Low-precision computing is essential for efficiently utilizing memory bandwidth and computing cores. While many mixed-precision algorithms have been developed for iterative sparse linear solvers, effectively leveraging half-precision (fp16) arithmetic remains challenging. This study introduces a novel nested Krylov approach that integrates the FGMRES and Richardson methods in a deeply nested structure, progressively reducing precision from double-precision to fp16 toward the innermost solver. To avoid meaningless computations beyond precision limits, the low-precision inner solvers perform only a few iterations per invocation, while the nested structure ensures their frequent execution. Numerical experiments show that incorporating fp16 into the approach directly enhances solver performance without compromising convergence, achieving speedups of up to 1.65 × and 2.42 × over double-precision and double-single mixed-precision implementations, respectively. Furthermore, the proposed method outperforms conventional mixed-precision Krylov solvers, CG, BiCGStab, and restarted FGMRES, by factors of up to 2.47, 2.74, and 69.10, respectively.

We consider solving a sparse linear system using integer (fixed-point) arithmetic. Integer arithmetic has attracted attention in scientific computing because of its high computational efficiency. Furthermore, considering the current circumstances of hardware development, integer arithmetic is expected to become increasingly important. Nevertheless, integer arithmetic has not been widely used for solving linear systems because it lacks robustness against overflow and underflow, making it hard to solve practical problems. Thus, we propose a new integer-based implementation framework for the Flexible GMRES (FGMRES) method, which enables integer-based solvers to solve linear systems with the same accuracy as conventional floating-point solvers. In addition, we propose an integer-only algebraic multigrid preconditioner. Combining it with the integer-based FGMRES framework, we develop an integer-based solver. Numerical experiments on CPUs showed that the developed integer-based solver has a comparable convergence rate to floating-point solvers. We also found the test cases where the integer-based solver runs faster than the floating-point solvers.

Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time and space complexity of dense matrix vector multiplication, and hence has been applied to numerous practical problems.

Abstract In this paper, we propose a new parallel ordering method to vectorize and parallelize the sparse triangular solver, which is called hierarchical block multi-color ordering. In this method, the parallel forward and backward substitutions can be vectorized while preserving the advantages of block multi-color ordering, that is, fast convergence and fewer thread synchronizations. To evaluate the proposed method in a parallel ICCG (Incomplete Cholesky Conjugate Gradient) solver, numerical tests were conducted using seven test matrices on three types of computational nodes. The numerical results indicate that the proposed method outperforms the conventional block and nodal multi-color ordering methods in 18 out of 21 test cases, which confirms the effectiveness of the method.

Our aim in this work is to improve the performance of the multi-threaded 3D FDTD solver using time-space tiling techniques that enable tile-level parallelization. The implementation of tile-level parallelization that we have used is based on the so-called diamond tiling technique. In this paper, we present a systematic manner for introducing time-space tiling techniques into the 3D FDTD solver and compare four different approaches. Our performance evaluation on a state-of-the-art multi-core processor demonstrated the effectiveness of the time-space tiling techniques with tile-level parallelism for the 3D FDTD method. For the problem with 2003 grid points, our implementation with two-dimensional tile-level parallelism achieved a speedup of 1.88 times over the naive implementation, while for the problem of 3003 grid points, our implementation with one-dimensional tile-level parallelism showed a speedup of 2.22 times. Both results are better than the speedup obtained from an implementation with intra-tile parallelization presented in a previous work.

This paper covers the multi-threaded parallel processing of a sparse triangular solver for a linear system with a sparse coefficient matrix, focusing on its application to a parallel ICCG solver. We propose algebraic block multi-color ordering, which is an enhanced version of block multi-color ordering for general unstructured analysis. We present blocking and coloring strategies that achieve a high cache hit ratio and fast convergence. Five numerical tests on a shared memory parallel computer verify that the computation time of the proposed method is between 1.7 and 2.6 times faster than that of the conventional multi-color ordering method.