Fast and Accurate Solution of Integral Formulations of Large MQS Problems Based on Hybrid OpenMP-MPI Parallelization  
[PERSON]  
1 Department of Electrical and Information Engineering, University of Cassino and Southern Lazio, 03043 Cassino, Italy; [EMAIL_ADDRESS]
2C.R.E.A.T.E. Consortium, 80125 Naples, Italy; [EMAIL_ADDRESS]
3 Fusion for Energy (F4E), 08019 Barcelona, Spain; [EMAIL_ADDRESS]
4 Department of Engineering, University of Campania \"[PERSON]\", 81031 Aversa, Italy 4  
S.V.V. The final goal of this work is the optimization of the solution of integral formulations of eddy current problems solved with direct methods, in applications where long transients must be studied, and where an effective pre-conditioning is not possible. The applications targeted here come from the world of the nuclear fusion machines, and specifically refer to the evaluation of the current density induced in the conducting structures as a consequence of an electromagnetic transient associated with plasma disruptive events. The proposed optimization is based on the parallelization of the tasks associated with the numerical implementation of the integral formulation, based on a hybrid OpenMP-MPI approach. The classical approach in parallel computing was based on the assumption that the systems were homogeneous, with each single node sharing the same characteristics: in this case, the computation time required by each node would be the same, if the same tasks were assigned to each single node. The present approach is based on the use of computational resources with high performance, made by cluster multicore systems [4], where variability and heterogeneity between the cores and between nodes introduces major issues, such as load imbalance, that strongly affect the effectiveness of parallel computation [5]. Optimizing the resources is of a primary importance for an efficient use of supercomputers, that are usually shared among several users. Traditional load-balancing techniques, such as those based on global re-partitioning of the threads, require a priori estimations on the time needed for completing each single task or, alternatively, require a dynamical reallocation of the loads based on a real-time check of the status of the nodes [6]. More recently, this problem has been addressed by using hybrid parallel programming paradigms, like those that adopt both a parametrized communication paradigm, such as the message passing interface, MPI [7], and an application programming interface, such as OpenMP [8; 9]. Such a hybrid programming approach has been used in several fields, such as computational fluid dynamics [10; 11], chemical simulations [12; 13], geotechnics [14], high efficiency video coding [15], and electromagnetic simulations [16; 17; 18], due to its very interesting performance [19]. As pointed out in [19; 20], the advantages of using a hybrid OpenMP-MPI programming approach are many, as follows: (i) suitability to the architecture of modern supercomputers, with interconnected shared memory nodes; (ii) improved scalability; (iii) optimization of the total consumed memory; (iv) reduction in the number of MPI processes. On the other hand, a few disadvantages can be observed, as follows: (i) complexity of the implementation; (ii) total gains in performance. These two issues require special care in the implementation of a hybrid approach, to handle the added extra costs to the existing code. The main contribution of this paper is that of defining and assessing the implementation of such an approach when solving the integral formulation of eddy current problems by means of a Galerkin-based finite elements method, such as CARIDDI code, which is widely used in the fusion community [21]. The proposed solution fully exploits the main advantages of both paradigms: the MPI approach is more efficient in the process-level parallelization, whereas the OpenMP approach optimizes the lower level parallelization.