A New Organization of Sparse Gauss Elimination for Solving PDEs

A new Gauss elimination algorithm is presented for solving sparse, nonsymmetric linear systems arising from partial differential equation (PDE) problems. It is particularly suitable for use on distributed memory message passing (DMMP) multiprocessor computers and it is presented and analyzed in this context. The objective of the algorithm is to exploit the sparsity (Le., reducing both computational and memory requirements) and sharply reduce the data structure manipulation overhead of standard sparse matrix algorithms. The algorithm is based on the nested dissection approach, which starts with a large set of very sparse, completely independent subsystems and progresses in stages to a single. nearly dense system at the last stage. The computational efforts of each stage are roughly equal (almost exactly equal for model problems), yet the data structures appropriate for the first and last stages are quite different. Thus we use different types of data structures and algorithm components at different stages of the solution. .. Supported by NSF grant CCR-8619817. ..... Supported in part by AFOSR grant 88-0243 and the Strategic Defense Initiative through ARO conlnlCl DAAG03-86-K-OI06. L INTRODUCTION Solving linear PDEs naturally generates large, sparse linear systems of equations to solve. These systems have structures which are not exploited by general purpose sparse matrix algorithms and we present a new organization of sparse Gauss elimination tailored to e;qIloit-these structures. We stan with some general background comments on sparse rnattix melhods for PDE problems. The linear systems are almost always created by the PDE solving system with the equations (manix rows) distributed among the processors. One has the freedom to choose the assignment of rows to processors, but one cannot choose to have columns assigned to processors wiilieut the high expense of performing a matrix transpose (or equivalent) on a DMMP machine. The linear systems are conunonly non-symmetric so that a solver of symmetric systems is applicable to a limited class of PDE problems and/or discretization methods. The lack of symmetry requires two data srructures on a DMMP machine, one each for the row and column sparsities. One can combine these in clever ways, but it is prohibitedly expensive to repeatedly obtain column sparsity information from the row sparsity structure. Note that similar sparsity patterns occur in other imponant applications (e.g., least squares problems [Rice, 1984]) and the considerations studied here for PDE problems are also relevant there. Symbolic factorization is not well suited for non-symmetric systems as. so far, the techniques generate much too large a data structure. Merging the symbolic factorization with the numerical computation is not inherently more expensive than doing these separately and, for non-symmetric systems, it allows the final data structure to be just the required size for the system. This dynamic data structure creation is used in the parallel sparse algorithm of [Mu and Rice, 1990.]. If nested dissection is performed geometrically rather than algebraically (as is natural in PDE problems), then a great deal of matrix structure is known "a priori" from the geometric structure and need not be explicitly expressed in the sparse matrix data structure. This idea is exploited in parallel sparse [Mu and Rice, 1989a] to some extent. There are two general row oriented organizations of Gauss elimination focusing on what happens when one eliminates an unknown from a pivot equation. Theyare/an·in and fan~our schemes. Another imponant organization is the mulrlfrontal scheme which is unknown oriented (using both rows and columns associated with unknowns). The fan-in organization processes the LV factorization row by row as follows.