Algorithmic patterns for H\mathcal{H}-matrices on many-core processors

In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix-vector products. The parallelization of $\mathcal{H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal{H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal{H}$ matrix construction and the fast matrix-vector product to many-core hardware. Here, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal{H}$ matrix library of this kind. We conclude this work by an in-depth performance analysis and a comparative performance study against a standard $\mathcal{H}$ matrix library, highlighting profound speedups of our many-core parallel approach

An adaptive hierarchical domain decomposition method for parallel contact dynamics simulations of granular materials

A fully parallel version of the contact dynamics (CD) method is presented in this paper. For large enough systems, 100% efficiency has been demonstrated for up to 256 processors using a hierarchical domain decomposition with dynamic load balancing. The iterative scheme to calculate the contact forces is left domain-wise sequential, with data exchange after each iteration step, which ensures its stability. The number of additional iterations required for convergence by the partially parallel updates at the domain boundaries becomes negligible with increasing number of particles, which allows for an effective parallelization. Compared to the sequential implementation, we found no influence of the parallelization on simulation results.Comment: 19 pages, 15 figures, published in Journal of Computational Physics (2011

A pattern language for parallelizing irregular algorithms

Dissertação apresentada na Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa para obtenção do grau de Mestre em Engenharia InformáticaIn irregular algorithms, data set’s dependences and distributions cannot be statically predicted. This class of algorithms tends to organize computations in terms of data locality instead of parallelizing control in multiple threads. Thus, opportunities for exploiting parallelism vary dynamically, according to how the algorithm changes data dependences. As such, effective parallelization of such algorithms requires new approaches that account for that dynamic nature. This dissertation addresses the problem of building efficient parallel implementations of irregular algorithms by proposing to extract, analyze and document patterns of concurrency and parallelism present in the Galois parallelization framework for irregular algorithms. Patterns capture formal representations of a tangible solution to a problem that arises in a well defined context within a specific domain. We document the said patterns in a pattern language, i.e., a set of inter-dependent patterns that compose well-documented template solutions that can be reused whenever a certain problem arises in a well-known context

A Sparse SCF algorithm and its parallel implementation: Application to DFTB

We present an algorithm and its parallel implementation for solving a self consistent problem as encountered in Hartree Fock or Density Functional Theory. The algorithm takes advantage of the sparsity of matrices through the use of local molecular orbitals. The implementation allows to exploit efficiently modern symmetric multiprocessing (SMP) computer architectures. As a first application, the algorithm is used within the density functional based tight binding method, for which most of the computational time is spent in the linear algebra routines (diagonalization of the Fock/Kohn-Sham matrix). We show that with this algorithm (i) single point calculations on very large systems (millions of atoms) can be performed on large SMP machines (ii) calculations involving intermediate size systems (1~000--100~000 atoms) are also strongly accelerated and can run efficiently on standard servers (iii) the error on the total energy due to the use of a cut-off in the molecular orbital coefficients can be controlled such that it remains smaller than the SCF convergence criterion.Comment: 13 pages, 11 figure

A Three-Level Parallelisation Scheme and Application to the Nelder-Mead Algorithm

We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of the second and third level starts to drop down after some critical parallelisation degree is reached. This weakness of the two-level template is addressed by introduction of one additional parallelisation level. As an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a few partial differential equations are solved numerically on the second level, and on the third level the parallel Wang's algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data