Synchronous parallel multisplitting method with convergence acceleration using a local Krylov-based minimization for solving linear systems

Abstract

Computer simulations of physical phenomena, such as heat transfer, often require the solution of linear equations. These linear equations occur in the form Ax $=\mathbf{b}$, where A is a matrix, $\mathbf{b}$ is a vector, and $\mathbf{x}$ is the vector of unknowns. Iterative methods are the most adapted to solve large linear systems because they can be easily parallelized. This paper presents a variant of the multisplitting iterative method with convergence acceleration using the Krylov-based minimization method. This paper particularly focuses on improving the convergence speed of the method with an implementation based on the PETSc (Portable Extensible Toolkit for Scientific Computation) library. This was achieved by reducing the need for synchronization - data exchange - during the minimization process and adding a preconditioner before the multisplitting method. All experiments were performed either over one or two sites of the Grid5000 platform and up to 128 cores were used. The results for solving a 2D Laplacian problem of size 10242 components, show a speed up of up to 23X and 86X when respectively compared to the algorithm in [8] and to the general multisplitting implementation.