Evaluating Performance of Mixed Precision Linear Solvers with Iterative Refinement


  • Boris I. Krasnopolsky Institute of Mechanics, Lomonosov Moscow State University
  • Alexey V. Medvedev Institute of Mechanics, Lomonosov Moscow State University




systems of linear algebraic equations, elliptic equations, algebraic multigrid methods, iterative refinement, mixed precision calculations


The solution of systems of linear algebraic equations is among the time-consuming problems when performing the numerical simulations. One of the possible ways of improving the corresponding solver performance is the use of reduced precision calculations, which, however, may affect the accuracy of the obtained solution. The current paper analyzes the potential of using the mixed precision iterative refinement procedure to solve the systems of equations occurring as a result of the discretization of elliptic differential equations. The paper compares several inner solver stopping criteria and proposes the one allowing to eliminate the residual deviation and minimize the number of extra iterations. The presented numerical calculation results demonstrate the efficiency of the adopted algorithm and show about the decrease in the solution time by a factor of 1.5 for the turbulent flow simulations when using the iterative refinement procedure to solve the corresponding pressure Poisson equation.


HYPRE: High performance preconditioners (2020). http://www.llnl.gov/CASC/hypre/, accessed: 2020-12-27

Abdelfattah, A., Anzt, H., Boman, E.G., et al.: A survey of numerical linear algebra methods utilizing mixed-precision arithmetic. The International Journal of High Performance Computing Applications 35(4), 344–369 (2021). https://doi.org/10.1177/10943420211003313

Anzt, H., Flegar, G., Novaković, V., et al.: Residual replacement in mixed-precision iterative refinement for sparse linear systems. In: Yokota, R., Weiland, M., Shalf, J., Alam, S. (eds.) High Performance Computing. pp. 554–561. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-02465-9_39

Baboulin, M., Buttari, A., Dongarra, J., et al.: Accelerating scientific computations with mixed precision algorithms. Computer Physics Communications 180(12), 2526–2533 (2009). https://doi.org/10.1016/j.cpc.2008.11.005

Clark, M., Babich, R., Barros, K., et al.: Solving lattice QCD systems of equations using mixed precision solvers on GPUs. Computer Physics Communications 181(9), 1517–1528 (2010). https://doi.org/10.1016/j.cpc.2010.05.002

Krasnopolsky, B.: An approach for accelerating incompressible turbulent flow simulations based on simultaneous modelling of multiple ensembles. Computer Physics Communications 229, 8–19 (2018). https://doi.org/10.1016/j.cpc.2018.03.023

Krasnopolsky, B.: Revisiting performance of BiCGStab methods for solving systems with multiple right-hand sides. Computers & Mathematics with Applications 79(9), 2574–2597 (2020). https://doi.org/10.1016/j.camwa.2019.11.025

Krasnopolsky, B., Medvedev, A.: Acceleration of large scale OpenFOAM simulations on distributed systems with multicore CPUs and GPUs. In: Parallel Computing: On the Road to Exascale. Advances in Parallel Computing, vol. 27, pp. 93–102 (2016). https://doi.org/10.3233/978-1-61499-621-7-93

Krasnopolsky, B., Medvedev, A.: XAMG: A library for solving linear systems with multiple right-hand side vectors. SoftwareX 14 (2021). https://doi.org/10.1016/j.softx.2021.100695

Krasnopolsky, B., Medvedev, A.: XAMG: Source code repository (2021). https://gitlab.com/xamg/xamg, accessed: 2021-03-31

Nikitin, N.: Finite-difference method for incompressible Navier-Stokes equations in arbitrary orthogonal curvilinear coordinates. Journal of Computational Physics 217, 759–781 (2006). https://doi.org/10.1016/j.jcp.2006.01.036

Saad, Y.: Iterative methods for sparse linear systems, 2nd edition. SIAM, Philadelpha, PA (2003)

Sleijpen, G.L.G., van der Vorst, H.A.: Reliable updated residuals in hybrid Bi-CG methods. Computing 56, 141–163 (1996). https://doi.org/10.1007/BF02309342

Sumiyoshi, Y., Fujii, A., Nukada, A., Tanaka, T.: Mixed-precision AMG method for many core accelerators. In: 21st European MPI Users’ Group Meeting, EuroMPI/ASIA ’14. pp. 127–132. ACM, New York, NY, USA (2014). https://doi.org/10.1145/2642769.2642794

van der Vorst, H.A.: BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). https://doi.org/10.1137/0913035

van der Vorst, H.A., Ye, Q.: Residual replacement strategies for Krylov subspace iterative methods for the convergence of true residuals. SIAM Journal on Scientific Computing 22(3), 835–852 (2000). https://doi.org/10.1137/S1064827599353865

Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Englewood Cliffs, Prentice-Hall, New Jersey (1963)

Williams, S., Waterman, A., Patterson, D.: Roofline: An insightful visual performance model for multicore architectures. Communications of the ACM 52(4), 65–76 (2009). https://doi.org/10.1145/1498765.1498785

Zarechnev, S., Krasnopolsky, B.: Improving performance of linear solvers by using indices compression for storing sparse matrices. In: Voevodin, V. (ed.) Russian Supercomputing Days: Proceedings of the International Conference. pp. 119–120. MAKS Press, Moscow (2020). https://doi.org/10.29003/m1406.RussianSCDays-2020




How to Cite

Krasnopolsky, B. I., & Medvedev, A. V. (2021). Evaluating Performance of Mixed Precision Linear Solvers with Iterative Refinement. Supercomputing Frontiers and Innovations, 8(3), 4–16. https://doi.org/10.14529/jsfi210301