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DOE OSTI2011

Parallelization of the Red-Black Algorithm on Solving the Second-Order PN Transport Equation with the Hybrid Finite Element Method

The Red-Black algorithm has been successfully applied on solving the second-order parity transport equation with the PN approximation in angle and the Hybrid Finite Element Method (HFEM) in space, i.e., the Variational Nodal Method (VNM) [1,2,3,4,5]. Any transport solving techniques, including the Red-Black algorithm, need to be parallelized in order to take the advantage of the development of supercomputers with multiple processors for the advanced modeling and simulation. To our knowledge, an attempt [6] was done to parallelize it, but it was devoted only to the z axis plans in three-dimensional calculations. General parallelization of the Red-Black algorithm with the spatial domain decomposition has not been reported in the literature. In this summary, we present our implementation of the parallelization of the Red-Black algorithm and its efficiency results.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2017

Families of Graph Algorithms: SSSP Case Study

Single-Source Shortest Paths (SSSP) is a well-studied graph problem. Examples of SSSP algorithms include the original Dijkstra’s algorithm and the parallel Δ-stepping and KLA-SSSP algorithms. In this paper, we use a novel Abstract Graph Machine (AGM) model to show that all these algorithms share a common logic and differ from one another by the order in which they perform work. We use the AGM model to thoroughly analyze the family of algorithms that arises from the common logic. We start with the basic algorithm without any ordering (Chaotic), and then we derive the existing and new algorithms by methodically exploring semantic and spatial ordering of work. Our experimental results show that new derived algorithms show better performance than the existing distributed memory parallel algorithms, especially at higher scales.

distributed-memory graph algorithms↗

DOE OSTI2018

Synchronization-Avoiding Graph Algorithms

Because they were developed for optimal sequential complexity, classical graph algorithms as found in textbooks have strictly-defined orders of operations. Enforcing a prescribed order of operations, or even an approximate order, in a distributed memory setting requires significant amounts of synchronization, which in turn can severely limit scalability. As a result, new algorithms are typically required to achieve scalable performance, even for solving well-known graph problems. Yet, even in these cases, parallel graph algorithms are written according to parallel programming models that evolved for, e.g., scientific computing, and that still have inherent, and scalability-limiting, amounts of synchronization. In this paper we present a new approach to parallel graph algorithms: synchronization-avoiding algorithms. To eliminate synchronization and its associated overhead, synchronization-avoiding algorithms perform work in an unordered and fully asynchronous fashion in such a way that the result is constantly refined toward its final state. “Wasted” work is minimized by locally prioritizing tasks using problem-dependent task utility metrics. We classify algorithms for graph applications into two broad categories: algorithms with monotonic updates (which evince global synchronization) and algorithms with nonmonotonic updates (which evince vertex-centric synchronization). We apply our approach to both classes and develop novel, synchronization-avoiding algorithms for solving exemplar problems: SSSP and connected components for the former, graph coloring for the latter. We demonstrate that eliminating synchronization in conjunction with effective scheduling policies and optimizations in the runtime results in improved scalability for both classes of algorithms.

Firoz, Jesun S.↗

DOE OSTI2018

A distributed-memory approximation algorithm for maximum weight perfect bipartite matching

We design and implement an efficient parallel approximation algorithm for the problem of maximum weight perfect matching in bipartite graphs, i.e. the problem of finding a set of non-adjacent edges that covers all vertices and has maximum weight. This problem differs from the maximum weight matching problem, for which scalable approximation algorithms are known. It is primarily motivated by finding good pivots in scalable sparse direct solvers before factorization where sequential implementations of maximum weight perfect matching algorithms, such as those available in MC64, are widely used due to the lack of scalable alternatives. To overcome this limitation, we propose a fully parallel distributed memory algorithm that first generates a perfect matching and then searches for weightaugmenting cycles of length four in parallel and iteratively augments the matching with a vertex disjoint set of such cycles. For most practical problems the weights of the perfect matchings generated by our algorithm are very close to the optimum. An efficient implementation of the algorithm scales up to 256 nodes (17,408 cores) on a Cray XC40 supercomputer and can solve instances that are too large to be handled by a single node using the sequential algorithm.

