A Parallel Graph Algorithm for Detecting Mesh Singularities in Distributed Memory Ice Sheet Simulations.
Abstract not provided.
Bogle, Ian↗
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Fast synchronization-free algorithms for parallel sparse triangular solves with multiple right-hand sides
The sparse triangular solve kernels, SpTRSV and SpTRSM, are important building blocks for a number of numerical linear algebra routines. Parallelizing SpTRSV and SpTRSM on today's many-core platforms, such as GPUs, is not an easy task since computing a component of the solution may depend on previously computed components, enforcing a degree of sequential processing. As a consequence, most existing work introduces a preprocessing stage to partition the components into a group of level-sets or colour-sets so that components within a set are independent and can be processed simultaneously during the subsequent solution stage. However, this class of methods requires a long preprocessing time as well as significant runtime synchronization over-heads between the sets. To address this, we
HPC↗
A Parallel Ghosting Algorithm for The Flexible Distributed Mesh Database
Critical to the scalability of parallel adaptive simulations are parallel control functions including load balancing, reduced inter-process communication and optimal data decomposition. In distributed meshes, many mesh-based applications frequently access neighborhood information for computational purposes which must be transmitted efficiently to avoid parallel performance degradation when the neighbors are on different processors. This article presents a parallel algorithm of creating and deleting data copies, referred to as ghost copies, which localize neighborhood data for computation purposes while minimizing inter-process communication. The key characteristics of the algorithm are: (1) It can create ghost copies of any permissible topological order in a 1D, 2D or 3D mesh based on selected adjacencies. (2) It exploits neighborhood communication patterns during the ghost creation process thus eliminating all-to-all communication. (3) For applications that need neighbors of neighbors, the algorithm can create n number of ghost layers up to a point where the whole partitioned mesh can be ghosted. Strong and weak scaling results are presented for the IBM BG/P and Cray XE6 architectures up to a core count of 32,768 processors. The algorithm also leads to scalable results when used in a parallel super-convergent patch recovery error estimator, an application that frequently accesses neighborhood data to carry out computation.
Mubarak, Misbah↗
A matrix-algebraic formulation of distributed-memory maximal cardinality matching algorithms in bipartite graphs
We describe parallel algorithms for computing maximal cardinality matching in a bipartite graph on distributed-memory systems. Unlike traditional algorithms that match one vertex at a time, our algorithms process many unmatched vertices simultaneously using a matrix-algebraic formulation of maximal matching. This generic matrix-algebraic framework is used to develop three efficient maximal matching algorithms with minimal changes. The newly developed algorithms have two benefits over existing graph-based algorithms. First, unlike existing parallel algorithms, cardinality of matching obtained by the new algorithms stays constant with increasing processor counts, which is important for predictable and reproducible performance. Second, relying on bulk-synchronous matrix operations, these algorithms expose a higher degree of parallelism on distributed-memory platforms than existing graph-based algorithms. We report high-performance implementations of three maximal matching algorithms using hybrid OpenMP-MPI and evaluate the performance of these algorithm using more than 35 real and randomly generated graphs. On real instances, our algorithms achieve up to 200 × speedup on 2048 cores of a Cray XC30 supercomputer. Even higher speedups are obtained on larger synthetically generated graphs where our algorithms show good scaling on up to 16,384 cores.
97 MATHEMATICS AND COMPUTING↗
p4est : Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees
We present scalable algorithms for parallel adaptive mesh refinement and coarsening (AMR), partitioning, and 2:1 balancing on computational domains composed of multiple connected two-dimensional quadtrees or three-dimensional octrees, referred to as a forest of octrees. By distributing the union of octants from all octrees in parallel, we combine the high scalability proven previously for adaptive single-octree algorithms with the geometric flexibility that can be achieved by arbitrarily connected hexahedral macromeshes, in which each macroelement is the root of an adapted octree. A key concept of our approach is an encoding scheme of the interoctree connectivity that permits arbitrary relative orientations between octrees. Based on this encoding we develop interoctree transformations of octants. These form the basis for high-level parallel octree algorithms, which are designed to interact with an application code such as a numerical solver for partial differential equations. We have implemented and tested these algorithms in the p4est software library. We demonstrate the parallel scalability of p4est on its own and in combination with two geophysics codes. Using p4est we generate and adapt multioctree meshes with up to $5.13\times10^{11}$ octants on as many as 220,320 CPU cores and execute the 2:1 balance algorithm in less than 10 seconds per million octants per process.
