Abstract
A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.
Highlights
The solution of sparse linear systems is of central importance to scientific and engineering computations and is the most time-consuming part, cf. [3,4,13]
Let us consider the sparse linear system resulting from Finite Difference (FD) discretization of three dimensional boundary value problems, i.e
Giannoutakis / Parallel preconditioned conjugate gradient square method and m, p are the semi-bandwidths, while u is the FD solution at the nodal points and s is a vector, of which the components result from a combination of source terms and imposed boundary conditions
Summary
The solution of sparse linear systems is of central importance to scientific and engineering computations and is the most time-consuming part, cf. [3,4,13]. The solution of sparse linear systems has been obtained by direct or iterative methods, cf [2,3,4,5,6,9,13]. Let us consider the sparse linear system resulting from Finite Difference (FD) discretization of three dimensional boundary value problems, i.e. where A is a sparse, diagonally dominant, positive definite, symmetric (n × n) matrix of the following form:. G.A. Gravvanis and K.M. Giannoutakis / Parallel preconditioned conjugate gradient square method (2). For symmetric positive definite problems, the rate of convergence of the conjugate gradient method depends on the distribution of the eigenvalues of the coefficient matrix. The preconditioned matrix will have a smaller spectral condition number, and the eigenvalues clustered around one, cf [2,6,13]. The performance and applicability of the new algorithmic schemes for solving 3D elliptic and parabolic P.D.E’s is discussed and numerical results are given
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