column of U = W (nens) − W (0) ∈ R n obs ×n obs and v i = e i is the i-th element of the canonical basis in R n obs . Thus, according to [20,Corollary 4], each linear system (32) can be solved with O(n 3 obs ) long operations, leading to a total of O n ens · n 3 obs . Therefore the computational cost of the analysis step is:…”

Section: Analysis Methodsmentioning

confidence: 99%

“…Moreover, according to [20,Theorem 3], when n obs n ens , the solution of linear system (32) can be computed with no more than O(n 2 ens · n obs + n 2 ens ) long operations. The resulting computational cost of the analysis step is:…”

Section: Analysis Methodsmentioning

confidence: 99%

“…Lastly, the stability conditions of Maponi's method are not discussed in [20]. Furthermore, the sequence of matrices W (k) are not proved to be non-singular, which is crucial for the well-performance of that method.…”

Section: Analysis Methodsmentioning

confidence: 99%

“…We apply the Sherman-Morrison formula (17) again. The solution of the linear system (20) can be obtained as follows:…”

Section: Iterative Implementation Of the Enkf Analysismentioning

confidence: 99%

See 4 more Smart Citations

An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula

2014

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Section: Analysis Methodsmentioning

confidence: 99%

Section: Analysis Methodsmentioning

confidence: 99%

Section: Analysis Methodsmentioning

confidence: 99%

Section: Analysis Methodsmentioning

confidence: 99%

“…We apply the Sherman-Morrison formula (17) again. The solution of the linear system (20) can be obtained as follows:…”

Section: Iterative Implementation Of the Enkf Analysismentioning

confidence: 99%

See 3 more Smart Citations

An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula

2014

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Incomplete inverse matrices

Filelis‐Papadopoulos

2021

Numerical Linear Algebra App

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The solution of large sparse linear systems is required in many scientific fields such as computational fluid dynamics, computational electromagnetism, computational finance, etc. The computation of the solution of these systems is performed with preconditioned iterative methods, which rely on effective preconditioning schemes. A new class of approximate inverses is proposed, namely incomplete inverse matrices, which are computed using a recursive Schur complement-based approach. This class of approximate inverses is based on a priori knowledge of a sparsity pattern. In order to have finer control over the density of the proposed approximate inverse, especially in the case of three-dimensional problems, on-the-fly filtration is used, resulting in substantial reduction in the number of nonzero elements. Implementation details and analysis for computing the proposed scheme are given. Numerical results depicting the effectiveness and applicability of the proposed scheme are also provided. K E Y W O R D Sapproximate inverse matrix, filtering, preconditioned iterative methods, sparsity patterns

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Adaptive Factored Incomplete Inverse Matrices

Filelis‐Papadopoulos

2024

Numerical Linear Algebra App

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The solution of large sparse linear systems is an essential part in many scientific fields. Numerical solution of such systems is usually performed using iterative methods in conjunction with effective preconditioning schemes. A new class of factored approximate inverses is proposed, namely adaptive factored incomplete inverse matrices, which are computed using a recursive Schur complement‐based approach. This class of approximate inverses does not require the knowledge of a sparsity pattern, which is formed adaptively during computation. For this to be possible, a flexible sparse storage scheme was designed. Several numerical dropping strategies are showcased and discussed, along with the effects of reordering to the preconditioner density and the corresponding convergence behavior. Dropping based on monitoring the growth of elements in the incomplete LU factorization is also presented and discussed along with improvements during computation of the successive Schur complements. A static sparsity pattern variant is also provided and discussed. Implementation details and analysis, for computing the proposed scheme, are given along with the computational complexity and memory requirements. Numerical results depicting the effectiveness and applicability of the proposed scheme are also presented.

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The solution of linear systems by using the Sherman–Morrison formula

Cited by 21 publications

References 10 publications

An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula

An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula

Incomplete inverse matrices

Adaptive Factored Incomplete Inverse Matrices

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