×

Incremental incomplete LU factorizations with applications. (English) Zbl 1240.65091

Summary: This paper addresses the problem of computing preconditioners for solving systems of linear equations with a sequence of slowly varying matrices. This problem arises in many important applications. For example, a common situation in computational fluid dynamics is when the equations change only slightly, possibly in some parts of the physical domain. In such situations, it is wasteful to recompute entirely any LU or ILU factorizations computed for the previous coefficient matrix. A number of techniques for computing incremental ILU factorizations are examined. For example, we consider methods based on approximate inverses as well as alternating techniques for updating the factors L and U of the factorization.

MSC:

65F08 Preconditioners for iterative methods
65F05 Direct numerical methods for linear systems and matrix inversion

Software:

CSparse
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Saad, Iterative Methods for Sparse Linear Systems (2003) · Zbl 1031.65046 · doi:10.1137/1.9780898718003
[2] Axelsson, Iterative Solution Methods (1994) · doi:10.1017/CBO9780511624100
[3] Birken, Preconditioner updates applied to CFD model problems, Applied Numerical Mathematics 58 (11) pp 1628– (2008) · Zbl 1148.76036
[4] Chan SM Brandwajn V Partial matrix refactorization 193 199
[5] Chehab, Differential equations and solution of linear systems, Numerical Algorithms 40 pp 103– (2005) · Zbl 1086.65026
[6] Chehab, Matrix differential equations and inverse preconditioners, Computational and Applied Mathematics 26 pp 95– (2007) · Zbl 1182.65057 · doi:10.1590/S0101-82052007000100005
[7] Benzi, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM Journal on Scientific Computing 17 pp 1135– (1996) · Zbl 0856.65019
[8] Benzi, Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics, Computational Methods in Applied Mechanics and Engineering 190 pp 6533– (2001) · Zbl 1021.74041
[9] Benzi, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM Journal on Scientific Computing 19 pp 968– (1998) · Zbl 0930.65027
[10] Kolotinina, Recent Advances in Iterative Methods, IMA Volumes in Mathematics and its Applications 60 (1994)
[11] Kolotilina, On a family of two-level preconditionings of the incomplete block factorization type, Soviet Journal of Numerical Analysis and Mathematical Modeling 1 pp 293– (1986) · Zbl 0825.65028
[12] Kolotilina, Factorized sparse approximate inverse preconditionings I. Theory, SIAM Journal on Matrix Analysis and Applications 14 pp 45– (1993) · Zbl 0767.65037
[13] Grote, Parallel Processing for Scientific Computing 2 pp 519– (1992)
[14] Grote, Parallel preconditionings with sparse approximate inverses, SIAM Journal on Scientific Computing 18 pp 838– (1997) · Zbl 0872.65031
[15] Chow, Approximate inverse preconditioners via sparse-sparse iterations, SIAM Journal on Scientific Computing 19 pp 995– (1998) · Zbl 0922.65034
[16] Chow, Approximate inverse techniques for block-partitioned matrices, SIAM Journal on Scientific Computing 18 pp 1657– (1997) · Zbl 0888.65035
[17] Holland, Sparse approximate inverses and target matrices, SIAM Journal on Scientific Computing 26 pp 1000– (2005) · Zbl 1077.65044
[18] Davis, Direct Methods for Sparse Linear Systems (2006) · Zbl 1119.65021 · doi:10.1137/1.9780898718881
[19] Gilbert, Sparse partial pivoting in time proportional to arithmetic operations, SIAM Journal on Scientific Computing 9 pp 862– (1988) · Zbl 0656.65036
[20] Tinney, Sparse vector methods, IEEE Transactions on Power Apparatus and Systems PAS-104 (2) pp 295– (1985)
[21] Golub, Matrix Computations (1996)
[22] Calgaro, An hybrid finite volume-finite element method for variable density incompressible flows, Journal of Computational Physics 227 pp 4671– (2008) · Zbl 1137.76037
[23] Strang, On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis 5 pp 506– (1968) · Zbl 0184.38503
[24] Eymard, Handbook of Numerical Analysis VII pp 713– (2000)
[25] LeVeque, Cambridge Texts in Applied Mathematics (2002)
[26] Girault, Springer Series in Computational Mathematics, in: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms (1986) · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[27] Ern, Éléments finis: théorie, applications, mise en oeuvre (2002)
[28] Guermond, A projection FEM for variable density incompressible flows, Journal of Computational Physics 165 pp 167– (2000) · Zbl 0994.76051
[29] Calgaro, A preconditioner for generalized saddle-point problems: application to 3d stationary Navier-Stokes equations, Numerical Methods for Partial Differential Equations 22 pp 1289– (2006) · Zbl 1370.76082
[30] Silvester, Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow, Journal of Computational and Applied Mathematics 128 pp 261– (2001) · Zbl 0983.76051
[31] Cahouet, Some fast 3D finite element solvers for the generalized Stokes problem, International Journal for Numerical Methods in Fluids 8 pp 865– (1988) · Zbl 0665.76038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.