Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Parallel Computing”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,495 records · Page 83

An active learning high-throughput microstructure calibration framework for solving inverse structure–process problems in materials informatics

Determining a process–structure–property relationship is the holy grail of materials science, where both computational prediction in the forward direction and materials design in the inverse direction are essential. Problems in materials design are often considered in the context of process–property linkage by bypassing the materials structure, or in the context of structure–property linkage as in microstructure-sensitive design problems. However, there is a lack of research effort in studying materials design problems in the context of process–structure linkage, which has a great implication in reverse engineering. In this paper, given a target microstructure, we propose an active learning high-throughput microstructure calibration framework to derive a set of processing parameters, which can produce an optimal microstructure that is statistically equivalent to the target microstructure. The proposed framework is formulated as a noisy multi-objective optimization problem, where each objective function measures a deterministic or statistical difference of the same microstructure descriptor between a candidate microstructure and a target microstructure. Furthermore, to significantly reduce the physical waiting wall-time, we enable the high-throughput feature of the microstructure calibration framework by adopting an asynchronously parallel Bayesian optimization to exploit high-performance computing resources. Case studies in additive manufacturing and grain growth are used to demonstrate the applicability of the proposed framework, where kinetic Monte Carlo (kMC) simulation is used as a forward predictive model, such that for a given target microstructure, the target processing parameters that produced this microstructure are successfully recovered.

36 MATERIALS SCIENCE↗

PyOMP: Multithreaded Parallel Programming in Python

We know that Python is a widely used language in scientific computing. When the goal is high performance, however, Python lags far behind low-level languages such as C and Fortran. To support applications that stress performance, Python needs to access the full capabilities of modern CPUs. That means support for parallel multithreading. In this paper, we describe PyOMP, a system that enables OpenMP in Python. Programmers write code in Python with OpenMP, Numba generates code that compiles to LLVM, and the resulting programs run with performance that approaches that from code written with C and OpenMP. In this paper we provide an update on the PyOMP project and explain how to install it and use it to write parallel multithreaded code in Python.

97 MATHEMATICS AND COMPUTING↗

Position Papers for the ASCR Workshop on the Management and Storage of Scientific Data

The purpose of this workshop is to identify priority research directions in the area of data management for high-performance and scientific computing above and beyond HPC’s traditional "the parallel file system is the data-management system" model. Supporting the breadth of the DOE mission, including the explosion of AI uses and the growing needs of experimental and observational science, motivates revisiting our assumptions about data management. There are many facets of this topic to explore including: (1) Interfaces for accessing data that resides on traditional persistent storage as well as memory devices; (2) Storage-system architecture design that supports scientific workflows on varied hierarchical storage and networking devices; (3) Devising metadata management infrastructure to support FAIR principles (Findability, Accessibility, Interoperability, and Reusability); (4) Capturing provenance information about scientific data; (5) Utilizing AI to learn I/O patterns of emerging workloads for efficient data management; (6) Providing data management support for AI and complex workflows; and (7) Understanding the overlap between traditional storage systems and I/O (SSIO) efforts and data management. While the program committee has identified these topics as important areas for discussion, we welcome position papers from the community that propose additional topics of interest for discussion at the workshop. The workshop agenda will include breakout sessions for discussing these and selected topic areas to inform priority research directions for data management for high-performance and scientific computing.

97 MATHEMATICS AND COMPUTING↗

Report for the ASCR Workshop on the Management and Storage of Scientific Data

The purpose of this workshop is to identify priority research directions in the area of data management for high-performance and scientific computing above and beyond HPC’s traditional "the parallel file system is the data-management system" model. Supporting the breadth of the DOE mission, including the explosion of AI uses and the growing needs of experimental and observational science, motivates revisiting our assumptions about data management. There are many facets of this topic to explore including: (1) Interfaces for accessing data that resides on traditional persistent storage as well as memory devices; (2) Storage-system architecture design that supports scientific workflows on varied hierarchical storage and networking devices; (3) Devising metadata management infrastructure to support FAIR principles (Findability, Accessibility, Interoperability, and Reusability); (4) Capturing provenance information about scientific data; (5) Utilizing AI to learn I/O patterns of emerging workloads for efficient data management; (6) Providing data management support for AI and complex workflows; and (7) Understanding the overlap between traditional storage systems and I/O (SSIO) efforts and data management. While the program committee has identified these topics as important areas for discussion, we welcome position papers from the community that propose additional topics of interest for discussion at the workshop. The workshop agenda will include breakout sessions for discussing these and selected topic areas to inform priority research directions for data management for high-performance and scientific computing.

