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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 1,063 records · Page 59

Final Report on XStack: Software Synthesis for High Productivity ExaScale Computing

The goal of the project was to develop a programming model that would significantly improve productivity in the high-performance computing domain by bringing together three components: a) Automated equivalence checking, b) Sketch-based program synthesis, and c) Autotuning. The report provides an executive summary of the research accomplished through this project. At the end of the report is appended a paper that describes in more detail the key technical accomplishments from this project, and which was published in SC 2014.

97 MATHEMATICS AND COMPUTING↗

VENTSAR V3.0: A Python GUI for Estimating Contaminant Concentrations on or Near Buildings Due to Building Effects and Plume Rise and For Calculating Inhalation and Plume Shine Doses

VENTSAR: Originated as VENTAX (Smith and Weber 1983) on the IBM Mainframe at the Savannah River Site (SRS) as a Fortran program. Updated to VENTSAR XL V1.0 (Simpkins 1997) as a spreadsheet version using macros created in Microsoft Excel. Later updated to VENTSAR XL V2.0 (Dixon 2018) as a spreadsheet executing a modified version of the original VENTAX Fortran code. Estimates contaminant concentrations on or near a building from a release at a nearby location. Calculates concentrations for a given meteorological exceedance probability or for a given stability and wind speed combination. Can model a single building with or without a penthouse on top or a ground location from either a stack or ground release. Plume rise can be considered. Contaminant releases can be chemical or radioactive with downwind concentrations determined at user-specified distances. Wind passing over and around buildings creates a complicated dispersion pattern. Air-intake vents may be located on building roofs or near the ground downwind of a release source. An estimation of pollutant concentrations on or near a structure is important in determining expected pollutant levels. Meteorological data are selected based on a specific area of the site. Fortran-based VENTAX able to make fast, complex mathematical calculations. Not user-friendly VENTSAR XL V1.0 no longer supported due to obsolete Excel macros. VENTSAR XL V2.0 an interim solution using an Excel spreadsheet to interface with the modified VENTAX Fortran code. Requires user to have Microsoft Excel. Not an intuitive interface for someone unfamiliar with VENTSAR. Does not perform dose calculations. VENTSAR V3.0 goal to combine the strengths of previous versions of VENTSAR into a single, powerful yet user-friendly program with a Graphical User Interface (GUI). Not dependent upon a specific program or plug-ins. Self-contained package to be used on any computer running Microsoft Windows. User-friendly, intuitive GUI created in Python for parameter input. Executes same modified VENTAX Fortran code as VENTSAR XL V2.0. Outputs formatted text file of completed calculations for easy review and dissemination. Performs inhalation and plume shine dose calculations for up to 11 user-selected nuclides from a nuclide dose factor library of almost 500 nuclides. 12 test cases were created for verification of V1.0 and V2.0. Same test cases run in V3.0 to verify correct performance. V3.0 produced similar results to V1.0. Confirmed GUI did not alter calculations in any way. Comparisons of the test case results confirm the V3.0 GUI does not influence the VENTSAR calculations. Simply passes same input parameters to VENTAX Fortran code for execution. Provides end user an easy-to-use, intuitive tool to quickly make building effect and plume rise calculations, as well as, inhalation and plume shine dose calculations. No experience with or working knowledge of Fortran, Excel, command line, or macros required. Self-contained package allows VENTSAR be deployed to any Windows computer without the need for specific programs or plug-ins.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Architecture for Quantum Time-dependent Circuits (ArQTIC) v1.0

ArQTiC is a domain-specific full-stack software package built for the dynamic simulations of materials on quantum computers. Its main contributions include providing a software library for high-level programming of such simulations on quantum computers, and providing post-processing capabilities that allow users to more efficiently analyze results from the quantum computer. Paired with the power to optimize and execute quantum circuits, ArQTiC opens the field of dynamic materials simulations on quantum computers to a broader community of scientists from a wider range of domain sciences, paving the way for accelerated progress towards physical quantum supremacy.

Bassman, Lindsay↗

Summer Proceedings 2018: The Center for Computing Research at Sandia National Laboratories

The Center for Computing Research (CCR) at Sandia National Laboratories organizes an active and productive summer program each year, in coordination with the Computer Science Research Institute (CSRI) and Cyber Engineering Research Institute (CERI). CERI focuses on open, exploratory research in cyber security in partnership with academia, industry, and government, and provides collaborators an accessible portal to Sandia's cybersecurity experts and facilities. Moreover, CERI provides an environment for visionary, threat-informed research on national cyber challenges. CSRI brings university faculty and students to Sandia National Laboratories for focused collaborative research on DOE computer and computational science problems. CSRI provides a mechanism by which university researchers learn about problems in computer and computational science at DOE Laboratories. Participants conduct leading - edge research, interact with scientists and engineers at the laboratories, and help transfer the results of their research to programs at the labs.

