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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

High-Fidelity Multiphysics Modeling of a Heat Pipe Microreactor Using BlueCrab

Researchers who are actively developing nuclear microreactors are planning to employ innovative designs and features using traditional commercial modeling tools that may be inadequate for their design and licensing activities. The codes developed under the U.S. Department of Energy Office of Nuclear Energy Advanced Modeling and Simulation (NEAMS) program provide flexibility in terms of geometry modeling and multiphysics coupling and are particularly well suited for modeling novel microreactor concepts. To test the maturity of these codes, this paper introduces a conceptual heat pipe microreactor (HP-MR) designed to gather various technologies of interest to microreactor developers such as control drums, heat pipes, and hydride moderators. Here, the objective of this effort is to demonstrate NEAMS tools capability to perform high-fidelity multiphysics simulations, using coupled neutronics (via the Griffin code), heat conduction (via the BISON code), heat pipe modeling (via the Sockeye code), and hydrogen redistribution in hydride metal moderator (via the SWIFT code). Codes are coupled in-memory through the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework, which permits flexible multiphysics data transfer schemes. The analysis confirmed two key aspects of the HP-MR concept: (1) its ability to follow the power load requested from the heat pipe and (2) its ability to avoid heat pipe cascading failure unless designed with high power close to operating failure limits of its heat pipes. The developed computational model was distributed publicly on the Virtual Test Bed for training purposes to accelerate adoption by industry and to provide a high-fidelity multiphysics solution for benchmarking against other tools. Additional multiphysics analyses including other transients and coupled physics were identified as necessary future work, together with a focus on validating multiphysics behavior against experiments.

Microreactor↗
DOE OSTI2020

GronOR: Massively Parallel and GPU-Accelerated Non-Orthogonal Configuration Interaction for Large Molecular Systems

GronOR is a program package for non-orthogonal configuration interaction calculations for an electronic wave function built in terms of anti-symmetrized products of multi-configuration molecular fragment wave functions. The two-electron integrals that have to be processed may be expressed in terms of atomic orbitals or in terms of an orbital basis determined from the molecular orbitals of the fragments. The code has been specifically designed for execution on distributed memory massively parallel and Graphics Processing Unit (GPU)-accelerated computer architectures, using an MPI+OpenACC/OpenMP programming approach. The task-based execution model used in the implementation allows for linear scaling with the number of nodes on the largest pre-exascale architectures available, provides hardware fault resiliency, and enables effective execution on systems with distinct central processing unit-only and GPU-accelerated partitions. The code interfaces with existing multi-configuration electronic structure codes that provide optimized molecular fragment orbitals, configuration interaction coefficients, and the required integrals. Algorithm and implementation details, parallel and accelerated performance benchmarks, and an analysis of the sensitivity of the accuracy of results and computational performance to thresholds used in the calculations are presented.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Performance Characteristics of the BlueField-2 SmartNIC

High-performance computing (HPC) researchers have long envisioned scenarios where application workflows could be improved through the use of programmable processing elements embedded in the network fabric. Recently, vendors have introduced programmable Smart Network Interface Cards (SmartNICs) that enable computations to be offloaded to the edge of the network. There is great interest in both the HPC and high-performance data analytics (HPDA) communities in understanding the roles these devices may play in the data paths of upcoming systems. This paper focuses on characterizing both the networking and computing aspects of NVIDIA’s new BlueField-2 SmartNIC when used in a 100Gb/s Ethernet environment. For the networking evaluation we conducted multiple transfer experiments between processors located at the host, the SmartNIC, and a remote host. These tests illuminate how much effort is required to saturate the network and help estimate the processing headroom available on the SmartNIC during transfers. For the computing evaluation we used the stress-ng benchmark to compare the BlueField-2 to other servers and place realistic bounds on the types of offload operations that are appropriate for the hardware. Our findings from this work indicate that while the BlueField-2 provides a flexible means of processing data at the network’s edge, great care must be taken to not overwhelm the hardware. While the host can easily saturate the network link, the SmartNIC’s embedded processors may not have enough computing resources to sustain more than half the expected bandwidth when using kernel-space packet processing. From a computational perspective, encryption operations, memory operations under contention, and on-card IPC operations on the SmartNIC perform significantly better than the general-purpose servers used for comparisons in our experiments. Therefore, applications that mainly focus on these operations may be good candidates for offloading to the SmartNIC.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Scaling Semantic Graph Databases in Size and Performance

