Hybrid methods based on LCG and GMRES

Abstract

Recently, the left conjugate gradient (LCG) method has been developed and applied in many fields, such as computing physics and thermal pollution. As a typical $Galerkin$ method, LCG is easy to get implemented and performs efficiently. LCG iteratively generates approximate solutions with residuals orthogonal to the Krylov subspaces. On the other hand, LCG lacks residual norm-minimization which is different from the famous generalized minimal residual method (GMRES). Although LCG and GMRES are based on different numerical principles, both algorithms exhibit similar behaviors in extensive computational experiments. To probe into the relationship between LCG and GMRES, we first demonstrate the mathematical equivalence of LCG and the full orthogonalization method (FOM). Then, the residual norm connection of LCG and GMRES is established as an easy consequence of relation between FOM and GMRES. Moreover, this paper presents two hybrid algorithms based on the LCG and GMRES. To unify the directness of LCG and optimality of GMRES, LCG residual norm-minimization (LCGR) algorithm is proposed. If orthogonalization is considered in the hybridization, LCG orthogonalization (LCGO) is designed. Numerical experiments show that two hybrid schemes do improve the performance of the standard LCG and GMRES, as well as the comparable methods such as nested GMRES (GMRESR) and generalized conjugate residual orthogonalization (GCRO).