Fast solvers for tridiagonal Toeplitz linear systems

Let A be a tridiagonal Toeplitz matrix denoted by A=Tritoep(β,α,γ). The matrix A is said to be: strictly diagonally dominant if |α|>|β|+|γ|, weakly diagonally dominant if |α|≥|β|+|γ|, subdiagonally dominant if |β|≥|α|+|γ|, and superdiagonally dominant if |γ|≥|α|+|β|. In this paper, we consider the solution of a tridiagonal Toeplitz system Ax=b, where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block 2×2 matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithmsNational Natural Science Foundation of China under Grant no. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, the research innovation program of Changsha University of Science and Technology for postgraduate students under Grant (CX2019SS34), and the Portuguese Funds through FCT-Fundação para a Ciência, within the Project UIDB/00013/2020 and UIDP/00013/202

A new approximate matrix factorization for implicit time integration in air pollution modeling

Implicit time stepping typically requires solution of one or several linear systems with a matrix I−τJ per time step where J is the Jacobian matrix. If solution of these systems is expensive, replacing I−τJ with its approximate matrix factorization (AMF) (I−τR)(I−τV), R+V=J, often leads to a good compromise between stability and accuracy of the time integration on the one hand and its efficiency on the other hand. For example, in air pollution modeling, AMF has been successfully used in the framework of Rosenbrock schemes. The standard AMF gives an approximation to I−τJ with the error τ2RV, which can be significant in norm. In this paper we propose a new AMF. In assumption that −V is an M-matrix, the error of the new AMF can be shown to have an upper bound τ||R||, while still being asymptotically $O(\tau^2)$. This new AMF, called AMF+, is equal in costs to standard AMF and, as both analysis and numerical experiments reveal, provides a better accuracy. We also report on our experience with another, cheaper AMF and with AMF-preconditioned GMRES

MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers

[EN] We consider the computation of a few eigenpairs of a generalized eigenvalue problem Ax = lambda Bx with block-tridiagonal matrices, not necessarily symmetric, in the context of Krylov methods. In this kind of computation, it is often necessary to solve a linear system of equations in each iteration of the eigensolver, for instance when B is not the identity matrix or when computing interior eigenvalues with the shift-and-invert spectral transformation. In this work, we aim to compare different direct linear solvers that can exploit the block-tridiagonal structure. Block cyclic reduction and the Spike algorithm are considered. A parallel implementation based on MPI is developed in the context of the SLEPc library. The use of GPU devices to accelerate local computations shows to be competitive for large block sizes.This work was supported by Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. Alejandro Lamas Davina was supported by the Spanish Ministry of Education, Culture and Sport through a grant with reference FPU13-06655.Lamas Daviña, A.; Roman, JE. (2018). MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers. Parallel Computing. 74:118-135. https://doi.org/10.1016/j.parco.2017.11.006S1181357

Time-asymptotic solutions of the Navier-Stokes equation for free shear flows using an alternating-direction implicit method

An uncoupled time asymptotic alternating direction implicit method for solving the Navier-Stokes equations was tested on two laminar parallel mixing flows. A constant total temperature was assumed in order to eliminate the need to solve the full energy equation; consequently, static temperature was evaluated by using algebraic relationship. For the mixing of two supersonic streams at a Reynolds number of 1,000, convergent solutions were obtained for a time step 5 times the maximum allowable size for an explicit method. The solution diverged for a time step 10 times the explicit limit. Improved convergence was obtained when upwind differencing was used for convective terms. Larger time steps were not possible with either upwind differencing or the diagonally dominant scheme. Artificial viscosity was added to the continuity equation in order to eliminate divergence for the mixing of a subsonic stream with a supersonic stream at a Reynolds number of 1,000

A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.The authors would like to thank the supports of the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

A communication-less parallel algorithm for tridiagonal Toeplitz systems

AbstractDiagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations

The numerical solution of sparse matrix equations by fast methods and associated computational techniques

The numerical solution of sparse matrix equations by fast methods and associated computational technique