An Efficient Algorithm For Simulating Fracture Using Large Fuse Networks

The high computational cost involved in modeling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve {\it a new large set of linear equations} every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algorithm based on the Sherman-Morrison-Woodbury formula for the simulation of progressive fracture in disordered quasi-brittle materials using discrete lattice networks. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to the same order as that of a simple {\it backsolve} (forward elimination and backward substitution) {\it using the already LU factored matrix}. That is, the computational cost is $O(nnz({\bf L}))$, where $nnz({\bf L})$ denotes the number of non-zeros of the Cholesky factorization ${\bf L}$ of the stiffness matrix ${\bf A}$. This algorithm using the direct sparse solver is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) iterative solvers, and eliminates the {\it critical slowing down} associated with the iterative solvers that is especially severe close to the critical points. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.Comment: 15 pages including 1 figure. On page pp11407 of the original paper (J. Phys. A: Math. Gen. 36 (2003) 11403-11412), Eqs. 11 and 12 were misprinted that went unnoticed during the proof reading stag

Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections

This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective in this context; nevertheless, their computation may be very expensive for large-scale problems, and resorting to approximations of them may be convenient. Here we propose a procedure for building inexact constraint preconditioners by updating a "seed" constraint preconditioner computed for a KKT matrix at a previous interior point iteration. These updates are obtained through low-rank corrections of the Schur complement of the (1,1) block of the seed preconditioner. The updated preconditioners are analyzed both theoretically and computationally. The results obtained show that our updating procedure, coupled with an adaptive strategy for determining whether to reinitialize or update the preconditioner, can enhance the performance of interior point methods on large problems.Comment: 22 page

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

Sparse symmetric indefinite linear systems of equations arise in numerous practical applications. In many situations, an iterative method is the method of choice but a preconditioner is normally required for it to be effective. In this paper, the focus is on a class of incomplete factorization algorithms that can be used to compute preconditioners for symmetric indefinite systems. A limited memory approach is employed that incorporates a number of new ideas with the goal of improving the stability, robustness and efficiency of the preconditioner. These include the monitoring of stability as the factorization proceeds and the incorporation of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real-world applications demonstrate the effectiveness of our approach

Efficient preconditioning for sequences of parametric complex symmetric linear systems

Solution of sequences of complex symmetric linear systems of the form Ajxj = bj, j = 0,..., s, Aj = A + αjEj, A Hermitian, E0, ..., E a complex diagonal matrices and α0, ..., αa scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi-Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then Aj is complex symmetric. The case A Hermitian positive semideflnite, Re(αj) ≥ 0 and such that the diagonal entries of E j, j = 0,..., s have nonnegative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A-1 are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches

On Updating Preconditioners for the Iterative Solution of Linear Systems

El tema principal de esta tesis es el desarrollo de técnicas de actualización de precondicionadores para resolver sistemas lineales de gran tamaño y dispersos Ax=b mediante el uso de métodos iterativos de Krylov. Se consideran dos tipos interesantes de problemas. En el primero se estudia la solución iterativa de sistemas lineales no singulares y antisimétricos, donde la matriz de coeficientes A tiene parte antisimétrica de rango bajo o puede aproximarse bien con una matriz antisimétrica de rango bajo. Sistemas como este surgen de la discretización de PDEs con ciertas condiciones de frontera de Neumann, la discretización de ecuaciones integrales y métodos de puntos interiores, por ejemplo, el problema de Bratu y la ecuación integral de Love. El segundo tipo de sistemas lineales considerados son problemas de mínimos cuadrados (LS) que se resuelven considerando la solución del sistema equivalente de ecuaciones normales. Concretamente, consideramos la solución de problemas LS modificados y de rango incompleto. Por problema LS modificado se entiende que el conjunto de ecuaciones lineales se actualiza con alguna información nueva, se agrega una nueva variable o, por el contrario, se elimina alguna información o variable del conjunto. En los problemas LS de rango deficiente, la matriz de coeficientes no tiene rango completo, lo que dificulta el cálculo de una factorización incompleta de las ecuaciones normales. Los problemas LS surgen en muchas aplicaciones a gran escala de la ciencia y la ingeniería como, por ejemplo, redes neuronales, programación lineal, sismología de exploración o procesamiento de imágenes. Los precondicionadores directos para métodos iterativos usados habitualmente son las factorizaciones incompletas LU, o de Cholesky cuando la matriz es simétrica definida positiva. La principal contribución de esta tesis es el desarrollo de técnicas de actualización de precondicionadores. Básicamente, el método consiste en el cálculo de una descomposición incompleta para un sistema lineal aumentado equivalente, que se utiliza como precondicionador para el problema original. El estudio teórico y los resultados numéricos presentados en esta tesis muestran el rendimiento de la técnica de precondicionamiento propuesta y su competitividad en comparación con otros métodos disponibles en la literatura para calcular precondicionadores para los problemas estudiados.The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax=b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of non-singular, non-symmetric linear systems where the coefficient matrix A has a skew-symmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that 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. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied.El tema principal d'esta tesi és actualitzar precondicionadors per a resoldre sistemes lineals grans i buits Ax=b per mitjà de l'ús de mètodes iteratius de Krylov. Es consideren dos tipus interessants de problemes. En el primer s'estudia la solució iterativa de sistemes lineals no singulars i antisimètrics, on la matriu de coeficients A té una part antisimètrica de baix rang, o bé pot aproximar-se amb una matriu antisimètrica de baix rang. Sistemes com este sorgixen de la discretització de PDEs amb certes condicions de frontera de Neumann, la discretització d'equacions integrals i mètodes de punts interiors, per exemple, el problema de Bratu i l'equació integral de Love. El segon tipus de sistemes lineals considerats, són problemes de mínims quadrats (LS) que es resolen considerant la solució del sistema equivalent d'equacions normals. Concretament, considerem la solució de problemes de LS modificats i de rang incomplet. Per problema LS modificat, s'entén que el conjunt d'equacions lineals s'actualitza amb alguna informació nova, s'agrega una nova variable o, al contrari, s'elimina alguna informació o variable del conjunt. En els problemes LS de rang deficient, la matriu de coeficients no té rang complet, la qual cosa dificultata el calcul d'una factorització incompleta de les equacions normals. Els problemes LS sorgixen en moltes aplicacions a gran escala de la ciència i l'enginyeria com, per exemple, xarxes neuronals, programació lineal, sismologia d'exploració o processament d'imatges. Els precondicionadors directes per a mètodes iteratius utilitzats més a sovint són les factoritzacions incompletes tipus ILU, o la factorització incompleta de Cholesky quan la matriu és simètrica definida positiva. La principal contribució d'esta tesi és el desenvolupament de tècniques d'actualització de precondicionadors. Bàsicament, el mètode consistix en el càlcul d'una descomposició incompleta per a un sistema lineal augmentat equivalent, que s'utilitza com a precondicionador pel problema original. L'estudi teòric i els resultats numèrics presentats en esta tesi mostren el rendiment de la tècnica de precondicionament proposta i la seua competitivitat en comparació amb altres mètodes disponibles en la literatura per a calcular precondicionadors per als problemes considerats.Guerrero Flores, DJ. (2018). On Updating Preconditioners for the Iterative Solution of Linear Systems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/10492

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