In this paper we present SGEM, a full software system for accelerating large-scale semantic graph databases on commodity clusters. Unlike current approaches, SGEM addresses semantic graph databases by only employing graph methods at all the levels of the stack. On one hand, this allows exploiting the space efficiency of graph data structures and the inherent parallelism of graph algorithms. These features adapt well to the increasing system memory and core counts of modern commodity clusters. On the other hand, however, these systems are optimized for regular computation and batched data transfers, while graph methods usually are irregular and generate fine-grained data accesses with poor spatial and temporal locality. Our framework comprises a SPARQL to data parallel C compiler, a library of parallel graph methods and a custom, multithreaded runtime system. We introduce our stack, motivate its advantages with respect to other solutions and show how we solved the challenges posed by irregular behaviors. We present the result of our software stack on the Berlin SPARQL benchmarks with datasets up to 10 billion triples (a triple corresponds to a graph edge), demonstrating scaling in dataset size and in performance as more nodes are added to the cluster.

Aggregation↗
NASA NTRS2000

Automatic Data Traffic Control on DSM Architecture

We study data traffic on distributed shared memory machines and conclude that data placement and grouping improve performance of scientific codes. We present several methods which user can employ to improve data traffic in his code. We report on implementation of a tool which detects the code fragments causing data congestions and advises user on improvements of data routing in these fragments. The capabilities of the tool include deduction of data alignment and affinity from the source code; detection of the code constructs having abnormally high cache or TLB misses; generation of data placement constructs. We demonstrate the capabilities of the tool on experiments with NAS parallel benchmarks and with a simple computational fluid dynamics application ARC3D.

Frumkin, Michael↗
DOE OSTI2020

The Locus Algorithm IV: Performance metrics of a grid computing system used to create catalogues of optimised pointings

This paper discusses the requirements for and performance metrics of the the Grid Computing system used to implement the Locus Algorithm to identify optimum pointings for differential photometry of 61,662,376 stars and 23,779 quasars. Initial operational tests indicated a need for a software system to analyse the data and a High Performance Computing system to run that software in a scalable manner. Practical assessments of the performance of the software in a serial computing environment were used to provide a benchmark against which the performance metrics of the HPC solution could be compared, as well as to indicate any bottlenecks in performance. These performance metrics indicated a distinct split in the performance dictated more by differences in the input data than by differences in the design of the systems used. This indicates a need for experimental analysis of system performance, and suggests that algorithmic complexity analyses may lead to incorrect or naive conclusions, especially in systems with high data I/O overhead such as grid computing. Further, it implies that systems which reduce or eliminate this bottleneck such as in-memory processing could lead to a substantial increase in performance.

Creaner, Oisín↗
NASA NTRS2006

A History of High-Performance Computing

Faster than most speedy computers. More powerful than its NASA data-processing predecessors. Able to leap large, mission-related computational problems in a single bound. Clearly, it s neither a bird nor a plane, nor does it need to don a red cape, because it s super in its own way. It's Columbia, NASA s newest supercomputer and one of the world s most powerful production/processing units. Named Columbia to honor the STS-107 Space Shuttle Columbia crewmembers, the new supercomputer is making it possible for NASA to achieve breakthroughs in science and engineering, fulfilling the Agency s missions, and, ultimately, the Vision for Space Exploration. Shortly after being built in 2004, Columbia achieved a benchmark rating of 51.9 teraflop/s on 10,240 processors, making it the world s fastest operational computer at the time of completion. Putting this speed into perspective, 20 years ago, the most powerful computer at NASA s Ames Research Center, home of the NASA Advanced Supercomputing Division (NAS), ran at a speed of about 1 gigaflop (one billion calculations per second). The Columbia supercomputer is 50,000 times faster than this computer and offers a tenfold increase in capacity over the prior system housed at Ames. What s more, Columbia is considered the world s largest Linux-based, shared-memory system. The system is offering immeasurable benefits to society and is the zenith of years of NASA/private industry collaboration that has spawned new generations of commercial, high-speed computing systems.

