Balanced incomplete factorization preconditioner with pivoting

[EN] In this work we study pivoting strategies for the preconditioner presented in Bru (SIAM J Sci Comput 30(5):2302-2318, 2008) which computes the LU factorization of a matrix A. This preconditioner is based on the Inverse Sherman Morrison (ISM) decomposition [Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula. Bru (SIAM J Sci Comput 25(2):701-715, 2003), that using recursion formulas derived from the Sherman-Morrison formula, obtains the direct and inverse LU factors of a matrix. We present a modification of the ISM decomposition that allows for pivoting, and so the computation of preconditioners for any nonsingular matrix. While the ISM algorithm at a given step computes only a new pair of vectors, the new pivoting algorithm in the k-th step also modifies all the remaining vectors from k + 1 to n. Thus, it can be seen as a right looking version of the ISM decomposition. 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.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The work was supported by Conselleria de Innovacion, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana (CIAICO/2021/162).Marín Mateos-Aparicio, J.; Mas Marí, J. (2023). Balanced incomplete factorization preconditioner with pivoting. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 117(1). https://doi.org/10.1007/s13398-022-01334-1117

Low-rank updates of balanced incomplete factorization preconditioners

[EN] Let Ax = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned terations. Consider the matrix B = A + PQT where P,Q ∈ Rn×k are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system Bx = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.This work was supported by the Spanish Ministerio de Economia y Competitividad under grant MTM2014-58159-P.Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2017). Low-rank updates of balanced incomplete factorization preconditioners. Numerical Algorithms. 74(2):337-370. https://doi.org/10.1007/s11075-016-0151-6S337370742Bellavia, S., Bertaccini, D., Morini, B.: Nonsymmetric preconditioner updates in Newton-Krylov methods for nonlinear systems. SIAM J. Sci. Comput. 33 (5), 2595–2619 (2011)Benzi, M., Bertaccini, D.: Approximate inverse preconditioning for shifted linear systems. BIT 43(2), 231–244 (2003)Bergamaschi, L., Bru, R., Martínez, A.: Low-rank update of preconditioners for the inexact Newton method with SPD Jacobian. Math. Comput. Model. 54, 1863–1873 (2011)Bergamaschi, L., Bru, R., Martínez, A., Mas, J., Putti, M.: Low-rank update of preconditioners for the nonlinear Richards Equation. Math. Comput. Model. 57, 1933–1941 (2013)Bergamaschi, L., Gondzio, J., Venturin, M., Zilli, G.: Inexact constraint preconditioners for linear systems arising in interior point methods. Comput. Optim. Appl. 36(2-3), 137–147 (2007)Beroiz, M., Hagstrom, T., Lau, S.R., Price, R.H.: Multidomain, sparse, spectral-tau method for helically symmetric flow. Comput. Fluids 102(0), 250–265 (2014)Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)Bollhöfer, M.: A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J. Sci. Comput. 25(1), 86–103 (2003)Bollhöfer, M., Saad, Y.: On the relations between ILUs and factored approximate inverses. SIAM. J. Matrix Anal. Appl. 24(1), 219–237 (2002)Bru, R., Cerdán, J., Marín, J., Mas, J.: Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula. SIAM J. Sci. Comput. 25(2), 701–715 (2003)Bru, R., Marín, J., Mas, J., Tůma, M.: Balanced incomplete factorization. SIAM J. Sci. Comput. 30(5), 2302–2318 (2008)Bru, R., Marín, J., Mas, J., Tůma, M.: Improved balanced incomplete factorization. SIAM J. Matrix Anal. Appl. 31(5), 2431–2452 (2010)Cerdán, J., Faraj, T., Malla, N., Marín, J., Mas, J.: Block approximate inverse preconditioners for sparse nonsymmetric linear systems. Electron. Trans. Numer. Anal. 37, 23–40 (2010)Cerdán, J., Marín, J., Mas, J., Tůma, M.: Block balanced incomplete factorization. Technical Report No. TR-IMM2015/04, Polytechnic University of Valencia, Spain (2015)Davis, T.A.: University of Florida Sparse Matrix Collection. available online at http://www.cise.ufl.edu/~davis/sparse/ , NA Digest, vol. 94, issue 42, October 1994.Tebbens, J.D., Tůma, M.: Efficient preconditioning of sequences of nonsymmetric linear systems. SIAM J. Sci Comput. 29(5), 1918–1941 (2007)Tebbens, J.D., Tůma, M.: Preconditioner updates for solving sequences of linear systems in matrix-free environment. Numer Linear Algebra Appl. 17, 997–1019 (2010)Embree, M., Sifuentes, J.A., Soodhalter, K.M., Szyld, D.B., Xue, F.: Short-term recurrence Krylov subspace methods for nearly hermitian matrices. SIAM.J. Matrix Anal. Appl. 33-2, 480–500 (2012)Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math. 64, 697–735 (2011)Gatto, P., Christiansen, R.E., Hesthaven, J.S.: A preconditioner based on a low-rank approximation with applications to topology optimization. Technical Report EPFL-ARTICLE-207108, École polytechnique fédérale de Lausanne, EPFL, CH-1015 Lausanne, 2015.Grasedyck, L., Hackbusch, W.: Construction and arithmetics of H-matrices. Computing 70(4), 295–334 (2003)Grasedyck, L., Kressner, D., Tobler, C.: A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36(1), 53–78 (2013)Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6(1), 229–269 (1997)Hager, W.W.: Updating the inverse of matrix. SIAM Rev. 31(2), 221–239 (1989)Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method. Fundamentals of algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2003)Saad, Y.: ILUT: a dual threshold incomplete L U factorization. Numer. Linear Algebra Appl. 1(4), 387–402 (1994)Saad, Y., Schulz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)Simoncini, V., Szyld, D.B.: The effect of non-optimal bases on the convergence of Krylov subspace methods. Numer Math. 100(4), 711–733 (2005)van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 12, 631–644 (1992

