Improved balanced incomplete factorization

[EN] . In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Mar´ın, J. Mas, and M. T˚uma, SIAM J. Sci. Comput., 30 (2008), pp. 2302– 2318]. The improvements are based on a deeper understanding of the inverse Sherman–Morrison (ISM) decomposition, and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors numerically influence each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systemsReceived by the editors January 26, 2009; accepted for publication (in revised form) by V. Simoncini June 1, 2010; published electronically August 12, 2010. This work was supported by Spanish grant MTM 2007-64477, by project IAA100300802 of the Grant Agency of the Academy of Sciences of the Czech Republic, and partially also by the International Collaboration Support M100300902 of AS CR.Bru García, R.; Marín Mateos-Aparicio, J.; Mas Marí, J.; Tuma, M. (2010). Improved balanced incomplete factorization. SIAM Journal on Matrix Analysis and Applications. 31(5):2431-2452. https://doi.org/10.1137/090747804S2431245231

Updating preconditioners for modified least squares problems

[EN] In this paper, we analyze how to update incomplete Cholesky preconditioners to solve least squares problems using iterative methods when the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Our proposed method computes a low-rank update of the preconditioner using a bordering method which is inexpensive compared with the cost of computing a new preconditioner. Moreover, the numerical experiments presented show that this strategy gives, in many cases, a better preconditioner than other choices, including the computation of a new preconditioner from scratch or reusing an existing one.Partially supported by Spanish Grants MTM2014-58159-P and MTM2015-68805-REDT.Marín Mateos-Aparicio, J.; Mas Marí, J.; Guerrero-Flores, DJ.; Hayami, K. (2017). Updating preconditioners for modified least squares problems. Numerical Algorithms. 75(2):491-508. https://doi.org/10.1007/s11075-017-0315-zS491508752Alexander, S.T., Pan, C.T., Plemmons, R.J.: Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing. Linear Algebra Appl. 98, 3–40 (1988)Andrew, R., Dingle, N.: Implementing QR factorization updating algorithms on GPUs. Parallel Comput. 40(7), 161–172 (2014). doi: 10.1016/j.parco.2014.03.003 . http://www.sciencedirect.com/science/article/pii/S0167819114000337 . 7th Workshop on Parallel Matrix Algorithms and ApplicationsBenzi, M., T˚uma, M.: A robust incomplete factorization preconditioner for positive definite matrices. Numer. Linear Algebra Appl. 10(5-6), 385–400 (2003)Benzi, M., Szyld, D.B., Van Duin, A.: Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comput. 20(5), 1652–1670 (1999)Björck, Å.: Numerical methods for Least Squares Problems. SIAM, Philadelphia (1996)Bru, R., Marín, J., Mas, J., T˚uma, M.: Preconditioned iterative methods for solving linear least squares problems. SIAM J. Sci. Comput. 36(4), A2002–A2022 (2014)Cerdán, J., Marín, J., Mas, J.: Low-rank upyears of balanced incomplete factorization preconditioners. Numer. Algorithms. doi: 10.1007/s11075-016-0151-6 (2016)Chambers, J.M.: Regression updating. J. Amer. Statist. Assoc. 66, 744–748 (1971)Davis, T.A., Hu, Y.: The university of florida sparse matrix collection. ACM trans. Math. Software 38(1), 1–25 (2011)Davis, T.A., Hager, W.W.: Modifying a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 20, 606–627 (1999)Davis, T.A., Hager, W.W.: Multiple-rank modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 22, 997–1013 (2001)Davis, T.A., Hager, W.W.: Row modification of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 26, 621–639 (2005)Hammarling, S., Lucas, C.: Updating the QR factorization and the least squares problem. Tech. rep., The University of Manchester, http://www.manchester.ac.uk/mims/eprints (2008)Olsson, O., Ivarsson, T.: Using the QR factorization to swiftly upyear least squares problems. Thesis report, Centre for Mathematical Sciences. The Faculty of Engineering at Lund University LTH (2014)Pothen, A., Fan, C.J.: Computing the block triangular form of a sparse matrix. ACM Trans. Math. Software 16, 303–324 (1990)Saad, Y.: ILUT: A dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 1(4), 387–402 (1994)Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Co., Boston (1996