97 MATHEMATICS AND COMPUTING↗

Hybrid RANS-LES of the Atmospheric Boundary Layer for Wind Farm Simulations: Preprint

Wind farm simulations often do not accurately represent wake-atmospheric boundary layer (ABL) interactions, blade boundary layer (BL) dynamics, and turbine-turbine interactions. In this work, we use Active Model Split (AMS), a new hybrid Reynolds-Averaged Navier Stokes (RANS)-large eddy simulation (LES) model, which is well suited to capture these effects because the model can (i) accurately simulate the ABL with the Coriolis effect, (ii) is accurate in adverse pressure gradients such as those near wind turbine blades, and (iii) has sufficiently low computational cost to simulate multiple turbines while resolving the blade BL. For simplicity and consistency we develop AMS to be used throughout the domain rather than in a zonal method. We implement our work in the massively parallel flow solver, Nalu-Wind, so that our model can access the compute resources needed for blade-resolved simulations of multiple wind turbines. To accomplish these aims, we modify the baseline AMS by changing the RANS contribution to SST k - omega with a length scale limiter, adding the Coriolis effect, and developing an appropriate wall treatment. We show that AMS of the ABL with the Coriolis effect matches LES reference results better than those obtained with RANS. We describe our plans to add buoyancy effects and wind turbines to our AMS simulations.

atmospheric boundary layer↗

Moment representation in the lattice Boltzmann method on massively parallel hardware

The widely-used lattice Boltzmann method (LBM) for computational fluid dynamics is highly scalable, but also significantly memory bandwidth-bound on current architectures. This paper presents a new regularized LBM implementation that reduces the memory footprint by only storing macroscopic, moment-based data. We show that the amount of data that must be stored in memory during a simulation is reduced by up to 47%. We also present a technique for cache-aware data re-utilization and show that optimizing cache utilization to limit data motion results in a similar improvement in time to solution. These new algorithms are implemented in the hemodynamics solver HARVEY and demonstrated using both idealized and realistic biological geometries. We develop a performance model for the moment representation algorithm and evaluate the performance on Summit.

Vardhan, Madhurima↗

Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks

Large-scale multiphysics simulations are computationally challenging due to the coupling of multiple processes with widely disparate time scales. The advent of exascale computing systems exacerbates these challenges since these systems enable ever-increasing size and complexity. In recent years, there has been renewed interest in developing multirate methods as a means to handle the large range of time scales, as these methods may afford greater accuracy and efficiency than more traditional approaches of using implicit-explicit (IMEX) and low-order operator splitting schemes. However, to date there have been few performance studies that compare different classes of multirate integrators on complex application problems. In this work, we study the performance of several newly developed multirate infinitesimal (MRI) methods, implemented in the SUNDIALS solver package, on two reacting flow model problems built on structured mesh frameworks. The first model revisits prior work on a compressible reacting flow problem with complex chemistry that is implemented using BoxLib but where we now include comparisons between a new explicit MRI scheme with the multirate spectral deferred correction (SDC) methods in the original paper. The second problem uses the same complex chemistry as the first problem, combined with a simplified flow model, but runs at a large spatial scale where explicit methods become infeasible due to stability constraints. Two recently developed IMEX MRI multirate methods are tested. These methods rely on advanced features of the AMReX framework on which the model is built, such as multilevel grids and multilevel preconditioners. The results from these two problems show that MRI multirate methods can offer significant performance benefits on complex multiphysics application problems and that these methods may be combined with advanced spatial discretization to compound the advantages of both.