Source record↗
DOE OSTI2025

A high-order computational framework for particle-resolved simulations of disperse multiphase flows

This work presents a high-order numerical approach for particle-resolved simulations of disperse multiphase flows, where the Navier-Stokes equations for fluid flow are solved using a high-order spectral element method in the Eulerian framework, and the particle phase is directly simulated with a discrete element method. The coupling between particles and fluids is explicitly handled using an adapted direct-forcing immersed boundary method. Unlike the conventional schemes, a high-order barycentric Lagrange interpolation method and a Gaussian projection kernel are used to ensure accurate momentum exchange between local boundary points and surrounding fluid nodes in the framework of high-order fluid solver. Benchmark tests of increasing complexity are conducted to demonstrate the accuracy and efficiency of our method. Here, it is found that our approach exhibits an excellent convergence performance, as the fluid element/grid is refined and the number of boundary points increases. Compared to conventional low-order methods, the proposed high-order framework enables the use of substantially larger fluid elements while maintaining high accuracy in modeling fluid-particle interactions, owing to the enhanced resolution of high-order basis functions. Moreover, since the primary unknowns are stored at element or grid nodes, the high-order approach offers improved efficiency in both CPU memory usage and total computational cost.

42 ENGINEERING↗
NASA NTRS2001

Automation of Data Traffic Control on DSM Architecture

The design of distributed shared memory (DSM) computers liberates users from the duty to distribute data across processors and allows for the incremental development of parallel programs using, for example, OpenMP or Java threads. DSM architecture greatly simplifies the development of parallel programs having good performance on a few processors. However, to achieve a good program scalability on DSM computers requires that the user understand data flow in the application and use various techniques to avoid data traffic congestions. In this paper we discuss a number of such techniques, including data blocking, data placement, data transposition and page size control and evaluate their efficiency on the NAS (NASA Advanced Supercomputing) Parallel Benchmarks. We also present a tool which automates the detection of constructs causing data congestions in Fortran array oriented codes and advises the user on code transformations for improving data traffic in the application.

Frumkin, Michael↗
NASA NTRS2003

Support of Multidimensional Parallelism in the OpenMP Programming Model

OpenMP is the current standard for shared-memory programming. While providing ease of parallel programming, the OpenMP programming model also has limitations which often effect the scalability of applications. Examples for these limitations are work distribution and point-to-point synchronization among threads. We propose extensions to the OpenMP programming model which allow the user to easily distribute the work in multiple dimensions and synchronize the workflow among the threads. The proposed extensions include four new constructs and the associated runtime library. They do not require changes to the source code and can be implemented based on the existing OpenMP standard. We illustrate the concept in a prototype translator and test with benchmark codes and a cloud modeling code.

Jin, Hao-Qiang↗
DOE OSTI2025

RingX: Scalable Parallel Attention for Long-Context Learning on HPC

The attention mechanism has become foundational for remarkable AI breakthroughs since the introduction of the Transformer, driving the demand for increasingly longer context to power frontier models such as large-scale reasoning language models and high-resolution image/video generators. However, its quadratic computational and memory complexities present substantial challenges. Current state-of-the-art parallel attention methods, such as ring attention, are widely adopted for long-context training but utilize a point-to-point communication strategy that fails to fully exploit the capabilities of modern HPC network architectures. In this work, we propose ringX, a scalable family of parallel attention methods optimized explicitly for HPC systems. By enhancing workload partitioning, refining communication patterns, and improving load balancing, ringX achieves up to 3.4 × speedup compared to conventional ring attention on the Frontier supercomputer. Optimized for both bi-directional and causal attention mechanisms, ringX demonstrates its effectiveness through training benchmarks of a Vision Transformer (ViT) on a climate dataset and a Generative Pre-Trained Transformer (GPT) model, Llama3 8B. Our method attains an end-to-end training speedup of approximately 1.5 × in both scenarios. To our knowledge, the achieved 38% model FLOPs utilization (MFU) for training Llama3 8B with a 1M-token sequence length on 4,096 GPUs represents one of the highest training efficiencies reported for long-context learning on HPC systems. Our code implementation is available at https://github.com/jqyin/ringX-attention.

