For relatively small system sizes, direct methods such as QR decomposition and LU factorization are generally the preferred approaches. The strategies adopted for solving such systems can be broadly classified into two categories: direct methods and iterative methods . Linear systems of equations are encountered in the gamut of applications areas within computational physics, from quantum to continuum to celestial mechanics. We thus find that AAR offers a robust and efficient alternative to current state-of-the-art solvers, with increasing advantages as the number of processors grows. Finally, in massively parallel applications to the Poisson equation, on up to 110,592 processors, we find that AAR shows superior strong and weak scaling to CG, with shorter minimum time to solution. In parallel applications to the Helmholtz and Poisson equations, we find that AAR shows superior strong and weak scaling to GMRES, Bi-CGSTAB, and Conjugate Gradient (CG) methods, using the same preconditioning, with consistently shorter times to solution at larger processor counts.
#Wise memory optimizer alternativeto serial
In serial applications to nonsymmetric systems, we find that AAR is comparably robust to GMRES, using the same preconditioning, while often outperforming it in time to solution and find AAR to be more robust than Bi-CGSTAB for the problems considered. Specifically, we generalize the recently proposed Alternating Anderson–Jacobi (AAJ) method (Pratapa et al., 2016) to include preconditioning, discuss efficient parallel implementation, and provide serial MATLAB and parallel C/C++ implementations. We present the Alternating Anderson–Richardson (AAR) method: an efficient and scalable alternative to preconditioned Krylov solvers for the solution of large, sparse linear systems on high performance computing platforms.