Mathematics↗
PANDORA: A Parallel Dendrogram Construction Algorithm for Single Linkage Clustering on GPU
This paper introduces Pandora, a parallel algorithm for computing dendrograms, the hierarchical cluster trees for single linkage clustering (SLC). Current parallel approaches construct dendrograms by partitioning a minimum spanning tree and removing edges. However, they struggle with skewed, hard-to-parallelize real-world dendrograms. Consequently, computing dendrograms is the sequential bottleneck in HDBSCAN*[21], a popular SLC variant. Pandora uses recursive tree contraction to address this limitation. Pandora contracts nodes to construct progressively smaller trees. It computes the smallest contracted dendrogram and expands it by inserting contracted edges. This recursive strategy is highly parallel, skew-independent, work-optimal, and well-suited for GPUs and multicores. We develop a performance portable implementation of Pandora in Kokkos[31] and evaluate its performance on multicore CPUs and multi-vendor GPUs (e.g., Nvidia, AMD) for dendrogram construction in HDBSCAN*. Multithreaded Pandora is 2.2x faster than the current best-multithreaded implementation. Our GPU version achieves 6-20x speedup on AMD GPUs and 10-37x on NVIDIA GPUs over multithreaded Pandora. Pandora removes HDBSCAN*’s sequential bottleneck, greatly boosting efficiency, particularly with GPUs.
Sao, Piyush↗
Massively parallel Monte Carlo for many-particle simulations on GPUs
Current trends in parallel processors call for the design of efficient massively parallel algorithms for scientific computing. Parallel algorithms for Monte Carlo simulations of thermodynamic ensembles of particles have received little attention because of the inherent serial nature of the statistical sampling. In this paper, we present a massively parallel method that obeys detailed balance and implement it for a system of hard disks on the GPU. We reproduce results of serial high-precision Monte Carlo runs to verify the method. This is a good test case because the hard disk equation of state over the range where the liquid transforms into the solid is particularly sensitive to small deviations away from the balance conditions. On a Tesla K20, our GPU implementation executes over one billion trial moves per second, which is 148 times faster than on a single Intel Xeon E5540 CPU core, enables 27 times better performance per dollar, and cuts energy usage by a factor of 13. With this improved performance we are able to calculate the equation of state for systems of up to one million hard disks. These large system sizes are required in order to probe the nature of the melting transition, which has been debated for the last forty years. In this paper we present the details of our computational method, and discuss the thermodynamics of hard disks separately in a companion paper.
97 MATHEMATICS AND COMPUTING↗
Development of a CPU/GPU portable software library for Lagrangian–Eulerian simulations of liquid sprays
The Lagrangian–Eulerian method is widely used for simulations of fuel sprays in turbulent combustion because of the advantage of treating the spray droplets as discrete points. One challenge of the Lagrangian–Eulerian method is the intense computational requirement when tracking the large number of Lagrangian particles needed for high fidelity. We have developed a performance-portable library, Grit, to track the Lagrangian particles in parallel on central processing unit (CPU) and graphics processing unit (GPU) accelerated high performance computing (HPC) architectures. Grit is a C++ library which employs Message Passing Interface (MPI) for distributed memory parallelism and Kokkos programming model for on-node shared memory parallelism with performance portability across different architectures of GPUs and multi-core/manycore CPUs. The parallel algorithms, key parallel kernels, and their performances on the pre-exascale supercomputer, Summit, are presented. Grit is coupled with a direct numerical simulation (DNS) solver, S3D, for multiphase simulations. A conservative formulation has been developed and implemented in Grit for phase coupling with thermodynamic consistency. The formulation separates the conservation of mass, momentum and energy from the physical models to prevent accidental violation of conservation laws due to inconsistent models. The formulation also enforces consistent definitions of enthalpies of the fuel for both phases and the latent heat of evaporation. Finally, simulations of turbulent particle-laden flow and the evaporation of dilute turbulent spray jet are performed to verify the software implementation, and to demonstrate the scalability of Grit for large-scale multiphase simulations.