Yin, Junqi [ORNL] (ORCID:0000000338435520)↗
NASA NTRS1993

Efficiently modeling neural networks on massively parallel computers

Neural networks are a very useful tool for analyzing and modeling complex real world systems. Applying neural network simulations to real world problems generally involves large amounts of data and massive amounts of computation. To efficiently handle the computational requirements of large problems, we have implemented at Los Alamos a highly efficient neural network compiler for serial computers, vector computers, vector parallel computers, and fine grain SIMD computers such as the CM-2 connection machine. This paper describes the mapping used by the compiler to implement feed-forward backpropagation neural networks for a SIMD (Single Instruction Multiple Data) architecture parallel computer. Thinking Machines Corporation has benchmarked our code at 1.3 billion interconnects per second (approximately 3 gigaflops) on a 64,000 processor CM-2 connection machine (Singer 1990). This mapping is applicable to other SIMD computers and can be implemented on MIMD computers such as the CM-5 connection machine. Our mapping has virtually no communications overhead with the exception of the communications required for a global summation across the processors (which has a sub-linear runtime growth on the order of O(log(number of processors)). We can efficiently model very large neural networks which have many neurons and interconnects and our mapping can extend to arbitrarily large networks (within memory limitations) by merging the memory space of separate processors with fast adjacent processor interprocessor communications. This paper will consider the simulation of only feed forward neural network although this method is extendable to recurrent networks.

Farber, Robert M.↗
NASA NTRS2000

Incremental Parallelization of Non-Data-Parallel Programs Using the Charon Message-Passing Library

Message passing is among the most popular techniques for parallelizing scientific programs on distributed-memory architectures. The reasons for its success are wide availability (MPI), efficiency, and full tuning control provided to the programmer. A major drawback, however, is that incremental parallelization, as offered by compiler directives, is not generally possible, because all data structures have to be changed throughout the program simultaneously. Charon remedies this situation through mappings between distributed and non-distributed data. It allows breaking up the parallelization into small steps, guaranteeing correctness at every stage. Several tools are available to help convert legacy codes into high-performance message-passing programs. They usually target data-parallel applications, whose loops carrying most of the work can be distributed among all processors without much dependency analysis. Others do a full dependency analysis and then convert the code virtually automatically. Even more toolkits are available that aid construction from scratch of message passing programs. None, however, allows piecemeal translation of codes with complex data dependencies (i.e. non-data-parallel programs) into message passing codes. The Charon library (available in both C and Fortran) provides incremental parallelization capabilities by linking legacy code arrays with distributed arrays. During the conversion process, non-distributed and distributed arrays exist side by side, and simple mapping functions allow the programmer to switch between the two in any location in the program. Charon also provides wrapper functions that leave the structure of the legacy code intact, but that allow execution on truly distributed data. Finally, the library provides a rich set of communication functions that support virtually all patterns of remote data demands in realistic structured grid scientific programs, including transposition, nearest-neighbor communication, pipelining, gather/scatter, and redistribution. At the end of the conversion process most intermediate Charon function calls will have been removed, the non-distributed arrays will have been deleted, and virtually the only remaining Charon functions calls are the high-level, highly optimized communications. Distribution of the data is under complete control of the programmer, although a wide range of useful distributions is easily available through predefined functions. A crucial aspect of the library is that it does not allocate space for distributed arrays, but accepts programmer-specified memory. This has two major consequences. First, codes parallelized using Charon do not suffer from encapsulation; user data is always directly accessible. This provides high efficiency, and also retains the possibility of using message passing directly for highly irregular communications. Second, non-distributed arrays can be interpreted as (trivial) distributions in the Charon sense, which allows them to be mapped to truly distributed arrays, and vice versa. This is the mechanism that enables incremental parallelization. In this paper we provide a brief introduction of the library and then focus on the actual steps in the parallelization process, using some representative examples from, among others, the NAS Parallel Benchmarks. We show how a complicated two-dimensional pipeline-the prototypical non-data-parallel algorithm- can be constructed with ease. To demonstrate the flexibility of the library, we give examples of the stepwise, efficient parallel implementation of nonlocal boundary conditions common in aircraft simulations, as well as the construction of the sequence of grids required for multigrid.