97 MATHEMATICS AND COMPUTING↗

PythonFOAM: In-situ data analyses with OpenFOAM and Python

Here, we outline the development of a general-purpose Python-based data analysis tool for OpenFOAM. Our implementation relies on the construction of OpenFOAM applications that have bindings to data analysis libraries in Python. Double precision data in OpenFOAM is cast to a NumPy array using the NumPy C-API and Python modules may then be used for arbitrary data analysis and manipulation on flow-field information. We highlight how the proposed wrapper may be used for an in-situ online singular value decomposition (SVD) implemented in Python and accessed from the OpenFOAM solver PimpleFOAM. Here, 'in-situ' refers to a programming paradigm that allows for a concurrent computation of the data analysis on the same computational resources utilized for the partial differential equation solver. In addition, to demonstrate parallel deployments, we deploy a distributed SVD, which collects snapshot data across the ranks of a distributed simulation to compute the global left singular vectors. Crucially, both OpenFOAM and Python share the same message passing interface (MPI) communicator for this deployment which allows Python objects and functions to exchange NumPy arrays across ranks. Subsequently, we provide scaling assessments of this distributed SVD on multiple nodes of Intel Broadwell and KNL architectures for canonical test cases such as the large eddy simulations of a backward facing step and a channel flow at friction Reynolds number of 395. Finally, we demonstrate the deployment of a deep neural network for compressing the flow-field information using an autoencoder to demonstrate an ability to use state-of-the-art machine learning tools in the Python ecosystem.

97 MATHEMATICS AND COMPUTING↗

Track reconstruction as a service for collider physics

Optimizing charged-particle track reconstruction algorithms is crucial for efficient event reconstruction in Large Hadron Collider (LHC) experiments due to their significant computational demands. Existing track reconstruction algorithms have been adapted to run on massively parallel coprocessors, such as graphics processing units (GPUs), to reduce processing time. Nevertheless, challenges remain in fully harnessing the computational capacity of coprocessors in a scalable and non-disruptive manner. This paper proposes an inference-as-a-service approach for particle tracking in high energy physics experiments. To evaluate the efficacy of this approach, two distinct tracking algorithms are tested: Patatrack, a rule-based algorithm, and Exa.TrkX, a machine learning-based algorithm. The as-a-service implementations show enhanced GPU utilization and can process requests from multiple CPU cores concurrently without increasing per-request latency. The impact of data transfer is minimal and insignificant compared to running on local coprocessors. This approach greatly improves the computational efficiency of charged particle tracking, providing a solution to the computing challenges anticipated in the High-Luminosity LHC era.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Ensuring statistical reproducibility of ocean model simulations in the age of hybrid computing

Novel high performance computing systems that feature hybrid architectures require large scale code refactoring to unravel underlying exploitable parallelism. Such redesign can often be accompanied with machine-precision changes as the order of computation cannot always be maintained. For chaotic systems like climate models, these round-off level differences can grow rapidly. Systematic errors may also manifest initially as machine-precision differences. Isolating genuine round off level differences from such errors remains a challenge. Here, we apply two-sample equality of distribution tests to evaluate statistical reproducibility of the ocean model component of US Department of Energy's Energy Exascale Earth System Model (E3SM). A 2-year control simulation ensemble is compared to a modified ensemble as a test case - after a known non-bit-for-bit change in a model component is introduced - to evaluate the null hypothesis that the two ensembles are statistically indistinguishable. To quantify the false negative rates of these tests, we conduct a formal power analysis using a targeted suite of short simulation ensembles. The ensemble suite contains several perturbed ensembles, each with a progressively different climate than the baseline ensemble - obtained by perturbing the magnitude of a single model tuning parameter, the Gent and McWilliams κ, in a controlled manner. The null hypothesis is evaluated for each of perturbed ensembles using these tests. The power analysis informs on the detection limits of the tests for given ensemble size allowing model developers to evaluate the impact of an introduced non-bit-for-bit change to the model.

Mahajan, Salil↗

Improving scalability of parallel CNN training by adaptively adjusting parameter update frequency

Synchronous SGD with data parallelism, the most popular parallelization strategy for CNN training, suffers from the expensive communication cost of averaging gradients among all workers. The iterative parameter updates of SGD cause frequent communications and it becomes the performance bottleneck. In this paper, we propose a lazy parameter update algorithm that adaptively adjusts the parameter update frequency to address the expensive communication cost issue. Our algorithm accumulates the gradients if the difference of the accumulated gradients and the latest gradients is sufficiently small. Here, the less frequent parameter updates reduce the per-iteration communication cost while maintaining the model accuracy. Our experimental results demonstrate that the lazy update method remarkably improves the scalability while maintaining the model accuracy. For ResNet50 training on ImageNet, the proposed algorithm achieves a significantly higher speedup (739.6 on 2048 Cori KNL nodes) as compared to the vanilla synchronous SGD (276.6) while the model accuracy is almost not affected (<0.2% difference).