97 MATHEMATICS AND COMPUTING↗
Basker: Parallel sparse LU factorization utilizing hierarchical parallelism and data layouts
Transient simulation in circuit simulation tools, such as SPICE and Xyce, depend on scalable and robust sparse LU factorizations for efficient numerical simulation of circuits and power grids. As the need for simulations of very large circuits grow, the prevalence of multicore architectures enable us to use shared memory parallel algorithms for such simulations. A parallel factorization is a critical component of such shared memory parallel simulations. We develop a parallel sparse factorization algorithm that can solve problems from circuit simulations efficiently, and map well to architectural features. This new factorization algorithm exposes hierarchical parallelism to accommodate irregular structure that arise in our target problems. It also uses a hierarchical two-dimensional data layout which reduces synchronization costs and maps to memory hierarchy found in multicore processors. We present an OpenMP based implementation of the parallel algorithm in a new multithreaded solver called Basker in the Trilinos framework. Here, we present performance evaluations of Basker on the Intel SandyBridge and Xeon Phi platforms using circuit and power grid matrices taken from the University of Florida sparse matrix collection and from Xyce circuit simulation. Basker achieves a geometric mean speedup of 5.91× on CPU (16 cores) and 7.4× on Xeon Phi (32 cores) relative to state-of-the-art solver KLU. Basker outperforms Intel MKL Pardiso solver (PMKL) by as much as 30× on CPU (16 cores) and 7.5× on Xeon Phi (32 cores) for low fill-in circuit matrices. Furthermore, Basker provides 5.4× speedup on a challenging matrix sequence taken from an actual Xyce simulation.
97 MATHEMATICS AND COMPUTING↗
A new augmentation based algorithm for extracting maximal chordal subgraphs
If every cycle of a graph is chordal length greater than three then it contains an edge between non-adjacent vertices. Chordal graphs are of interest both theoretically, since they admit polynomial time solutions to a range of NP-hard graph problems, and practically, since they arise in many applications including sparse linear algebra, computer vision, and computational biology. A maximal chordal subgraph is a chordal subgraph that is not a proper subgraph of any other chordal subgraph. Existing algorithms for computing maximal chordal subgraphs depend on dynamically ordering the vertices, which is an inherently sequential process and therefore limits the algorithms’ parallelizability. In our paper we explore techniques to develop a scalable parallel algorithm for extracting a maximal chordal subgraph. We demonstrate that an earlier attempt at developing a parallel algorithm may induce a non-optimal vertex ordering and is therefore not guaranteed to terminate with a maximal chordal subgraph. We then give a new algorithm that first computes and then repeatedly augments a spanning chordal subgraph. After proving that the algorithm terminates with a maximal chordal subgraph, we then demonstrate that this algorithm is more amenable to parallelization and that the parallel version also terminates with a maximal chordal subgraph. That said, the complexity of the new algorithm is higher than that of the previous parallel algorithm, although the earlier algorithm computes a chordal subgraph which is not guaranteed to be maximal. Finally, we experimented with our augmentation-based algorithm on both synthetic and real-world graphs. We provide scalability results and also explore the effect of different choices for the initial spanning chordal subgraph on both the running time and on the number of edges in the maximal chordal subgraph.
97 MATHEMATICS AND COMPUTING↗
Efficient Scalable Parallel Higher Order Direct MoM-SIE Method With Hierarchically Semiseparable Structures for 3-D Scattering
A novel fast scalable parallel algorithm is proposed for the solution of large 3-D scattering problems based on: 1) the double (geometrical and current-approximation) higher order (DHO) method of moments (MoM) in the surface integral equation (SIE) formulation and 2) a direct solver for dense linear systems utilizing hierarchically semiseparable (HSS) structures. Namely, an HSS matrix representation is used for compression, factorization, and solution of the system matrix. In addition, a rank-revealing QR decomposition for memory compression is used, with a stopping criterion in terms of the relative rank tolerance value. One method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the HSS algorithm accuracy and parallelization. Numerical examples show how the accuracy of the DHO HSS-MoM-SIE method is easily controllable by using the relative tolerance for the matrix compression. Furthermore, the examples demonstrate low memory consumption, as well as much faster simulation time, when compared to the direct LU decomposition. The method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low-order discretizations. Finally, great scalability of the algorithm is demonstrated on more than one thousand processes.
