Parallel Algorithm Of Krylov Subspace Method For Solving Large Sparse Linear Systems

At present, a hotspot problem to be researched in the large-scale science com-putation is to seek high performance parallel methods for solving large sparse linear systems. Krylov subspace algorithms is one of high performance iterative method-s for solving large sparse linear systems. On the distributed parallel computation environment, global communications which must be conducted by inner product-s calculations became a bottleneck of Krylov subspace algorithms can be highly efficient parallel computation.In order to reduce the number of global communication brought by the in-ner product calculations and improve the parallel performance. Two kinds of Bi-Conjugate residual (BiCR) product-type algorithms which called Stabilized CRS method(SCRS) and Based on Associate Residual BiCRSTAB2 Method (BiCRSTAB-2AR) was researched. These methods are obtained by reconstructing on the CRS algorithm and BiCRSTAB2 algorithm, respectively. On the distributed parallel environments, we get the improved algorithm of parallclization to solve large un-symmetric sparse linear systems by changing the computing sequence of SCRS method and BiCRSTAB2AR method, respectively. Then we obtain the improved Bi-CRSTAB2AR(hereinafter referred to as IBiCRSTAB2AR) and improved Stabilized Conjugate Residual Squared (hereinafter referred to as ISCRS). We reduce three global communication points to one by reconstructing and changing the comput-ing sequence of BiCRSTAB2AR and SCRS methods and replace the discrete inner products by sequence inner product, respectively. Such that all inner products and matrix-vector multiplications can compute at the same time. which eliminate the data dependency between inner products and matrix-vector multiplications and im-prove the efficient of parallel computation.Compared to the improvement of global communication performance,the incre-ment of computation work in the improved methods can be neglected. Performance analysis shows that IBiCRSTAB2AR and ISCRS methods do better in speedup-s, parallelism and scalability than BiCRSTAB2 and SCRS methods, Respectively. Compare with BiCRSTAB2AR and SCRS method the performance improvement ra-tio of IBiCRSTAB-2AR and ISCRS methods have obtained 66.7% and 75%. Respcc- tively. Isoefficiency function analysis also show that IBiCRSTAB2AR and ISCRS methods have better scalability than BiCRSTAB2 and SCRS methods. Respective-ly. Furthermore, According to numerical experiment, we compare BiCRSTAB2AR with BiCRSTA-B2 and SCRS with CRS, CGS and BiCGSTAB methods. The numerical experiment results show that BiCRSTAB2AR. and SCRS have faster con-vergence speed and better convergence effect than BiCRSTAB2 and CRS,CGS and BiCGSTAB2 methods. Respectively. The parallel numerical experiment results show IBiCRSTAB2AR and ISCRS methods do better in specdups, parallelism and scal-ability than BiCRSTAB2 and SCRS methods. Respectively. The results are agree well with the theoretical analysis.