97 MATHEMATICS AND COMPUTING↗

Computer Security for I and C Systems at Nuclear Facilities - 116

Cyber-attacks on Instrumentation and Control (I and C) systems may jeopardize the safety and security of nuclear facilities by contributing to sabotage or aiding in the unauthorized removal of nuclear material. The effects of cyber-attacks on I and C systems important to safety to safety may result in a wide range of consequences, such as a temporary loss of process control or unacceptable radiological consequences. Public awareness of cyber-attacks that affect I and C systems may also undermine confidence in the safety and security of nuclear facilities. The International Atomic Energy Agency (IAEA) recognizes the need for the guidance on the protection of I and C systems that provide safety, security, or auxiliary functions at nuclear facilities from cyber-attacks. To meet this need, The IAEA has developed a new Nuclear Security Series publication with the provisional title Computer Security of Instrumentation and Control Systems at Nuclear Facilities which builds upon and expands the guidance published in Nuclear Security Series No. 17 Computer Security at Nuclear Facilities (2011) . This publication will be formally released in 2017 as Technical Guidance and will be the second international consensus publication dedicated to Computer Security at Nuclear Facilities. It was developed in parallel with the new IAEA Nuclear Safety publications providing recommendations on I and C systems at research reactors (RR) and nuclear power plants (NPP) . The purpose of this paper is to provide an overview of this publication and its application. The paper will also discuss its relation to other IAEA Nuclear Security Series (NSS) publications on computer security. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

HydraGNN_Predictive_GFM_2024 - Ensemble of predictive graph foundation models for ground state atomistic materials modeling

