There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In this paper, we analyze the main features and discuss the tuning of the algorithms for the direct solution of sparse linear systems on distributed memory computers developed in the context of a long term European research project. The algorithms use a multifrontal approach and are especially designed to cover a large class of problems. The problems can be symmetric positive definite, general symmetric, or unsymmetric matrices, both possibly rank deficient, and they can be provided by the user in several formats. The algorithms achieve high performance by exploiting parallelism coming from the sparsity in the problem and that available for dense matrices. The algorithms use a dynamic distributed task scheduling technique to accommodate numerical pivoting and to allow the migration of computational tasks to lightly loaded processors. Large computational tasks are divided into subtasks to enhance parallelism. Asynchronous communication is used throughout the solution process to efficiently overlap communication with computation.
We illustrate our design choices by experimental results obtained on an SGI Origin 2000 and an IBM SP2 for test matrices provided by industrial partners in the PARASOL project.

/pdf/a-fully-asynchronous-multifrontal-solver-using-distributed-1xlwbmzqj5.pdf

A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling

The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5.

https://publications.lib.chalmers.se/records/fulltext/228672/local_228672.pdf

The FEniCS Project Version 1.5

In the field of HPC, the current hardware trend is to design multiprocessor architectures featuring heterogeneous technologies such as specialized coprocessors (e.g. Cell/BE) or data-parallel accelerators (e.g. GPUs). Approaching the theoretical performance of these architectures is a complex issue. Indeed, substantial efforts have already been devoted to efficiently offload parts of the computations. However, designing an execution model that unifies all computing units and associated embedded memory remains a main challenge. We therefore designed StarPU, an original runtime system providing a high-level, unified execution model tightly coupled with an expressive data management library. The main goal of StarPU is to provide numerical kernel designers with a convenient way to generate parallel tasks over heterogeneous hardware on the one hand, and easily develop and tune powerful scheduling algorithms on the other hand. We have developed several strategies that can be selected seamlessly at run-time, and we have analyzed their efficiency on several algorithms running simultaneously over multiple cores and a GPU. In addition to substantial improvements regarding execution times, we have obtained consistent superlinear parallelism by actually exploiting the heterogeneous nature of the machine. We eventually show that our dynamic approach competes with the highly optimized MAGMA library and overcomes the limitations of the corresponding static scheduling in a portable way. Copyright © 2010 John Wiley & Sons, Ltd.

/pdf/starpu-a-unified-platform-for-task-scheduling-on-2qdx6n5x2r.pdf

StarPU: a unified platform for task scheduling on heterogeneous multicore architectures

We consider the problem of designing a dynamic scheduling strategy that takes into account both workload and memory information in the context of the parallel multifrontal factorization. The originality of our approach is that we base our estimations (work and memory) on a static optimistic scenario during the analysis phase. This scenario is then used during the factorization phase to constrain the dynamic decisions that compute fully irregular partitions in order to better balance the workload. We show that our new scheduling algorithm significantly improves both the memory behaviour and the factorization time. We give experimental results for large challenging real-life 3D problems on 64 and 128 processors.

/pdf/hybrid-scheduling-for-the-parallel-solution-of-linear-7h62u6cen9.pdf

Hybrid scheduling for the parallel solution of linear systems

Solving large sparse symmetric positive definite systems of linear equations is a crucial and time-consuming step, arising in many scientific and engineering applications. The block partitioning and scheduling problem for sparse parallel factorization without pivoting is considered. There are two major aims to this study: the scalability of the parallel solver, and the compromise between memory overhead and efficiency. Parallel experiments on a large collection of irregular industrial problems validate our approach.

PASTIX: a high-performance parallel direct solver for sparse symmetric positive definite systems

PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems

Among existing preconditioners, the level-of-fill ILU has been quite popular as a general-purpose technique. Experimental observations have shown that, when coupled with block techniques, these methods can be quite effective in solving realistic problems arising from various applications. In this work, we consider an extension of this kind of method which is suitable for parallel environments. Our method is developed from the framework of high performance sparse direct solvers. The main idea we propose is to define an adaptive blockwise incomplete factorization that is much more accurate (and numerically more robust) than the scalar incomplete factorizations commonly used to precondition iterative solvers. These requirements lead to a robust class of parallel preconditioners based on generalized versions of block ILU techniques.

On finding approximate supernodes for an efficient block-ILU(k) factorization

We presen t and analyze a general algorithm which computes an efficient static scheduling of block computations for a parallel L:D:Lt factorization of sparse symmetric positive definite systems based on a combination of 1D and 2D block distributions. Our solver uses a supernodal fan-in approach and is fully driven by this scheduling. We give an overview of the algorithm and present performance results and comparisons with PSPASES on an IBM-SP2 with 120 MHz Pow er2SC nodes for a collection of irregular problems.

PaStiX: A Parallel Sparse Direct Solver Based on a Static Scheduling for Mixed 1D/2D Block Distributions

Since the last decade, most of the supercomputer architectures are based on clusters of SMP nodes. In those architectures the exchanges between processors are made through shared memory when the processors are located on a same SMP node and through the network otherwise. Generally, the MPI implementations provided by the constructor on those machines are adapted to this situation and take advantage of the shared memory to treat messages between processors in a same SMP node. Nevertheless, this transparent approach to exploit shared memory does not avoid the storage of the extra-structures needed to manage efficiently the communications between processors. For high performance parallel direct solvers, the storage of these extra-structures can become a bottleneck. In this paper, we propose an hybrid MPI-thread implementation of a parallel direct solver and analyse the benefits of this approach in terms of memory and run-time performances.

Pascal Hénon

Papers

PASTIX: a high-performance parallel direct solver for sparse symmetric positive definite systems

PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems

On finding approximate supernodes for an efficient block-ILU(k) factorization

PaStiX: A Parallel Sparse Direct Solver Based on a Static Scheduling for Mixed 1D/2D Block Distributions

On using an hybrid MPI-Thread programming for the implementation of a parallel sparse direct solver on a network of SMP nodes