346 research outputs found
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Solving large scale linear programming
The interior point method (IPM) is now well established as a competitive technique for solving very large scale linear programming problems. The leading variant of the interior point method is the primal dual - predictor corrector algorithm due to Mehrotra. The main computational steps of this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations.
We describe an implementation of the predictor corrector IPM algorithm on MasPar, a massively parallel SIMD computer. At the heart of the implemen-tation is a parallel Cholesky factorization algorithm for sparse matrices. Our implementation uses a new scheme of mapping the matrix onto the processor grid of the MasPar, that results in a more efficient Cholesky factorization than previously suggested schemes.
The IPM implementation uses the parallel unit of MasPar to speed up the factorization and other computationally intensive parts of the IPM. An impor-tant part of this implementation is the judicious division of data and computation between the front-end computer, that runs the main IPM algorithm, and the par-allel unit. Performanc
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A Parallel Direct Method for Finite Element Electromagnetic Computations Based on Domain Decomposition
High performance parallel computing and direct (factorization-based) solution methods have been the two main trends in electromagnetic computations in recent years. When time-harmonic (frequency-domain) Maxwell\u27s equation are directly discretized with the Finite Element Method (FEM) or other Partial Differential Equation (PDE) methods, the resulting linear system of equations is sparse and indefinite, thus harder to efficiently factorize serially or in parallel than alternative methods e.g. integral equation solutions, that result in dense linear systems. State-of-the-art sparse matrix direct solvers such as MUMPS and PARDISO don\u27t scale favorably, have low parallel efficiency and high memory footprint. This work introduces a new class of sparse direct solvers based on domain decomposition method, termed Direct Domain Decomposition Method (D3M), which is reliable, memory efficient, and offers very good parallel scalability for arbitrary 3D FEM problems.
Unlike recent trends in approximate/low-rank solvers, this method focuses on `numerically exact\u27 solution methods as they are more reliable for complex `real-life\u27 models. The proposed method leverages physical insights at every stage of the development through a new symmetric domain decomposition method (DDM) with one set of Lagrange multipliers. Applying a special regularization scheme at the interfaces, either artificial loss or gain is introduced to each domain to eliminate non-physical internal resonances. A block-wise recursive algorithm based on Takahashi relationship is proposed for the efficient computation of discrete Dirichlet-to-Neumann (DtN) map to reduce the volumetric problem from all domains into an auxiliary surfacial problem defined on the domain interfaces only. Numerical results show up to 50% run-time saving in DtN map computation using the proposed block-wise recursive algorithm compared to alternative approaches. The auxiliary unknowns on the domain interfaces form a considerably (approximately an order of magnitude) smaller block-wise sparse matrix, which is efficiently factorized using a customized block LDL factorization with restricted pivoting to ensure stability.
The parallelization of the proposed D3M is realized based on Directed Acyclic Graph (DAG). Recent advances in parallel dense direct solvers, have shifted toward parallel implementation that rely on DAG scheduling to achieve highly efficient asynchronous parallel execution. However, adaptation of such schemes to sparse matrices is harder and often impractical. In D3M, computation of each domain\u27s discrete DtN map ``embarrassingly parallel\u27\u27, whereas the customized block LDLT is suitable for a block directed acyclic graph (B-DAG) task scheduling, similar to that used in dense matrix parallel direct solvers. In this approach, computations are represented as a sequence of small tasks that operate on domains of DDM or dense matrix blocks of the reduced matrix. These tasks can be statically scheduled for parallel execution using their DAG dependencies and weights that depend on estimates of computation and communication costs.
Comparisons with state-of-the-art exact direct solvers on electrically large problems suggest up to 20% better parallel efficiency, 30% - 3X less memory and slightly faster in runtime, while maintaining the same accuracy
PT-Scotch: A tool for efficient parallel graph ordering
The parallel ordering of large graphs is a difficult problem, because on the
one hand minimum degree algorithms do not parallelize well, and on the other
hand the obtainment of high quality orderings with the nested dissection
algorithm requires efficient graph bipartitioning heuristics, the best
sequential implementations of which are also hard to parallelize. This paper
presents a set of algorithms, implemented in the PT-Scotch software package,
which allows one to order large graphs in parallel, yielding orderings the
quality of which is only slightly worse than the one of state-of-the-art
sequential algorithms. Our implementation uses the classical nested dissection
approach but relies on several novel features to solve the parallel graph
bipartitioning problem. Thanks to these improvements, PT-Scotch produces
consistently better orderings than ParMeTiS on large numbers of processors
Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers
The linear equations that arise in interior methods for constrained
optimization are sparse symmetric indefinite and become extremely
ill-conditioned as the interior method converges. These linear systems present
a challenge for existing solver frameworks based on sparse LU or LDL^T
decompositions. We benchmark five well known direct linear solver packages
using matrices extracted from power grid optimization problems. The achieved
solution accuracy varies greatly among the packages. None of the tested
packages delivers significant GPU acceleration for our test cases
Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers
In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration
Parallel density matrix propagation in spin dynamics simulations
Several methods for density matrix propagation in distributed computing
environments, such as clusters and graphics processing units, are proposed and
evaluated. It is demonstrated that the large communication overhead associated
with each propagation step (two-sided multiplication of the density matrix by
an exponential propagator and its conjugate) may be avoided and the simulation
recast in a form that requires virtually no inter-thread communication. Good
scaling is demonstrated on a 128-core (16 nodes, 8 cores each) cluster.Comment: Submitted for publicatio
Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing
Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM
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