We provide the ensemble of fifteen pre-trained graph foundation models (GFMs) for atomistic materials modeling applications. Each one of the fifteen GFMs has been trained on five open-source datasets that (once aggregated) amount to over 154 million atomistic structures, which cover over two-thirds of the natural elements of the periodic table and that comprises a broad set of organic and inorganic compounds. This vast set of atomistic structures comprises ground state configurations that are dynamically stable (i.e., equilibrated structures with atomic forces approximately close to zero values) as well as dynamically unstable structures (i.e., non-equilibrium structures with non-negligible non-zero values of atomic forces). The ensemble of datasets aggregated does NOT include excited states. The datasets have been curated to remove atomistic structures with spectral norm of the force tensor above 100 eV/angstrom. Moreover, a linear term of the energy was computed for each dataset using a linear regression model that uses the chemical concentration of each natural element as regressor. The linear term predicted by the linear regression model has been subtracted from each original energy value to perform a re-alignment of the energy values across different electronic structures approximation theories performed to generate the diverse multi-source, multi-fidelity datasets. The folder "ADIOS_files" contains the set of pre-processed datasets in Adaptable I/O System (ADIOS) format (https://www.exascaleproject.org/research-project/adios/) that have been used for the development and training of GFMs in this work. The "ADIOS_files" directory contains 6 sub-directories named as follows: - ANI1x-v3.bp - MPTrj-v3.bp - OC2020-20M-v3.bp - OC2020-v3.bp - OC2022-v3.bp - qm7x-v3.bp Each sub-directory contains the pre-processed datasets converted in Adaptable I/O System (ADIOS) format (https://www.exascaleproject.org/research-project/adios/) that have been used to the development, training, and performance testing of the ensemble go predictive graph foundation models. Each GFM was developed using HydraGNN (https://github.com/ORNL/HydraGNN) as underlying graph neural network (GNN) architecture. The multi-task learning (MTL) capability of HydraGNN was used to simultaneously train the GFMs on labeled values for direct predictions of energy (a total system property of an atomistic structure that measures the chemical stability) and atomic forces (an atomic level property of an atomistic structure that measures the dynamical stability). The hyper parameters of the GFM have been tuned using scalable hyperparameter optimization (HPO) algorithms implemented in the software DeepHyper (https://github.com/deephyper/deephyper). The pre-training of each HPO trial was performed using distributed data parallelism (DDP) to scale the training across 128 compute nodes of the exascale OLCF supercomputer Frontier. Each HPO trial was trained only for 10 epochs and an early stopping was performed to avoid wasting significant computational resources on GNN architectures that were clearly underperforming. For each HPO trial, the 'omnistat' tool developed by (AMD Research - Advanced Micro Device) was used to measure the total energy consumption in kWh. The ensemble of GFMs was obtained by selecting the fifteen best performing HPO trials. Four models have been selected for their clear advantage in accuracy, and these are the GFMs with IDs 229, 156, 147, 260. Additional eleven models have been selected based on judicious balance between accuracy and energy consumption needed for training, and these are the GFMs with IDs 165, 78, 137, 1, 175, 171, 181, 67, 179, 167, 351. Each selected GFM of the ensemble was continued to cumulate a total of at most 30 epochs. In some cases, the total number of epochs actually performed was les than 30 due to two combined factors: (1) the size of the GFM (i.e., the number of model parameters to train) and (2) the total wall-clock time for which the computational resources could be allocated on OLCF-Frontier. The "Ensemble_of_models" directory contains 15 sub-directories named as follows: - gfm_0.229 - gfm_0.156 - gfm_0.147 - gfm_0.260 - gfm_0.165 - gfm_0.78 - gfm_0.137 - gfm_0.1 - gfm_0.175 - gfm_0.171 - gfm_0.181 - gfm_0.67 - gfm_0.179 - gfm_0.167 - gfm_0.351 Each one of these sub-directories refers to one of the fifteen HPO trials that have been selected to continue the pre-training with at most 30 epochs. With each sub-directory associated with a specific HPO trial, the following files can be found: - config.json: file for argument parsing to develop and train an HydraGNN architecture - gfm_0.ID_epoch_N.pk: file with model parameters for HPO ID trial after N epochs of training The ensemble of fifteen GFM architectures was used for (1) ensemble averaging to stabilize the predictions of energy and atomic forces after pre-training for post-processing analysis and (2) ensemble uncertainty quantification (UQ). The code used to develop, pre-train, and load the pre-trained models for post-processing analysis is available on the ORNL-GitHub at the following link: https://github.com/ORNL/HydraGNN/tree/Predictive_GFM_2024

36 MATERIALS SCIENCE↗

A parallel hierarchical blocked adaptive cross approximation algorithm

This article presents a low-rank decomposition algorithm based on subsampling of matrix entries. The proposed algorithm first computes rank-revealing decompositions of submatrices with a blocked adaptive cross approximation (BACA) algorithm, and then applies a hierarchical merge operation via truncated singular value decompositions (H-BACA). The proposed algorithm significantly improves the convergence of the baseline ACA algorithm and achieves reduced computational complexity compared to the traditional decompositions such as rank-revealing QR. Numerical results demonstrate the efficiency, accuracy, and parallel scalability of the proposed algorithm.

97 MATHEMATICS AND COMPUTING↗

Distributed-Memory Sparse Deep Neural Network Inference Using Global Arrays

Partitioned Global Address Space (PGAS) models exhibit tremendous promise in developing efficient and productive distributed-memory parallel applications. They have been used extensively in scientific computations due to conveniently offering a ``shared-memory''-like model and convenient interfaces that separate communication with synchronization. Traditionally, PGAS communication models have been applied to dense/contiguously distributed data, but most modern applications depict varied levels of sparsity. Existing PGAS models require certain adaptations to support distributed sparse computations, since associated computations often require matrix arithmetic, in addition to data movement. The Global Arrays toolkit from Pacific Northwest National Laboratory (PNNL) is one of the earliest PGAS models to combine one-sided data communication and distributed matrix operations and is still used in the popular NWChem quantum chemistry suite. Recently, we have expanded the Global Arrays toolkit to support common sparse operations, like sparse matrix-dense matrix multiplies (SpMM), sparse matrix-sparse matrix multiplication (SpGEMM) and Sampled Dense-Dense Matrix Multiplication (SDDMM). As it turns out, these operations are the bedrock of sparse Deep Learning (DL); sparse deep neural networks and Graph Neural Networks (GNNs) have gained increasing attention recently in achieving speedups on training and inference with reduced memory footprints. Unlike scientific applications in High Performance Computing (HPC), modern (distributed-memory capable) DL toolkits often rely on non-standardized and closed-source vendor software optimizations, creating challenges in software-hardware co-design at scale. Our goal is to support a variety of distributed-memory sparse matrix operations and helper functions in the newly created Sparse Global Arrays (SGA), such that it is possible to build portable and productive Machine Learning scenarios for algorithm/software and hardware codesign purposes. Contemporary data-parallel schemes for training/inference are undergoing a major overhaul since model replication limits scalability and causes resource inefficiencies. As such, we have adopted tensor parallelism in decomposing the model and inputs, to mitigate memory issues. Current implementation is built on top of MPI and uses CPUs to maximize the portability across the platforms.

