Project Team Graal

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Section: New Results

Parallel Sparse Direct Solvers and Combinatorial Scientific Computing

Participants : Maurice Brémond, Guillaume Joslin, Johannes Langguth, Jean-Yves L'Excellent, Mohamed Sid-Lakhdar, Bora Uçar.

Parallel computation of entries of the inverse of a sparse matrix

Following last year's work on computing entries of the inverse of a sparse matrix in a serial, in-core or out-of-core environment, and that was implemented in Mumps , we have pursued work to address this issue in a parallel environment. In such this case, it has been shown that minimizing the number of operations (or the number of accesses to the factors) and balancing the work between the processors are contradictory objectives. Several ideas have been investigated and implemented in order to deal with this issue and to reach high speed-ups. Experimental results are promising and show good speed-ups on relatively small number of processors (up to 16) when dealing with large blocks of sparse right-hand sides, while we used to experience speed-downs before.

Multithreaded parallelism for the MUMPS solver

Apart from using message-passing, we have in the past only exploited multicore parallelism through threaded libraries (e.g. BLAS: Basic Linear Algebra Subroutines), and a few OpenMP directives. We are currently investigating the combination of this fork-join model with threaded parallelism resulting from the task graph, which, in our context, is a tree. To do so, and in order to also target NUMA architectures, we apply ideas from distributed-memory environments to multithreaded environments. Simulations based on benchmarks followed by a first prototype implementation have validated this approach for some classes of matrices on small numbers of cores. We are currently revisiting this implementation and plan to pursue experiments on larger numbers of cores with larger classes of matrices. This starting work was done in the context of a master thesis and is the object of a starting PhD thesis. In a distributed-memory environments, it will be combined with parallelism based on message passing, where the scalability of the existing communication schemes should also be addressed. Both directions will be followed in order to face the multicore (r)evolution.

Low-rank approximations

Low-rank approximations are commonly used to compress the representation of data structures. The loss of information induced is often negligible and can be controlled. Although the dense internal datastructures involved in a multifrontal method, the so-called frontal matrices or fronts, are full-rank, they can be represented by a set of low-rank matrices. Applying to our context the notion of geometric clustering used by Bebendorf to define hierarchical matrices, we have shown that the efficiency of this representation to reduce the complexity of both the factorization and solve phases strongly depends on how variables are grouped. The proposed approach can be used either to accelerate the factorization and solution phases or to build a preconditioner. The ultimate goal of this work is to extend the features of the Mumps solver to exploit low-rank properties.

This work, and the work described in the two previous paragraphs are in the context of a collaboration with ENSEEIHT-IRIT and with the partners involved in the Mumps project (see Section  5.2 ).

On partitioning problems with complex objectives

Hypergraph and graph partitioning tools are used to partition work for efficient parallelization of many sparse matrix computations. Most of the time, the objective function that is reduced by these tools relates to reducing the communication requirements, and the balancing constraints satisfied by these tools relate to balancing the work or memory requirements. Sometimes, the objective sought for having balance is a complex function of a partition. We mention some important class of parallel sparse matrix computations that have such balance objectives. For these cases, the current state of the art partitioning tools fall short of being adequate. To the best of our knowledge, there is only a single algorithmic framework in the literature to address such balance objectives. We propose another algorithmic framework to tackle complex objectives and experimentally investigate the proposed framework.

On the Use of Cluster-Based Partial Message Logging to Improve Fault Tolerance for MPI HPC Applications

Fault tolerance is becoming a major concern in HPC systems. The two traditional approaches for message passing applications, coordinated checkpointing and message logging, have severe scalability issues. Coordinated checkpointing protocols make all processes roll back after a failure. Message logging protocols log a huge amount of data and can induce an overhead on communication performance. Hierarchical rollback-recovery protocols based on the combination of coordinated checkpointing and message logging are an alternative. These partial message logging protocols are based on process clustering: only messages between clusters are logged to limit the consequence of a failure to one cluster. These protocols would work efficiently only if one can find clusters of processes in the applications such that the ratio of logged messages is very low. We study the communication patterns of message passing HPC applications to show that partial message logging is suitable in most cases. We propose a partitioning algorithm to find suitable clusters of processes given the communication pattern of an application. Finally, we evaluate the efficiency of partial message logging using two state of the art protocols on a set of representative applications.

Integrated data placement and task assignment for scientific workflows in clouds

We consider the problem of optimizing the execution of data-intensive scientific workflows in the Cloud. We address the problem under the following scenario. The tasks of the workflows communicate through files; the output of a task is used by another task as an input file and if these tasks are assigned on different execution sites, a file transfer is necessary. The output files are to be stored at a site. Each execution site is to be assigned a certain percentage of the files and tasks. These percentages, called target weights, are pre-determined and reflect either user preferences or the storage capacity and computing power of the sites. The aim is to place the data files into and assign the tasks to the execution sites so as to reduce the cost associated with the file transfers, while complying with the target weights. To do this, we model the workflow as a hypergraph and with a hypergraph-partitioning-based formulation, we propose a heuristic which generates data placement and task assignment schemes simultaneously. We report simulation results on a number of real-life and synthetically generated scientific workflows. Our results show that the proposed heuristic is fast, and can find mappings and assignments which reduce file transfers, while respecting the target weights.

