XAMG: A library for solving linear systems with multiple right-hand side vectors

Abstract

This paper presents the XAMG library for solving large sparse systems of linear algebraic equations with multiple right-hand side vectors. The library specializes, but is not limited, to the solution of linear systems obtained from the discretization of elliptic differential equations. A corresponding set of numerical methods includes Krylov subspace, algebraic multigrid, Jacobi, Gauss–Seidel, and Chebyshev iterative methods. The parallelization is implemented with MPI+POSIX shared memory hybrid programming model, which introduces a three-level hierarchical decomposition using the corresponding per-level synchronization and communication primitives. The code contains a number of optimizations, including the multilevel data segmentation, compression of indices, mixed-precision floating-point calculations, vector status flags, and others. The XAMG library uses the program code of the well-known hypre library to construct the multigrid matrix hierarchy. The XAMG’s own implementation for the solve phase of the iterative methods provides up to a twofold speedup compared to hypre for the tests performed. Additionally, XAMG provides extended functionality to solve systems with multiple right-hand side vectors.

Keywords

• Systems of linear algebraic equations
• Krylov subspace iterative methods
• Algebraic multigrid method
• Multiple right-hand sides
• Hybrid programming model
• MPI+POSIX shared memory