Azad, Ariful↗

NASA NTRS1997

Computational Issues in Damping Identification for Large Scale Problems

Two damping identification methods are tested for efficiency in large-scale applications. One is an iterative routine, and the other a least squares method. Numerical simulations have been performed on multiple degree-of-freedom models to test the effectiveness of the algorithm and the usefulness of parallel computation for the problems. High Performance Fortran is used to parallelize the algorithm. Tests were performed using the IBM-SP2 at NASA Ames Research Center. The least squares method tested incurs high communication costs, which reduces the benefit of high performance computing. This method's memory requirement grows at a very rapid rate meaning that larger problems can quickly exceed available computer memory. The iterative method's memory requirement grows at a much slower pace and is able to handle problems with 500+ degrees of freedom on a single processor. This method benefits from parallelization, and significant speedup can he seen for problems of 100+ degrees-of-freedom.

Pilkey, Deborah L.↗

NASA NTRS1996

Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Cai, Xiao-Chuan↗

DOE OSTI2020

Parallel performance of algebraic multigrid domain decomposition

Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large-scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to grow on modern machines. This article examines the design, implementation, and parallel performance of a novel algorithm, algebraic multigrid domain decomposition (AMG-DD), designed specifically to limit communication. The goal of AMG-DD is to provide a low-communication alternative to standard AMG V-cycles by trading some additional computational overhead for a significant reduction in communication cost. Numerical results show that AMG-DD achieves superior accuracy per communication cost compared with AMG, and speedup over AMG is demonstrated on a large GPU cluster.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1998

Globalized Newton-Krylov-Schwarz Algorithms and Software for Parallel Implicit CFD

Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, "routine" parallelization is essential. The pseudo-transient matrix-free Newton-Krylov-Schwarz (Psi-NKS) algorithmic framework is presented as an answer. We show that, for the classical problem of three-dimensional transonic Euler flow about an M6 wing, Psi-NKS can simultaneously deliver: globalized, asymptotically rapid convergence through adaptive pseudo- transient continuation and Newton's method-, reasonable parallelizability for an implicit method through deferred synchronization and favorable communication-to-computation scaling in the Krylov linear solver; and high per- processor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of Psi-NKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of Psi-NKS, and we describe a freely available, MPI-based portable parallel software implementation of the solver employed here.

Gropp, W. D.↗

DOE OSTI2016

Array-based, parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE's using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework "Mesh Oriented dAtaBase" (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. Furthermore, we also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Revisiting Parallel Cyclic Reduction and Parallel Prefix-Based Algorithms for Block Tridiagonal System of Equations

Simulations that require solutions of block tridiagonal systems of equations rely on fast parallel solvers for runtime efficiency. Leading parallel solvers that are highly effective for general systems of equations, dense or sparse, are limited in scalability when applied to block tridiagonal systems. This paper presents scalability results as well as detailed analyses of two parallel solvers that exploit the special structure of block tridiagonal matrices to deliver superior performance, often by orders of magnitude. A rigorous analysis of their relative parallel runtimes is shown to reveal the existence of a critical block size that separates the parameter space spanned by the number of block rows, the block size and the processor count, into distinct regions that favor one or the other of the two solvers. Dependence of this critical block size on the above parameters as well as on machine-specific constants is established. These formal insights are supported by empirical results on up to 2,048 cores of a Cray XT4 system. To the best of our knowledge, this is the highest reported scalability for parallel block tridiagonal solvers to date.

