Recent advances in direct methods for solving unsymmetric sparse systems of linear equations

Abstract

During the past few years, algorithmic improvements alone have reduced the time required for the direct solution of unsymmetric sparse systems of linear equations by almost an order of magnitude. This paper compares the performance of some well-known software packages for solving general sparse systems. In particular, it demonstrates the consistently high level of performance achieved by WSMP-the most recent of such solvers. It compares the various algorithmic components of these solvers and discusses their impact on solver performance. Our experiments show that the algorithmic choices made in WSMP enable it to run more than twice as fast as the best among similar solvers and that WSMP can factor some of the largest sparse matrices available from real applications in only a few seconds on a 4-CPU workstation. Thus, the combination of advances in hardware and algorithms makes it possible to solve those general sparse linear systems quickly and easily that might have been considered too large until recently.

Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

“…A detailed comparison can be found in [12,13]. A "fail" indicates that the solver ran out of memory, e.g.…”

“…All experiments reported in this paper with PARDISO were conducted with a nested dissection algorithm [14]. Like other modern sparse factorization packages [2,5,7,8,13,16], PARDISO takes advantage of the supernode technology -adjacent groups of rows and columns with the same structure in the factors L and U are treated as one supernode. An interchange among these rows of a supernode has no effect on the overall fill-in and this is the mechanism for finding a suitable pivot Table 1.…”

“…The best time is shown in boldface, the second best time is underlined, and the best operation count is indicated by . The last row shows the approximate smallest relative pivot threshold that yielded a residual norm close to machine precision after iterative refinement for each package ( [12,13]). …”

Improved hybrid elements for structural analysis

“…The displacement-based and hybrid n-noded (where n is either 8 or 27) brick elements are denoted by Bn and Sn respectively, the displacement-based and hybrid n-noded wedge elements (where n is either 6 or 18) are denoted by Bn and Wn respectively, while the displacement-based and hybrid n-noded (where n is either 4 or 9) axisymmetric elements are denoted by Bn and An respectively. The WSMP sparse matrix solver [Gupta 2000;2002] is used. Full integration is used to construct the element stiffness matrices in all cases.…”

An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

“…Uniform meshes and time steps are used in all the examples. Full integration is used to evaluate all the integrals arising in the formulation of the elements, and the WSMP sparse matrix solver [Gupta 2000;2002] is used to solve the system of equations. A Saint-Venant-Kirchhoff material model is used in all the examples unless otherwise stated.…”