A General Framework for Reproducible Parallel Preconditioned Krylov Methods Using Three BLAS Variants

Krylov subspace methods are crucial techniques for solving linear systems. To effectively tackle large-scale linear systems, parallelism techniques are frequently utilized. However, the use of parallelism can increase the non-associativity of floating-point operations, potentially resulting in non-reproducibility of the computations. In order to address the issue of non-reproducibility in parallel preconditioned Krylov subspace iterative methods, we investigated the sources of irreproducibility and solutions to achieve reproducibility. The main reason for the irreproducibility in iterative methods is attributed to distributed dot products. To tackle this, we investigated three reproducible Basic Linear Algebra Subprograms (BLAS) as replacements for standard distributed dot products. We utilized OzBLAS, ExBLAS, and ReproBLAS to ensure reproducibility in iterative methods. We combined Jacobi preconditioners and tested and verified on three iterative methods: CG, BiCGSTAB and GMRES. Numerical experiments demonstrate that the iterative method based on OzBLAS is reproducible in a multi-threaded environment. The iterative method based on ExBLAS and ReproBLAS achieves reproducibility in a multi-process environment. Our evaluation indicated that ReproBLAS performed well in terms of performance, robustness, and parallelism. Therefore, we employ a reproducible iterative method based on ReproBLAS to solve the large-scale algebraic equations resulting from the discretization of the two-dimensional three-temperature heat conduction equation, which arises from practical application problems in the field of laser fusion radiation hydrodynamics. Numerical experiments demonstrate the reproducible solution of the three-temperature heat conduction algebraic equations.