Seal, Sudip K↗

DOE OSTI2023

Massively-parallel Lagrangian particle code and applications

Massively-parallel, distributed-memory algorithms for the Lagrangian particle hydrodynamic method (Samulyak et al., 2018) have been developed, verified, and implemented. The key component of parallel algorithms is a particle management module that includes a parallel construction of octree databases, dynamic adaptation and refinement of octrees, and particle migration between parallel subdomains. The particle management module is based on the p4est (parallel forest of k-trees) library. The massively-parallel Lagrangian particle code has been applied to a variety of fundamental science and applied problems. A summary of Lagrangian particle code applications to the injection of impurities into thermonuclear fusion devices and to the simulation of supersonic hydrogen jets in support of laser-plasma wakefield acceleration research has also been presented.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1998

Parallelization of the Pipelined Thomas Algorithm

In this study the following questions are addressed. Is it possible to improve the parallelization efficiency of the Thomas algorithm? How should the Thomas algorithm be formulated in order to get solved lines that are used as data for other computational tasks while processors are idle? To answer these questions, two-step pipelined algorithms (PAs) are introduced formally. It is shown that the idle processor time is invariant with respect to the order of backward and forward steps in PAs starting from one outermost processor. The advantage of PAs starting from two outermost processors is small. Versions of the pipelined Thomas algorithms considered here fall into the category of PAs. These results show that the parallelization efficiency of the Thomas algorithm cannot be improved directly. However, the processor idle time can be used if some data has been computed by the time processors become idle. To achieve this goal the Immediate Backward pipelined Thomas Algorithm (IB-PTA) is developed in this article. The backward step is computed immediately after the forward step has been completed for the first portion of lines. This enables the completion of the Thomas algorithm for some of these lines before processors become idle. An algorithm for generating a static processor schedule recursively is developed. This schedule is used to switch between forward and backward computations and to control communications between processors. The advantage of the IB-PTA over the basic PTA is the presence of solved lines, which are available for other computations, by the time processors become idle.

Povitsky, A.↗

NASA NTRS2013

Experiments with a Parallel Multi-Objective Evolutionary Algorithm for Scheduling

Evolutionary multi-objective algorithms have great potential for scheduling in those situations where tradeoffs among competing objectives represent a key requirement. One challenge, however, is runtime performance, as a consequence of evolving not just a single schedule, but an entire population, while attempting to sample the Pareto frontier as accurately and uniformly as possible. The growing availability of multi-core processors in end user workstations, and even laptops, has raised the question of the extent to which such hardware can be used to speed up evolutionary algorithms. In this paper we report on early experiments in parallelizing a Generalized Differential Evolution (GDE) algorithm for scheduling long-range activities on NASA's Deep Space Network. Initial results show that significant speedups can be achieved, but that performance does not necessarily improve as more cores are utilized. We describe our preliminary results and some initial suggestions from parallelizing the GDE algorithm. Directions for future work are outlined.

scheduling↗

DOE OSTI2021

ParChain: a framework for parallel hierarchical agglomerative clustering using nearest-neighbor chain

This paper studies the hierarchical clustering problem, where the goal is to produce a dendrogram that represents clusters at varying scales of a data set. We propose the ParChain framework for designing parallel hierarchical agglomerative clustering (HAC) algorithms, and using the framework we obtain novel parallel algorithms for the complete linkage, average linkage, and Ward's linkage criteria. Compared to most previous parallel HAC algorithms, which require quadratic memory, our new algorithms require only linear memory, and are scalable to large data sets. ParChain is based on our parallelization of the nearest-neighbor chain algorithm, and enables multiple clusters to be merged on every round. We introduce two key optimizations that are critical for efficiency: a range query optimization that reduces the number of distance computations required when finding nearest neighbors of clusters, and a caching optimization that stores a subset of previously computed distances, which are likely to be reused. Experimentally, we show that our highly-optimized implementations using 48 cores with two-way hyper-threading achieve 5.8--110.1x speedup over state-of-the-art parallel HAC algorithms and achieve 13.75--54.23x self-relative speedup. Compared to state-of-the-art algorithms, our algorithms require up to 237.3x less space. Our algorithms are able to scale to data set sizes with tens of millions of points, which existing algorithms are not able to handle.