UMPa: A Multi-objective, multi-level partitioner for communication minimization

We propose a directed hypergraph model and a refinement heuristic to distribute communicating tasks among the processing units in a distributed memory setting. The aim is to achieve load balance and minimize the maximum data sent by a processing unit. We also take two other communication metrics into account with a tie-breaking scheme. With this approach, task distributions causing an excessive use of network or a bottleneck processor which participates to almost all of the communication are avoided. We show on a large number of problem instances that our model improves the maximum data sent by a processor up to 34% for parallel environments with 4,16,64 and 256 processing units compared to the state of the art which only minimizes the total communication volume.

A Divisive clustering technique for maximizing the modularity

We present a new graph clustering algorithm aimed at obtaining clusterings of high modularity. The algorithm pursues a divisive clustering approach and using established graph partitioning algorithms and techniques to compute recursive bipartitions of the input as well as to refine clusters. Experimental evaluation shows that the modularity scores obtained compare favorably to many previous approaches. In the majority of test cases, the algorithm outperformed the best known alternatives. In particular, among 13 problem instances common in the literature, the proposed algorithm improves the best known modularity in 9 cases.

Constructing elimination trees for sparse unsymmetric matrices

The elimination tree model for sparse unsymmetric matrices and an algorithm for constructing it have been recently proposed [Eisenstat and Liu, SIAM J. Matrix Anal. Appl., 26 (2005) and 29 (2008)]. The construction algorithm has a worst case time complexity 𝒪(mn) for an n×n unsymmetric matrix having m nonzeros. We propose another algorithm that has a worst case time complexity of 𝒪(mlogn).

Multithreaded clustering for multi-level hypergraph partitioning

Requirements for efficient parallelization of many complex and irregular applications can be cast as a hypergraph partitioning problem. The current-state-of-the art software libraries that provide tool support for the hypergraph partitioning problem are designed and implemented before the game-changing advancements in multi-core computing. Hence, analyzing the structure of those tools for designing multithreaded versions of the algorithms is a crucial tasks. The most successful partitioning tools are based on the multi-level approach. In this approach, a given hypergraph is coarsened to a much smaller one, a partition is obtained on the the smallest hypergraph, and that partition is projected to the original hypergraph while refining it on the intermediate hypergraphs. The coarsening operation corresponds to clustering the vertices of a hypergraph and is the most time consuming task in a multi-level partitioning tool. We present three efficient multithreaded clustering algorithms which are very suited for multi-level partitioners. We compare their performance with that of the ones currently used in today's hypergraph partitioners. We show on a large number of real life hypergraphs that our implementations, integrated into a commonly used partitioning library PaToH, achieve good speedups without reducing the clustering quality.

Partitioning, ordering, and load balancing in a hierarchically parallel hybrid linear solver

PDSLin is a general-purpose algebraic parallel hybrid (direct/iterative) linear solver based on the Schur complement method. The most challenging step of the solver is the computation of a preconditioner based on an approximate global Schur complement. We investigate two combinatorial problems to enhance PDSLin's performance at this step. The first is a multi-constraint partitioning problem to balance the workload while computing the preconditioner in parallel. For this, we describe and evaluate a number of graph and hypergraph partitioning algorithms to satisfy our particular objective and constraints. The second problem is to reorder the sparse right-hand side vectors to improve the data access locality during the parallel solution of a sparse triangular system with multiple right-hand sides. This is needed to eliminate the unknowns associated with the interface in PDSLin. We study two reordering techniques: one based on a postordering of the elimination tree and the other based on a hypergraph partitioning. To demonstrate the effect of these techniques on the performance of PDSLin, we present the numerical results of solving large-scale linear systems arising from numerical simulations of modeling accelerator cavities and of modeling fusion devices.

Experiments on push-relabel-based maximum cardinality matching algorithms for bipartite graphs

We report on careful implementations of several push-relabel-based algorithms for solving the problem of finding a maximum cardinality matching in a bipartite graph and compare them with fast augmenting-path-based algorithms. We analyze the algorithms using a common base for all implementations and compare their relative performance and stability on a wide range of graphs. The effect of a set of known initialization heuristics on the performance of matching algorithms is also investigated. Our results identify a variant of the push-relabel algorithm and a variant of the augmenting-path-based algorithm as the fastest with proper initialization heuristics, while the push-relabel based one having a better worst case performance.

Towards a scalable hybrid linear solver based on combinatorial algorithms

The availability of large-scale computing platforms comprised of tens of thousands of multicore processors motivates the need for the next generation of highly scalable sparse linear system solvers. These solvers must optimize parallel performance, processor (serial) performance, as well as memory requirements, while being robust across broad classes of applications and systems. In this study, we present a hybrid parallel solver that combines the desirable characteristics of direct methods (robustness) and effective iterative solvers (low computational cost), while alleviating their drawbacks (memory requirements, lack of robustness). We discuss several combinatorial problems that arise in the design of this hybrid solver, present algorithms to solve these combinatorial problems, and demonstrate their impact on a large-scale three-dimensional PDE-constrained optimization problem.