VanderWijngaart, Rob F.↗
DOE OSTI2019

Nonintrusive reduced order modeling framework for quasigeostrophic turbulence

In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed in this work exploits the time series prediction capability of long short-term memory (LSTM) recurrent neural network architecture such that (1) in the training phase, the LSTM model is trained on the modal coefficients extracted from the high-resolution data snapshots using proper orthogonal decomposition (POD) transform, and (2) in the testing phase, the trained model predicts the modal coefficients for the total time recursively based on the initial time history. Hence, no prior information about the underlying governing equations is required to generate the ROM. To illustrate the predictive performance of the proposed framework, the mean flow fields and time series response of the field values are reconstructed from the predicted modal coefficients by using an inverse POD transform. As a representative benchmark test case, we consider a two-dimensional quasigeostrophic ocean circulation model which, in general, displays an enormous range of fluctuating spatial and temporal scales. We first demonstrate that the conventional Galerkin projection-based reduced order modeling of such systems requires a high number of POD modes to obtain a stable flow physics. In addition, ROM-Galerkin projection (ROM-GP) does not seem to capture the intermittent bursts appearing in the dynamics of the first few most energetic modes. However, the proposed nonintrusive ROM framework based on LSTM (ROM-LSTM) yields a stable solution even for a small number of POD modes. We also observe that the ROM-LSTM model is able to capture quasiperiodic intermittent bursts accurately, and yields a stable and accurate mean flow dynamics using the time history of a few previous time states, denoted as the lookback time window in this paper. We show several features of ROM-LSTM framework such as significantly higher accuracy than ROM-GP, and faster performance using larger time step size. Throughout the paper, we demonstrate our findings in terms of time series evolution of the field values and mean flow patterns, which suggest that the proposed fully nonintrusive ROM framework is robust and capable of predicting chaotic nonlinear fluid flows in an extremely efficient way compared to the conventional projection-based ROM framework.

42 ENGINEERING↗
DOE OSTI2014

Comparing the Predicted Neutron Yields and Radioactivities of Two Accelerators in Taiwan: 235-MeV Proton Cyclotron and 3-GeV Electron Synchrotron - Paper 27

Two major accelerator facilities in Taiwan are about to open. The first proton treatment center equipped with a 235 MeV proton cyclotron and four gantry rooms in Linkou Chang Gung Memorial Hospital is currently under beam commissioning and clinical trials are expected to start soon. In addition, the second light source facility, named Taiwan Photon Source (TPS), in National Synchrotron Radiation Research Center is now approaching its final assembly and will be ready for beam commissioning in this year. These two facilities mark significant milestones in accelerator development and application of Taiwan. Operation of these accelerators could potentially produce intense secondary neutrons and serious material activation. Both issues are crucial and must be carefully evaluated for the purpose of radiation safety, especially in a densely populated hospital or in a crowded user experimental floor. Monte Carlo simulations with evaluated cross sections or benchmarked nuclear models are considered as the most reliable method for accelerator shielding designs or related analyses. This study predicted and compared the magnitudes of potential neutron production and induced radioactivity due to the operation of these two accelerators. For conservative consideration, the primary particle energy and full beam output were assumed to hit a thick target for maximum radiation production. In order to ensure the quality of predictions, much effort has been devoted to associated benchmark calculations, targeting those bombardment experiments involving with the same particle types and similar beam energies and targets. (authors)