Computer Science↗

NASA NTRS1995

A Parallel Incompressible Navier-Stokes Solver With a Parallel Multigrid Elliptic Kernal

The paper describes numerical algorithms and parallel implementations of a time-dependent, incompressible Navier-Stokes flow solver and a multigrid elliptic solver, which is also used as a computation kernal in the flow solver. The implemented solvers are numerically stable and computationally efficient, and they scale well to a large number of processors for problems with moderate granularity.

solver Navier-Stokes solver multigrid elliptic sol↗

NASA NTRS1987

Algorithmic commonalities in the parallel environment

The ultimate aim of this project was to analyze procedures from substantially different application areas to discover what is either common or peculiar in the process of conversion to the Massively Parallel Processor (MPP). Three areas were identified: molecular dynamic simulation, production systems (rule systems), and various graphics and vision algorithms. To date, only selected graphics procedures have been investigated. They are the most readily available, and produce the most visible results. These include simple polygon patch rendering, raycasting against a constructive solid geometric model, and stochastic or fractal based textured surface algorithms. Only the simplest of conversion strategies, mapping a major loop to the array, has been investigated so far. It is not entirely satisfactory.

Mcanulty, Michael A.↗

DOE OSTI2017

Frog: Asynchronous Graph Processing on GPU with Hybrid Coloring Model

GPUs have been increasingly used to accelerate graph processing for complicated computational problems regarding graph theory. Many parallel graph algorithms adopt the asynchronous computing model to accelerate the iterative convergence. Unfortunately, the consistent asynchronous computing requires locking or atomic operations, leading to significant penalties/overheads when implemented on GPUs. As such, coloring algorithm is adopted to separate the vertices with potential updating conflicts, guaranteeing the consistency/correctness of the parallel processing. Common coloring algorithms, however, may suffer from low parallelism because of a large number of colors generally required for processing a large-scale graph with billions of vertices. We propose a light-weight asynchronous processing framework called Frog with a preprocessing/hybrid coloring model. The fundamental idea is based on Pareto principle (or 80-20 rule) about coloring algorithms as we observed through masses of realworld graph coloring cases. We find that a majority of vertices (about 80%) are colored with only a few colors, such that they can be read and updated in a very high degree of parallelism without violating the sequential consistency. Accordingly, our solution separates the processing of the vertices based on the distribution of colors. In this work, we mainly answer three questions: (1) how to partition the vertices in a sparse graph with maximized parallelism, (2) how to process large-scale graphs that cannot fit into GPU memory, and (3) how to reduce the overhead of data transfers on PCIe while processing each partition. Here, we conduct experiments on real-world data (Amazon, DBLP, YouTube, RoadNet-CA, WikiTalk and Twitter) to evaluate our approach and make comparisons with well-known non-preprocessed (such as Totem, Medusa, MapGraph and Gunrock) and preprocessed (Cusha) approaches, by testing four classical algorithms (BFS, PageRank, SSSP and CC). On all the tested applications and datasets, Frog is able to significantly outperform existing GPU-based graph processing systems except Gunrock and MapGraph. MapGraph gets better performance than Frog when running BFS on RoadNet-CA. The comparison between Gunrock and Frog is inconclusive. Frog can outperform Gunrock more than 1.04X when running PageRank and SSSP, while the advantage of Frog is not obvious when running BFS and CC on some datasets especially for RoadNet-CA.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A GPU‐Accelerated Generative Adversarial Model for Causal Inference

We develop a GPU-accelerated machine learning generative adversarial model designed to facilitate causal inferences from observational data. Our model's theoretical framework is conceptualized in a manner that is amenable to being operable and scalable for high-performance computing platforms. We leverage GPU acceleration to develop a parallel evolutionary algorithm to achieve large-scale parallel computation of the model within a now widely accessible computing platform. This capability both enhances computational speedup and efficiency and also extends the use of the model to a broader range of substantive research domains while maintaining the underlying theoretical properties of the model.

GPU↗