Multiple-rank modifications of a sparse Cholesky factorization.

Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LDL T or LL T , we develop sparse techniques for updating the factorization after either adding a collection of columns to A or deleting a collection of columns from A. Our techniques are based on an analysis and manipulation of the underlying graph structure, using the framework developed in an earlier paper on rank-1 modifications [T. A. Davis and W. W. Hager, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606-627]. Computationally, the multiple-rank update has better memory traffic and executes much faster than an equivalent series of rank-1 updates since the multiple-rank update makes one pass through L computing the new entries, while a series of rank-1 updates requires multiple passes through L

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 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

A sparse decomposition of low rank symmetric positive semi-definite matrices

Suppose that $A \in \mathbb{R}^{N \times N}$ is symmetric positive semidefinite with rank $K \le N$. Our goal is to decompose $A$ into $K$ rank-one matrices $\sum_{k=1}^K g_k g_k^T$ where the modes $\{g_{k}\}_{k=1}^K$ are required to be as sparse as possible. In contrast to eigen decomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where $A$ is the covariance function and is intractable to solve in general. In this paper, we partition the indices from 1 to $N$ into several patches and propose to quantify the sparseness of a vector by the number of patches on which it is nonzero, which is called patch-wise sparseness. Our aim is to find the decomposition which minimizes the total patch-wise sparseness of the decomposed modes. We propose a domain-decomposition type method, called intrinsic sparse mode decomposition (ISMD), which follows the "local-modes-construction + patching-up" procedure. The key step in the ISMD is to construct local pieces of the intrinsic sparse modes by a joint diagonalization problem. Thereafter a pivoted Cholesky decomposition is utilized to glue these local pieces together. Optimal sparse decomposition, consistency with different domain decomposition and robustness to small perturbation are proved under the so called regular-sparse assumption (see Definition 1.2). We provide simulation results to show the efficiency and robustness of the ISMD. We also compare the ISMD to other existing methods, e.g., eigen decomposition, pivoted Cholesky decomposition and convex relaxation of sparse principal component analysis [25] and [40]

An efficient null space inexact Newton method for hydraulic simulation of water distribution networks

Null space Newton algorithms are efficient in solving the nonlinear equations arising in hydraulic analysis of water distribution networks. In this article, we propose and evaluate an inexact Newton method that relies on partial updates of the network pipes' frictional headloss computations to solve the linear systems more efficiently and with numerical reliability. The update set parameters are studied to propose appropriate values. Different null space basis generation schemes are analysed to choose methods for sparse and well-conditioned null space bases resulting in a smaller update set. The Newton steps are computed in the null space by solving sparse, symmetric positive definite systems with sparse Cholesky factorizations. By using the constant structure of the null space system matrices, a single symbolic factorization in the Cholesky decomposition is used multiple times, reducing the computational cost of linear solves. The algorithms and analyses are validated using medium to large-scale water network models.Comment: 15 pages, 9 figures, Preprint extension of Abraham and Stoianov, 2015 (https://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0001089), September 2015. Includes extended exposition, additional case studies and new simulations and analysi