Reproducibility of Parallel Preconditioned Conjugate Gradient in Hybrid Programming Environments

Roman Iakymchuk (1, 2) , Maria Barreda (3) , Stef Graillat (1) , José I Aliaga (3) , Enrique S Quintana-Ortí (4)

1 PEQUAN - Performance et Qualité des Algorithmes Numériques
2 Fraunhofer ITWM - Fraunhofer Institute of Industrial Mathematics
3 Universidad Jaime I de Catellon
4 UPV - Universitat Politècnica de València = Universitad Politecnica de Valencia = Polytechnic University of Valencia

The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the other relies upon floating-point expansions and, hence, expands the intermediate precision. Instead of converting the entire solver into its ExBLAS-related implementation, we identify those parts that violate reproducibility/non-associativity, secure them, and combine this with the sequential executions. These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these approaches on two modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of less than 37.7 % on 768 cores.

Mots clés: reproducibility; accuracy; OpenMP tasks; MPI; Preconditioned Conjugate Gradient; fused multiply-add; long accumulator; floating-point expansion

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Arithmétique des ordinateurs; Analyse numérique [math.NA]; Logiciel mathématique [cs.MS]

Roman Iakymchuk, Maria Barreda, Stef Graillat, José I Aliaga, Enrique S Quintana-Ortí. Reproducibility of Parallel Preconditioned Conjugate Gradient in Hybrid Programming Environments. 2020. ⟨hal-02427795v2⟩