97 MATHEMATICS AND COMPUTING↗
Extreme-Scale Bayesian Inference for Uncertainty Quantification of Complex Simulations
Uncertainty quantification (UQ)—that is, quantifying uncertainties in complex mathematical models and their large-scale computational implementations—is widely viewed as one of the outstanding challenges facing the field of CS&E over the coming decade. The EUREKA project set to address the most difficult class of UQ problems: those for which both the underlying PDE model as well as the uncertain parameters are of extreme scale. In the project we worked on these extreme-scale challenges in the following four areas: 1. Scalable parallel algorithms for sampling and characterizing the posterior distribution that exploit the structure of the underlying PDEs and parameter-to-observable map. These include structure-exploiting versions of the randomized maximum likelihood method, which aims to overcome the intractability of employing conventional MCMC methods for solving extreme-scale Bayesian inversion problems by appealing to and adapting ideas from large-scale PDE-constrained optimization, which have been very successful at exploring high-dimensional spaces. 2. Scalable parallel algorithms for construction of prior and likelihood functions based on learning methods and non-parametric density estimation. Constructing problem-specific priors remains a critical challenge in Bayesian inference, and more so in high dimensions. Another challenge is construction of likelihood functions that capture unmodeled couplings between observations and parameters. We will create parallel algorithms for non-parametric density estimation using high dimensional N-body methods and combine them with supervised learning techniques for the construction of priors and likelihood functions. 3. Bayesian inadequacy models, which augment physics models with stochastic models that represent their imperfections. The success of the Bayesian inference framework depends on the ability to represent the uncertainty due to imperfections of the mathematical model of the phenomena of interest. This is a central challenge in UQ, especially for large-scale models. We propose to develop the mathematical tools to address these challenges in the context of extreme-scale problems. 4. Parallel scalable algorithms for Bayesian optimal experimental design (OED). Bayesian inversion yields quantified uncertainties in the model parameters, which can be propagated forward through the model to yield uncertainty in outputs of interest. This opens the way for designing new experiments to reduce the uncertainties in the model parameters and model predictions. Such experimental design problems have been intractable for large-scale problems using conventional methods; we will create OED algorithms that exploit the structure of the PDE model and the parameter-to-output map to overcome these challenges. Parallel algorithms for these four problems were created, analyzed, prototyped, implemented, tuned, and scaled up for leading-edge supercomputers, including UT-Austin’s own 10 petaflops Stampede system, ANL’s Mira system, and ORNL’s Titan system. While our focus is on fundamental mathematical/computational methods and algorithms, we will assess our methods on model problems derived from several DOE mission applications, including multiscale mechanics and ice sheet dynamics.
97 MATHEMATICS AND COMPUTING↗
Phase locking of a seven-channel continuous wave fibre laser system by a stochastic parallel gradient algorithm
A seven-channel fibre laser system operated by the master oscillator – multichannel power amplifier scheme is the phase locked using a stochastic parallel gradient algorithm. The phase modulators on lithium niobate crystals are controlled by a multichannel electronic unit with the microcontroller processing signals in real time. The dynamic phase locking of the laser system with the bandwidth of 14 kHz is demonstrated, the time of phasing is 3 – 4 ms. (fibre and integrated-optical structures)
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Development of Variational Guiding Center Algorithms for Parallel Calculations in Experimental Magnetic Equilibria
Structure-preserving algorithms obtained via discrete variational principles exhibit strong promise for the calculation of guiding center test particle trajectories. The non-canonical Hamiltonian structure of the guiding center equations forms a novel and challenging context for geometric integration. To demonstrate the practical relevance of these methods, a prototypical variational midpoint algorithm is applied to an experimental magnetic equilibrium. The stability characteristics, conservation properties, and implementation requirements associated with the variational algorithms are addressed. Furthermore, computational run time is reduced for large numbers of particles by parallelizing the calculation on GPU hardware.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Parareal in Time for Dynamic Simulations of Power Systems
In recent years, there have been significant developments in parallel algorithms and high performance parallel computing platforms. Parareal in time algorithm has become popular for long transient simulations (e.g., molecular dynamics, fusion, reacting flows). Parareal is a parallel algorithm which divides the time interval into sub-intervals and solves them concurrently. This paper investigates the applicability of the parareal algorithm to power system dynamic simulations. Preliminary results on the application of parareal for multi-machine power systems are reported in this paper. Two widely used test systems, WECC 3-generator 9-bus system, New England 10-generator 39- bus system, is used to explore the effectiveness of the parareal. Severe 3 phase bus faults are simulated using both the classical and detailed models of multi-machine power systems. Actual Speedup of 5-7 times is observed assuming ideal parallelization. It has been observed that the speedup factors of the order of 20 can be achieved by using fast coarse approximations of power system models. Dependency of parareal convergence on fault duration and location has been observed.