A Survey on Nikodým and Vitali-Hahn-Saks Properties

[EN] Let ba(A) be the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra A of subsets of Omega and endowed with the variation norm. . A subset B of A is a Nikod ́ym set for ba(A) if each B-pointwise bounded subset M of ba(A) is uniformly bounded on A and B is a strong Nikod´ym set for ba(A) if each increasing covering (Bm)1 m=1 of B contains a Bn which is a Nikod´ym set for ba(A). If, additionally, the Nikod´ym subset B verifies that the sequential B-pointwise convergence in ba(A) implies weak convergence then B has the Vitali-Hahn-Saks property, (VHS ) in brief, and B has the strong (VHS ) property if for each increasing covering (Bm)1m=1 of B there exists Bq that has (VHS ) property Motivated by Valdivia result that every -algebra has strong Nikod´ym property and by his 2013 open question concerning that if Nikod´ym property in an algebra of subsets implies strong Nikod´ym property we survey this Valdivia theorem and we get that in a strong Nikod´ym set the (VHS ) property implies the strong (VHS ) property.Research supported for the second named author by Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation & Universities of Spain.López Alfonso, S.; López-Pellicer, M.; Mas Marí, J. (2021). A Survey on Nikodým and Vitali-Hahn-Saks Properties. Montes Taurus Journal of Pure and Applied Mathematics. 3(3):112-121. http://hdl.handle.net/10251/181186S1121213

An Algorithm for the Single-Input Pole Assignment Problem

[EN] B. N. Datta and K. Datta have proposed an efficient parallel algorithm for the single-input pole assignment problem when the spectrum set to be assigned is pairwise distinct and disjoint from the spectrum of the dynamical matrix. This paper first presents a theoretical analysis of that algorithm and then presents the necessary modifications on it for an arbitrary spectrum. Based on the modified algorithm, the authors present two new algorithms for the problem and give some results on controllability that are of independent interests.Bru, R.; Mas Marí, J.; Urbano Salvador, AM. (1994). An Algorithm for the Single-Input Pole Assignment Problem. SIAM Journal on Matrix Analysis and Applications. 15(2):393-407. doi:10.1137/S0895479889161125S39340715

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

Block approximate inverse preconditioners for sparse nonsymmetric linear systems

[EN] In this paper block approximate inverse preconditioners to solve sparse nonsymmetric linear systems with iterative Krylov subspace methods are studied. The computation of the preconditioners involves consecutive updates of variable rank of an initial and nonsingular matrix A0 and the application of the Sherman-MorrisonWoodbury formula to compute an approximate inverse decomposition of the updated matrices. Therefore, they are generalizations of the preconditioner presented in Bru et al. [SIAM J. Sci. Comput., 25 (2003), pp. 701¿715]. The stability of the preconditioners is studied and it is shown that their computation is breakdown-free for H-matrices. To test the performance the results of numerical experiments obtained for a representative set of matrices are presented.Cerdán Soriano, JM.; Faraj El Guelei, T.; Malla Martínez, N.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2010). Block approximate inverse preconditioners for sparse nonsymmetric linear systems. ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS. 12(37):23-40. http://hdl.handle.net/10251/99451S2340123