A comparative study of sparse approximate inverse preconditioners

Benzi, Michele; Tůma, Miroslav

doi:10.1016/s0168-9274(98)00118-4

Cited by 224 publications

(208 citation statements)

References 68 publications

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“…More recently, sparse approximate inverses have become an interesting approach to precondition conjugate gradient iterations [13,14]. Contrary to incomplete factorizations, the computation of approximate inverses can be parallelized in a straightforward way.…”

Section: Incomplete Cholesky Factorizationsmentioning

confidence: 99%

Numerical performance of incomplete factorizations for 3D transient convection–diffusion problems

Ferran

Sandoval

2007

Advances in Engineering Software

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Many environmental processes can be modelled as transient convection-diffusion-reaction problems. This is the case, for instance, of the operation of activated-carbon filters. For industrial applications there is a growing demand for 3D simulations, so efficient linear solvers are a major concern. We have compared the numerical performance of two families of incomplete Cholesky factorizations as preconditioners of conjugate gradient iterations: drop-tolerance and prescribed-memory strategies. Numerical examples show that the former are computationally more efficient, but the latter may be preferable due to their predictable memory requirements.

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Section: Incomplete Cholesky Factorizationsmentioning

confidence: 99%

Numerical performance of incomplete factorizations for 3D transient convection–diffusion problems

Ferran

Sandoval

2007

Advances in Engineering Software

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“…Even for symmetric positive definite matrices, existence of the standard IC factorization is guaranteed only for some special classes of matrices (see, e.g., [14]). In the symmetric positive definite case, variants of IC have been developed to avoid ill-conditioning and breakdown (see, e.g., [15,16]). Nevertheless, usually these modifications are expensive and introduce additional parameters to be chosen in the algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…For a given square matrix A, there exist several proposals for constructing robust sparse inverse approximations which are based on optimization techniques, mainly based on minimizing the Frobenius norm of the residual (I − XA) over a set P of matrices with a certain sparsity pattern; see e.g., [5,6,15,[17][18][19][20][21][22]. An advantage of these approximate inverse preconditioners is that the process of building them, as well as applying them, is well suited for parallel platforms.…”

Section: Introductionmentioning

confidence: 99%

Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

Chehab

Raydan

2016

Mathematics

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Abstract:We focus on inverse preconditioners based on minimizing F(X) = 1 − cos(XA, I), where XA is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods to minimize F(X) on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of F(X) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.

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“…Therefore it is necessary to first compress the force-displacement matrix. We note that there have been considerable efforts in directly approximating an inverse matrix using a sparse matrix [32][33][34]. However they are developed to obtain better pre-conditioners for fast convergence.…”

Section: Clustering Methodsmentioning

confidence: 99%

An efficient large deformation method using domain decomposition

Huang

Liu

Bao

et al. 2006

Computers & Graphics

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Efficiently simulating large deformations of flexible objects is a challenging problem in computer graphics. In this paper, we present a physically based approach to this problem, using the linear elasticity model and a finite elements method. To handle large deformations in the linear elasticity model, we exploit the domain decomposition method, based on the observation that each sub-domain undergoes a relatively small local deformation, involving a global rigid transformation. In order to efficiently solve the deformation at each simulation time step, we pre-compute the object responses in terms of displacement accelerations to the forces acting on each node, yielding a force-displacement matrix. However, the force-displacement matrix could be too large to handle for densely tessellated objects. To address this problem, we present two methods. The first method exploits spatial coherence to compress the force-displacement matrix using the clustered principal component analysis method; and the second method pre-computes only the force-displacement vectors for the boundary vertices of the sub-domains and resorts to the Cholesky factorization to solve the acceleration for the internal vertices of the sub-domains. Finally, we present some experimental results to show the large deformation effects and fast performance on complex large scale objects under interactive user manipulations.

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A comparative study of sparse approximate inverse preconditioners

Cited by 224 publications

References 68 publications

Numerical performance of incomplete factorizations for 3D transient convection–diffusion problems

Numerical performance of incomplete factorizations for 3D transient convection–diffusion problems

Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

An efficient large deformation method using domain decomposition

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