43 PARTICLE ACCELERATORS↗
DOE OSTI2015

Latent uncertainties of the precalculated track Monte Carlo method

Purpose: While significant progress has been made in speeding up Monte Carlo (MC) dose calculation methods, they remain too time-consuming for the purpose of inverse planning. To achieve clinically usable calculation speeds, a precalculated Monte Carlo (PMC) algorithm for proton and electron transport was developed to run on graphics processing units (GPUs). The algorithm utilizes pregenerated particle track data from conventional MC codes for different materials such as water, bone, and lung to produce dose distributions in voxelized phantoms. While PMC methods have been described in the past, an explicit quantification of the latent uncertainty arising from the limited number of unique tracks in the pregenerated track bank is missing from the paper. With a proper uncertainty analysis, an optimal number of tracks in the pregenerated track bank can be selected for a desired dose calculation uncertainty. Methods: Particle tracks were pregenerated for electrons and protons using EGSnrc and GEANT4 and saved in a database. The PMC algorithm for track selection, rotation, and transport was implemented on the Compute Unified Device Architecture (CUDA) 4.0 programming framework. PMC dose distributions were calculated in a variety of media and compared to benchmark dose distributions simulated from the corresponding general-purpose MC codes in the same conditions. A latent uncertainty metric was defined and analysis was performed by varying the pregenerated track bank size and the number of simulated primary particle histories and comparing dose values to a “ground truth” benchmark dose distribution calculated to 0.04% average uncertainty in voxels with dose greater than 20% of D{sub max}. Efficiency metrics were calculated against benchmark MC codes on a single CPU core with no variance reduction. Results: Dose distributions generated using PMC and benchmark MC codes were compared and found to be within 2% of each other in voxels with dose values greater than 20% of the maximum dose. In proton calculations, a small (≤1 mm) distance-to-agreement error was observed at the Bragg peak. Latent uncertainty was characterized for electrons and found to follow a Poisson distribution with the number of unique tracks per energy. A track bank of 12 energies and 60000 unique tracks per pregenerated energy in water had a size of 2.4 GB and achieved a latent uncertainty of approximately 1% at an optimal efficiency gain over DOSXYZnrc. Larger track banks produced a lower latent uncertainty at the cost of increased memory consumption. Using an NVIDIA GTX 590, efficiency analysis showed a 807 × efficiency increase over DOSXYZnrc for 16 MeV electrons in water and 508 × for 16 MeV electrons in bone. Conclusions: The PMC method can calculate dose distributions for electrons and protons to a statistical uncertainty of 1% with a large efficiency gain over conventional MC codes. Before performing clinical dose calculations, models to calculate dose contributions from uncharged particles must be implemented. Following the successful implementation of these models, the PMC method will be evaluated as a candidate for inverse planning of modulated electron radiation therapy and scanned proton beams.

07 ISOTOPE AND RADIATION SOURCES↗
NASA NTRS2006

Programs for Testing Processor-in-Memory Computing Systems

The Multithreaded Microbenchmarks for Processor-In-Memory (PIM) Compilers, Simulators, and Hardware are computer programs arranged in a series for use in testing the performances of PIM computing systems, including compilers, simulators, and hardware. The programs at the beginning of the series test basic functionality; the programs at subsequent positions in the series test increasingly complex functionality. The programs are intended to be used while designing a PIM system, and can be used to verify that compilers, simulators, and hardware work correctly. The programs can also be used to enable designers of these system components to examine tradeoffs in implementation. Finally, these programs can be run on non-PIM hardware (either single-threaded or multithreaded) using the POSIX pthreads standard to verify that the benchmarks themselves operate correctly. [POSIX (Portable Operating System Interface for UNIX) is a set of standards that define how programs and operating systems interact with each other. pthreads is a library of pre-emptive thread routines that comply with one of the POSIX standards.]

Katz, Daniel S.↗
DOE OSTI2014

HPGMG

HPGMG is a scalable reimplementation of miniGMG (see previous software disclosure) and is a compact geometric multigrid (MG benchmark designed to proxy the performance characteristics of the solves found in adaptive mesh refinement multigrid (AMR MG) applications. It solves the equation a*alpha*u - b div beta grad u = f on a large cubical domain using cell-centered values and either homogenous dirichlet or periodic boundaries. a and b are scalar constants, alpha and beta are space varying constants, f is the right hand side, and u is the solution. The cubical domain is divided into cubical subdomains which are distributed across the supercomputer. Whereas miniGMG enforced a static spatial decomposition across all levels in the MG V-cycle, HPGMG is free to redistribute data at each level including the possibility of moving all data to a single process (memory permitting). Thus, where miniGMG's U-cycle demanded a coarse grid solve across the entire supercomputer (a performance bottleneck), HPGMG can avoid this by combining distributed data and continuing the V-cycle. Technically, HPGMG can still implement a U-cycle or a Krylov bottom solve, but this solve can now be done on a global problem of only 2^3-8^3 cells. Additionally, HPGMG supports the MG F-cycle which avoids the iterative process of the v-cycle with a direct approach that can reach the discretization error in one pass. This provides a substantial performance boosts and opens avenues for exploiting systems that include hierarchical memories or heterogeneous processors. Like miniGMG, the righthand side f in HPGMG is generated from a continuous function for u. Thus, one can solve.

Williams, Samuel↗