Section: New Results

Efficient algorithmic for load balancing and code coupling in complex simulations

Dynamic load balancing for massively parallel coupled codes

In the field of scientific computing, load balancing is a major issue that determines the performance of parallel applications. Nowadays, simulations of real-life problems are becoming more and more complex, involving numerous coupled codes, representing different models. In this context, reaching high performance can be a great challenge. In the PhD of Maria Predari (started in october 2013), we develop new graph partitioning techniques, called co-partitioning, that address the problem of load balancing for two coupled codes: the key idea is to perform a "coupling-aware" partitioning, instead of partitioning these codes independently, as it is usually done. More precisely, we propose to enrich the classic graph model with interedges, that represent the coupled code interactions. We describe two new algorithms, called AWARE and PROJREPART, and compare them to the currently used approach (called NAIVE). In recent experimental results, we notice that both AWARE and PROJREPART algorithms succeed to balance the computational load in the coupling phase and in some cases they succeed to reduce the coupling communications costs. Surprisingly we notice that our algorithms do not degrade the global graph edgecut, despite the additional constraints that they impose. In future work, we aim at validating our results on real-life cases in the field of aeronautic propulsion. In order to achieve that, we plan to integrate our algorithms within the Scotch framework. Finally, our algorithms should be implemented in parallel and should be extended in order to manage more complex applications with more than two interacting models.

Graph partitioning for hybrid solvers

Nested Dissection has been introduced by A. George and is a very popular heuristic for sparse matrix ordering before numerical factorization. It allows to maximize the number of parallel tasks, while reducing the fill-in and the operation count. The basic standard idea is to build a "small separator" $S$ of the graph associated with the matrix in order to split the remaining vertices in two parts $P_0$ and $P_1$ of "almost equal size". The vertices of the separator $S$ are ordered with the largest indices, and then the same method is applied recursively on the two sub-graphs induced by $P_0$ and $P_1$. At the end, if $k$ levels of recursion are done, we get $2^k$ sets of independent vertices separated from each other by $2^k-1$ separators. However, if we examine precisely the complexity analysis for the estimation of asymptotic bounds for fill-in or operation count when using Nested Dissection ordering, we can notice that the size of the halo of the separated sub-graphs (set of external vertices belonging to an old separator and previously ordered) plays a crucial role in the asymptotic behavior achieved. In the perfect case, we need halo vertices to be balanced among parts. Considering now hybrid methods mixing both direct and iterative solvers such as HIPS , MaPHyS , obtaining a domain decomposition leading to a good balancing of both the size of domain interiors and the Scalable numerical schemes for scientific applications size of interfaces is a key point for load balancing and efficiency in a parallel context. This leads to the same issue: balancing the halo vertices to get balanced interfaces. For this purpose, we revisit the algorithm introduced by Lipton, Rose and Tarjan which performed the recursion of nested dissection in a different manner: at each level, we apply recursively the method to the sub-graphs But, for each sub-graph, we keep track of halo vertices. We have implemented that in the Scotch framework, and have studied its main algorithm to build a separator, called greedy graph growing.

This work is developed in the framework of Astrid Casadei's PhD. These contributions have been presented at the international conference HIPC 2014 [29] in Goa.