Determinación de H-matrices

Cuando la matriz de comparación de una matriz A, M(A), es M-matriz, se dice que A es H-matriz. Este tipo de matrices aparece, por ejemplo, en la discretización por elementos finitos de ciertas ecuaciones parabólicas no lineales. Además, esta característica garantiza la existencia de precondicionadores para la resolución de sistemas lineales por métodos iterativos de tipo Krylov. Aunque son muchas las caracterizaciones de H-matriz que provienen de las de M-matriz invertible, una H-matriz invertible puede tener una matriz de comparación singular. En este trabajo utilizamos una caracterización de H-matriz con elementos diagonales no nulos, tanto si M(A) es invertible como singular, basada en que ρ ≤ 1, siendo ρ el radio espectral de la matriz de Jacobi de M(A). Proponemos entonces algoritmos para acotar el valor de ρ y concluir si la matriz A es H-matriz o no. Se proponen algoritmos para aproximar el valor de ρ y el vector de Perron asociado y se demuestra su convergencia para matrices irreducibles

Preconditioners for nonsymmetric linear systems with low-rank skew-symmetric part

[EN] We present a preconditioning technique for solving nonsymmetric linear systems Ax = b, where the coefficient matrix A has a skew-symmetric part that can be well approximated with a skew-symmetric low-rank matrix. The method consists of updating a preconditioner obtained from the symmetric part of A. We present some results concerning to the approximation properties of the preconditioner and the spectral properties of the preconditioning technique. The results of the numerical experiments performed show that our strategy is competitive compared with some specific methods. (C) 2018 Elsevier B.V. All rights reserved.This work was supported by the Spanish Ministerio de Economia y Competitividad under grants MTM2014-58159-P and MTM2015-68805-REDT.Cerdán Soriano, JM.; Guerrero-Flores, DJ.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2018). Preconditioners for nonsymmetric linear systems with low-rank skew-symmetric part. Journal of Computational and Applied Mathematics. 343:318-327. https://doi.org/10.1016/j.cam.2018.04.023S318327343J. Sifuentes, Preconditioned iterative methods for inhomogeneous acoustic scattering applications (Ph.D. thesis), 2010.Beckermann, B., & Reichel, L. (2008). The Arnoldi Process and GMRES for Nearly Symmetric Matrices. SIAM Journal on Matrix Analysis and Applications, 30(1), 102-120. doi:10.1137/060668274Embree, M., Sifuentes, J. A., Soodhalter, K. M., Szyld, D. B., & Xue, F. (2012). Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices. SIAM Journal on Matrix Analysis and Applications, 33(2), 480-500. doi:10.1137/110851006Saad, Y., & Schultz, M. H. (1986). GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing, 7(3), 856-869. doi:10.1137/0907058Cerdán, J., Marín, J., & Mas, J. (2016). Low-rank updates of balanced incomplete factorization preconditioners. Numerical Algorithms, 74(2), 337-370. doi:10.1007/s11075-016-0151-6Van der Vorst, H. A. (1992). Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing, 13(2), 631-644. doi:10.1137/0913035Bergamaschi, L., Gondzio, J., Venturin, M., & Zilli, G. (2007). Inexact constraint preconditioners for linear systems arising in interior point methods. Computational Optimization and Applications, 36(2-3), 137-147. doi:10.1007/s10589-006-9001-0Davis, T. A., & Hu, Y. (2011). The university of Florida sparse matrix collection. ACM Transactions on Mathematical Software, 38(1), 1-25. doi:10.1145/2049662.2049663Berry, M. W., Pulatova, S. A., & Stewart, G. W. (2005). Algorithm 844. ACM Transactions on Mathematical Software, 31(2), 252-269. doi:10.1145/1067967.1067972Stewart, G. W. (1999). Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix. Numerische Mathematik, 83(2), 313-323. doi:10.1007/s002110050451Saad, Y. (1994). ILUT: A dual threshold incomplete LU factorization. Numerical Linear Algebra with Applications, 1(4), 387-402. doi:10.1002/nla.168001040

Low-rank update of preconditioners for the nonlinear Richard's equation

Preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite (SPD) Jacobian. Following the theoretical work in Bergamaschi et al. (2011) [4] we start from a given approximation of the inverse of the initial Jacobian, and we construct a sequence of preconditioners by means of a low rank update, for the linearized systems arising in the Picard Newton solution of the nonlinear discretized Richards equation. Numerical results onto a very large and realistic test case show that the proposed approach is more efficient, in terms of iteration number and CPU time, as compared to computing the preconditioner of choice at every nonlinear iteration.The support of the CARIPARO Foundation (Grant NPDE: Non-linear Partial Differential Equations: models, analysis, and control - theoretic problems), and of the Spanish DGI grant MTM2010-18674 is acknowledged.Bergamaschi, L.; Bru García, R.; Martínez Calomardo, Á.; Mas Marí, J.; Putti, M. (2013). Low-rank update of preconditioners for the nonlinear Richard's equation. Mathematical and Computer Modelling. 57(7):1933-1941. https://doi.org/10.1016/j.mcm.2012.01.013S1933194157

