Variable Block CG Algorithms for Solving Large Sparse Symmetric Positive Definite Linear Systems on Parallel Computers, I: General Iterative Scheme

• Authors: A. A. Nikishin;A. Yu. Yeremin
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 1995

This paper considers a new approach to construction of efficient parallel solution methods of large sparse SPD linear systems. This approach is based on the so-called variable block CG method, a generalization of the standard block CG method, where it is possible to reduce the iteration block size adaptively (at any iteration) by construction of an $A$-orthogonal projector without restarts and without algebraic convergence of residual vectors. It enables one to find the constructive compromise between the required resource of parallelism, the resulting convergence rate, and the serial arithmetic costs of one block iteration to minimize the total parallel solution time. The orthogonality and minimization properties of the variable CG method are established and the theoretical analysis of the convergence rate is presented. The results of numerical experiments with large FE systems coming from $h$- and $p$-approximations of three-dimensional equilibrium equations for linear elastic orthotropic materials show that the convergence rate of the variable block CG method is comparable to that of the standard block CG method even when utilizing a large block size, while the total serial arithmetic costs of the variable block CG method are comparable or even smaller than those of the corresponding point CG method.