Flexible Preconditioning GPBiCG And Parallel BiCG Methods

In the field of scientific and engineering calculating, a lot of large-scale sci-entific computing problems are usually attributed to the solution of large sparse linear systems. The solution of linear systems has become the bottleneck of the whole numerical simulation since it takes a large percentage of the whole CPU time. The key to solve such problems and develop related high performance soft-ware is to design the high efficient linear solver. Currently, there are two methods to reduce the solution time for the given Krylov subspace methods. One is par-allel computing while the other is to make use of preconditioning techniques.Two problems are studied in this work:one is to derive the BiCG algorithm again under the FLAME framework, reorder the inner product so as to reduce the number of global communication points and improve the parallel scalability of the new method; the other is to propose the flexible preconditioned GPBiCG method for nonsymmetric linear systems.First, we introduce the FLAME framework and use it to derive Krylov sub-space methods. In this case, we consider the biconjugate gradient (BiCG)method. The property of vector sequences, the orthogonality of the residuals and the bi-conjugation of the search direction. must be used to derive the algorithm. There might be several different BiCG algorithms but they are mathematically equiv-alent to each other. We choose the one that is suitable for distributed memory parallel computing. We make some theoretical analysis and also conduct some numerical experiments to clarify that the modified algorithm has reduced the time of global communication and improved the parallel scalability.Next, we present a flexible version of GPBiCG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to use any iterative method. For example, the standard GPBiCG algorithm itself can be used as a preconditioner, as can other Krylov subspace methods or splitting methods. Numerical experiments are conducted for flexible GPBiCG for a few matrices including some nonsymmetric matrices. These experiments illustrate the conver-gence and robustness of the flexible iterative method.