Efficient Acceleration Framework for Complex‐Valued Linear Systems: Alternating Anderson Accelerating Preconditioned Richardson Approach

Abstract

ABSTRACTInspired by the efficiency of Anderson acceleration (AA) and the fact that its un‐truncated version is essentially equivalent to the generalized minimal residual method (GMRES) for solving linear systems, in this study, the preconditioned alternating Anderson acceleration Richardson (PA3R) approach is proposed as an acceleration framework, and its relationship to AA and GMRES is examined from a spectrum perspective. Next, the convergence condition of the PA3R approach is demonstrated to be (where is the splitting of system matrix ) for a general system matrix, which always holds for the mature splitting iteration methods. This new approach is applied to four well‐established splitting methods to solve complex symmetric linear systems. Furthermore, the optimal relaxation parameters are investigated, showing that the eigenvalue distribution region of the iteration matrices of the resulting methods are approximately half of that of the original methods. The numerical results indicate that for examples with dimensions exceeding 4 million, the use of the PA3R approach results in a significant efficiency improvement especially in terms of CPU time compared with the non‐PA3R version.