Optimized Schwarz method without overlap for the gravitational potential equation on cluster of graphics processing unit

Abstract

Many engineering and scientific problems need to solve boundary value problems for partial differential equations or systems of them. For most cases, to obtain the solution with desired precision and in acceptable time, the only practical way is to harness the power of parallel processing. In this paper, we present some effective applications of parallel processing based on hybrid CPU/GPU domain decomposition method. Within the family of domain decomposition methods, the so-called optimized Schwarz methods have proven to have good convergence behaviour compared to classical Schwarz methods. The price for this feature is the need to transfer more physical information between subdomain interfaces. For solving large systems of linear algebraic equations resulting from the finite element discretization of the subproblem for each subdomain, Krylov method is often a good choice. Since the overall efficiency of such methods depends on effective calculation of sparse matrix–vector product, approaches that use graphics processing unit (GPU) instead of central processing unit (CPU) for such task look very promising. In this paper, we discuss effective implementation of algebraic operations for iterative Krylov methods on GPU. In order to ensure good performance for the non-overlapping Schwarz method, we propose to use optimized conditions obtained by a stochastic technique based on the covariance matrix adaptation evolution strategy. The performance, robustness, and accuracy of the proposed approach are demonstrated for the solution of the gravitational potential equation for the data acquired from the geological survey of Chicxulub crater.

Keywords:

• optimized Schwarz method
• domain decomposition method
• Krylov methods
• GPU
• CUDA
• gravitational potential equation

1999 AMS Subject Classifications:

• 49M27
• 65F10
• 68W10
• 97R60
• 65Y05

Acknowledgements

The authors are grateful to the CUDA Research Center at Ecole Centrale Paris for the computer facilities and computer time used during this long-term research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge partial financial support from the OpenGPU project (2012-2014), and CRESTA (Collaborative Research into Exascale Systemware, Tools and Applications).