Multiprecision Block-Jacobi for iterative triangular solves

Recent research efforts have shown that Jacobi and block-Jacobi relaxation methods can be used as an effective and highly parallel approach for the solution of sparse triangular linear systems arising in the application of ILU-type preconditioners. Simultaneously, a few independent works have focused on designing efficient high performance adaptive-precision block-Jacobi preconditioning (block-diagonal scaling), in the context of the iterative solution of sparse linear systems, on manycore architectures. In this paper, we bridge the gap between relaxation methods based on regular splittings and preconditioners by demonstrating that iterative refinement can be leveraged to construct a relaxation method from the preconditioner. In addition, we exploit this insight to construct a highly-efficient sparse triangular system solver for graphics processors that combines iterative refinement with the block-Jacobi preconditioner available in the Ginkgo library.