Reproducibility Strategies for Parallel Preconditioned Conjugate Gradient

Roman Iakymchuk (1, 2) , Maria Barreda (3) , Matthias Wiesenberger (4) , José I Aliaga (3) , Enrique S Quintana-Ortí (5)

1 Fraunhofer ITWM - Fraunhofer Institute of Industrial Mathematics
2 PEQUAN - Performance et Qualité des Algorithmes Numériques
3 Universidad Jaime I de Catellon
4 DTU - Danmarks Tekniske Universitet = Technical University of Denmark
5 University of Valencia,Valencia

The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29 % (ExBLAS) and 4 % (lightweight) on 768 processes.

Mots clés: Reproducibility; accuracy; floating-point expansion; long accumulator; fused multiply-add; preconditioned Conjugate Gradient; High-Performance Computing

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Arithmétique des ordinateurs; Optimisation et contrôle [math.OC]; Logiciel mathématique [cs.MS]

Roman Iakymchuk, Maria Barreda, Matthias Wiesenberger, José I Aliaga, Enrique S Quintana-Ortí. Reproducibility Strategies for Parallel Preconditioned Conjugate Gradient. 2019. ⟨hal-02391618v1⟩