A New Approach to Construction of Efficient Iterative Schemes for Massively Parallel Applications: Variable Block CG and BiCG Methods and Variable Block Arnoldi Procedure

The iterative solution of large sparse symmetric and unsymmetric linear systems comprises the most time consuming stage when solving many computationally intensive 3D industrial problems. The standard approach to construction of efficient parallel solution methods consists in designing first of all efficient parallel preconditioners. Unfortunately, it is very difficult to find a constructive compromise between parallel properties, preconditioning quality and arithmetic costs for constructing the preconditioner. Another way to enhance the parallelism of the iterative solution methods is related to consideration of block iterative schemes like block CG method introduced by O`Leary and Underwood. Unfortunately, there does not exist any ease approach to choose the optimal block size of such schemes which is optimal with respect to using parallelism, the resulting convergence rate, and the arithmetic costs of one block iteration. We introduce the so called variable Block CG methods and the Variable Block Arnoldi procedure where we can adaptively reduce the current block size without any restarts. This makes it possible to reduce substantially the block size while preserving at the same time the high convergence rate of the block iterative scheme with the original large constant block size.