Pivoting in ISM factorizations

[EN] In this work we study pivoting strategies for the preconditioner presented Balanced Incomplete Preconditioner (SIAM J Sci Comput 30(5):2302¿2318, 2008) which computes the LU factorization of a matrix A. We present a modification of the algorithm that allows for pivoting, and so the computation of preconditioners for any nonsingular matrix. The results of numerical experiments with ill-conditioned and highly indefinite matrices arising from different applications show the robustness of the new algorithm, since it is able to solve problems that are not possible to solve otherwise.Mas Marí, J.; Marín Mateos-Aparicio, J. (2022). Pivoting in ISM factorizations. Universitat Politècnica de València. 160-165. http://hdl.handle.net/10251/19241616016

Preconditioners for rank deficient least squares problems

[EN] In this paper we present a method for computing sparse preconditioners for iteratively solving rank deficient least squares problems (LS) by the LSMR method. The main idea of the method proposed is to update an incomplete factorization computed for a regularized problem to recover the solution of the original one. The numerical experiments for a wide set of matrices arising from different science and engineering applications show that the preconditioner proposed, in most cases, can be successfully applied to accelerate the convergence of the iterative Krylov subspace method.This work was supported by the Spanish Ministerio de Economia, Industria y Competitividad, Spain under grants MTM2017-85669-P and MTM2017-90682-REDT.Cerdán Soriano, JM.; Guerrero, D.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2020). Preconditioners for rank deficient least squares problems. Journal of Computational and Applied Mathematics. 372:1-11. https://doi.org/10.1016/j.cam.2019.112621S111372Paige, C. C., & Saunders, M. A. (1982). LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. ACM Transactions on Mathematical Software, 8(1), 43-71. doi:10.1145/355984.355989Paige, C. C., & Saunders, M. A. (1982). Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems. ACM Transactions on Mathematical Software, 8(2), 195-209. doi:10.1145/355993.356000Golub, G., & Kahan, W. (1965). Calculating the Singular Values and Pseudo-Inverse of a Matrix. Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 2(2), 205-224. doi:10.1137/0702016Fong, D. C.-L., & Saunders, M. (2011). LSMR: An Iterative Algorithm for Sparse Least-Squares Problems. SIAM Journal on Scientific Computing, 33(5), 2950-2971. doi:10.1137/10079687xScott, J. (2017). On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems. SIAM Journal on Scientific Computing, 39(4), C319-C339. doi:10.1137/16m1065380HSL, A collection of Fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk/.Li, N., & Saad, Y. (2006). MIQR: A Multilevel Incomplete QR Preconditioner for Large Sparse Least‐Squares Problems. SIAM Journal on Matrix Analysis and Applications, 28(2), 524-550. doi:10.1137/050633032Benzi, M., & T?ma, M. (2003). A robust incomplete factorization preconditioner for positive definite matrices. Numerical Linear Algebra with Applications, 10(5-6), 385-400. doi:10.1002/nla.320Hayami, K., Yin, J.-F., & Ito, T. (2010). GMRES Methods for Least Squares Problems. SIAM Journal on Matrix Analysis and Applications, 31(5), 2400-2430. doi:10.1137/070696313R. Bru, J. Marín, J. Mas, M. Tůma, Preconditioned iterative methods for solving linear least squares problems, SIAM J. Sci. Comput. 36 (4).Gould, N., & Scott, J. (2017). The State-of-the-Art of Preconditioners for Sparse Linear Least-Squares Problems. ACM Transactions on Mathematical Software, 43(4), 1-35. doi:10.1145/3014057Cerdán, J., Marín, J., & Mas, J. (2016). Low-rank updates of balanced incomplete factorization preconditioners. Numerical Algorithms, 74(2), 337-370. doi:10.1007/s11075-016-0151-6Davis, T. A., & Hu, Y. (2011). The university of Florida sparse matrix collection. ACM Transactions on Mathematical Software, 38(1), 1-25. doi:10.1145/2049662.2049663Pothen, A., & Fan, C.-J. (1990). Computing the block triangular form of a sparse matrix. ACM Transactions on Mathematical Software, 16(4), 303-324. doi:10.1145/98267.98287Arridge, S. R., Betcke, M. M., & Harhanen, L. (2014). Iterated preconditioned LSQR method for inverse problems on unstructured grids. Inverse Problems, 30(7), 075009. doi:10.1088/0266-5611/30/7/07500