Gurrala, Gurunath↗
Phasing of seven-channel fibre laser radiation with dynamic turbulent phase distortions using a stochastic parallel gradient algorithm at a bandwidth of 450 kHz
We have demonstrated an experimental setup for the coherent phasing of a seven-channel fibre laser system ( λ = 1064 nm) in a scheme comprising a master oscillator and a set of parallel amplifiers with lithium niobate-based fibre-optic phase modulators. Using a stochastic parallel gradient algorithm, an instrumental phase modulator control unit ensures a bandwidth of the system up to 450 kHz. The effectiveness of phasing of light transmitted through a turbulent medium with a characteristic time scale τ{sub turb} has been studied experimentally as a function of phasing time τ{sub ph}. The results demonstrate that the average Strehl ratio begins to rise at τ{sub turb}/τ{sub ph} ⩾ 2 and that the effectiveness of compensation for dynamic phase distortions in the beam propagation path rises sharply at τ{sub turb}/τ{sub ph} ≈ 20. For τ{sub turb}/τ{sub ph} ⩾ 30 – 40, the average Strehl ratio remains constant at the level reached. (control of laser radiation parameters)
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Theoretically and practically efficient parallel nucleus decomposition
This paper studies the nucleus decomposition problem, which has been shown to be useful in finding dense substructures in graphs. We present a novel parallel algorithm that is efficient both in theory and in practice. Our algorithm achieves a work complexity matching the best sequential algorithm while also having low depth (parallel running time), which significantly improves upon the only existing parallel nucleus decomposition algorithm (Sariyüce et al. , PVLDB 2018). The key to the theoretical efficiency of our algorithm is a new lemma that bounds the amount of work done when peeling cliques from the graph, combined with the use of a theoretically-efficient parallel algorithms for clique listing and bucketing. We introduce several new practical optimizations, including a new multi-level hash table structure to store information on cliques space-efficiently and a technique for traversing this structure cache-efficiently. On a 30-core machine with two-way hyper-threading on real-world graphs, we achieve up to a 55x speedup over the state-of-the-art parallel nucleus decomposition algorithm by Sariyüce et al. , and up to a 40x self-relative parallel speedup. We are able to efficiently compute larger nucleus decompositions than prior work on several million-scale graphs for the first time.
Computer Science↗
Scheduling and Performance of Asynchronous Tasks in Fortran 2018 with FEATS
Most parallel scientific programs contain compiler directives (pragmas) such as those from OpenMP (Hermanns in Parallel programming in Fortran 95 using openMP, 2002. School of Aeronautical Engineering, Universidad Politécnica de Madrid, España, 2011), explicit calls to runtime library procedures such as those implementing the Message Passing Interface (MPI) (in A message-passing interface standard version 4.0, 2021. https://www.mpi-forum.org/docs/mpi-4.0/mpi40-report.pdf), or compiler-specific language extensions such as those provided by CUDA (Ruetsch and Fatica in CUDA Fortran for scientists and engineers: best practices for efficient CUDA Fortran programming, Elsevier, 2013). By contrast, the recent Fortran standards empower developers to express parallel algorithms without directly referencing lower-level parallel programming models (Numrich in Parallel programming with co-arrays, CRC Press, 2018, and Curcic in Modern Fortran: building efficient parallel applications, Manning Publications, 2020). Fortran’s parallel features place the language within the Partitioned Global Address Space (PGAS) class of programming models. When writing programs that exploit data parallelism, application developers often find it straightforward to develop custom parallel algorithms. Problems involving complex, heterogeneous, staged calculations, however, pose much greater challenges. Such applications require careful coordination of tasks in a manner that respects dependencies prescribed by a directed acyclic graph. When rolling one’s own solution proves difficult, extending a customizable framework becomes attractive. Further, the paper presents the design, implementation, and use of the Framework for Extensible Asynchronous Task Scheduling (FEATS), which we believe to be the first task scheduling tool written in modern Fortran. We describe the benefits and compromises associated with choosing Fortran as the implementation language, and we propose ways in which future Fortran standards can best support the use case in this paper.
97 MATHEMATICS AND COMPUTING↗