Distributed computing, machine learning↗

AutoSIM

AutoSIM enables several instances of an executable to run in parallel on a directory of input files and consolidate results, at scale on the world's fastest supercomputers. The software accomplishes this by balancing compression, transmission, and computation of input files, executables, and output files to allow scalable parallel runs of any executable. The software breaks-up simulations into a configurable number of "churns" or sequential instances of executable invocations on a configurable number of cores/node. It sends the executable, supporting files, and appropriate number of input files to each node for independent execution, and then transfers the executable results to a shared disk for post-processing analysis. This MPI-based code has been demonstrated using EnergyPlus, DOE's flagship whole-building simulation engine, but was architected to allow scalable runtime of any software for any application (e.g. sensitivity analysis and uncertainty quantification). The code has also been demonstrated to scale on the Titan supercomputer using 131,072 processors to run 524,288 EnergyPlus annual simulations, and write 45TB of data to disk in 68 mins 08 secs.

New, Joshua R↗

HyperSpace: Distributed Bayesian Hyperparameter Optimization

As machine learning models continue to increase in complexity, so does the potential number of free model parameters commonly known as hyperparameters. While there has been considerable progress toward finding optimal configurations of these hyperparameters, many optimization procedures are treated as black boxes. We believe optimization methods should not only return a set of optimized hyperparameters, but also give insight into the effects of model hyperparameter settings. To this end, we present HyperSpace, a parallel implementation of Bayesian sequential model-based optimization. HyperSpace leverages high performance computing (HPC) resources to better understand unknown, potentially non-convex hyperparameter search spaces. We show that it is possible to learn the dependencies between model hyperparameters through the optimization process. By partitioning large search spaces and running many optimization procedures in parallel, we also show that it is possible to discover families of good hyperparameter settings over a variety of models including unsupervised clustering, regression, and classification tasks.

Computer Science↗

Monte Carlo explicitly correlated second-order many-body perturbation theory

A stochastic algorithm is proposed and implemented that computes a basis-set-incompleteness (F12) correction to an ab initio second-order many-body perturbation energy as a short sum of 6- to 15-dimensional integrals of Gaussian-type orbitals, an explicit function of the electron-electron distance (geminal), and its associated excitation amplitudes held fixed at the values suggested by Ten-no. The integrals are directly evaluated (without a resolution-of-the-identity approximation or an auxiliary basis set) by the Metropolis Monte Carlo method. Applications of this method to 17 molecular correlation energies and 12 gas-phase reaction energies reveal that both the nonvariational and variational formulas for the correction give reliable correlation energies (98% or higher) and reaction energies (within 2 kJ mol -1 with a smaller statistical uncertainty) near the complete-basis-set limits by using just the aug-cc-pVDZ basis set. The nonvariational formula is found to be 2–10 times less expensive to evaluate than the variational one, though the latter yields energies that are bounded from below and is, therefore, slightly but systematically more accurate for energy differences. Being capable of using virtually any geminal form, the method confirms the best overall performance of the Slater-type geminal among 6 forms satisfying the same cusp conditions. Lastly, not having to precompute lower-dimensional integrals analytically, to store them on disk, or to transform them in a nonscalable dense-matrix-multiplication algorithm, the method scales favorably with both system size and computer size; the cost increases only as O(n 4 ) with the number of orbitals (n), and its parallel efficiency reaches 99.9% of the ideal case on going from 16 to 4096 computer processors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