97 MATHEMATICS AND COMPUTING↗

Fast Approximate Union Volume in High Dimensions with Line Samples

The classical problem of calculating the volume of the union of d-dimensional balls is known as "Union Volume." We present line-sampling approximation algorithms for Union Volume. Our methods may be extended to other Boolean operations, such as setminus; or to other shapes, such as hyper-rectangles. The deterministic, exact approaches for Union Volume do not scale well to high dimensions. However, we adapt several of these exact approaches to approximation algorithms based on sampling. We perform local sampling within each ball using lines. We have several variations, depending on how the overlapping volume is partitioned, and depending on whether radial, axis-aligned, or other line patterns are used. Our variations fall within the family of Monte Carlo sampling, and hence have about the same theoretical convergence rate, 1 /$\sqrt{M}$, where M is the number of samples. In our limited experiments, line-sampling proved more accurate per unit work than point samples, because a line sample provides more information, and the analytic equation for a sphere makes the calculation almost as fast. We performed a limited empirical study of the efficiency of these variations. We suggest a more extensive study for future work. We speculate that different ball arrangements, differentiated by the distribution of overlaps in terms of volume and degree, will benefit the most from patterns of line samples that preferentially capture those overlaps. Acknowledgement We thank Karl Bringman for explaining his BF-ApproxUnion (ApproxUnion) algorithm [3] to us. We thank Josiah Manson for pointing out that spoke darts oversample the center and we might get a better answer by uniform sampling. We thank Vijay Natarajan for suggesting random chord sampling. The authors are grateful to Brian Adams, Keith Dalbey, and Vicente Romero for useful technical discussions. This work was sponsored by the Laboratory Directed Research and Development (LDRD) Program at Sandia National Laboratories. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR), Applied Mathematics Program. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

97 MATHEMATICS AND COMPUTING↗

GEO2D - Two-Dimensional Computer Model of a Ground Source Heat Pump System

This file contains a zipped file that contains many files required to run GEO2D. GEO2D is a computer code for simulating ground source heat pump (GSHP) systems in two-dimensions. GEO2D performs a detailed finite difference simulation of the heat transfer occurring within the working fluid, the tube wall, the grout, and the ground. Both horizontal and vertical wells can be simulated with this program, but it should be noted that the vertical wall is modeled as a single tube. This program also models the heat pump in conjunction with the heat transfer occurring. GEO2D simulates the heat pump and ground loop as a system. Many results are produced by GEO2D as a function of time and position, such as heat transfer rates, temperatures and heat pump performance. On top of this information from an economic comparison between the geothermal system simulated and a comparable air heat pump systems or a comparable gas, oil or propane heating systems with a vapor compression air conditioner. The version of GEO2D in the attached file has been coupled to the DOE heating and cooling load software called ENERGYPLUS. This is a great convenience for the user because heating and cooling loads are an input to GEO2D. GEO2D is a user friendly program that uses a graphical user interface for inputs and outputs. These make entering data simple and they produce many plotted results that are easy to understand. In order to run GEO2D access to MATLAB is required. If this program is not available on your computer you can download the program MCRInstaller.exe, the 64 bit version, from the MATLAB website or from the Geothermal Data Repository. This is a free download which will enable you to run GEO2D.

15 GEOTHERMAL ENERGY↗

Vibrational Signatures of Electronic Properties in Renewable-Energy Catalysis

The objective of this research program is to discover and develop new approaches for ab initio computational simulations of molecular motion. Specifically, this program aims to decipher the connection between the underlying electronic structure and the accordant molecular vibrations in reactive ions and radicals for renewable-energy purposes. Recent experimental progress in ion sources and optical spectroscopies has unearthed considerable new inner-sphere detail for these complexes, but the connection between these spectral signatures and mechanistic information often remains elusive, to the continued frustration of experimentalists. For this purpose, new anharmonic vibrational frequency methods, along with a publicly deployed software package, will be developed. Working closely with committed experimental collaborators, this conceptual and computational framework will be used to explain the results of new spectroscopy experiments, focusing specifically on the inner-shell mechanisms of renewable-energy catalysis. The oxidation half of catalytic water-splitting chemistry will be a central focus, along with fundamental studies of the manner in which strong ions and radicals activate solvent as a chemical species. The resulting products of the research program will include openly available software and algorithms for the ab initio simulation of challenging vibrational spectra, as well as critical mechanistic insight into energy-focused catalytic processes that are opaque to other existing analytical techniques.

42 ENGINEERING↗

Improving robustness and computational efficiency using modern C++

For nearly two decades, the C++ programming language has been the dominant programming language for experimental HEP. The publication of ISO/IEC 14882:2011, the current version of the international standard for the C++ programming language, makes available a variety of language and library facilities for improving the robustness, expressiveness, and computational efficiency of C++ code. However, much of the C++ written by the experimental HEP community does not take advantage of the features of the language to obtain these benefits, either due to lack of familiarity with these features or concern that these features must somehow be computationally inefficient. In this paper, we address some of the features of modern C+-+, and show how they can be used to make programs that are both robust and computationally efficient. We compare and contrast simple yet realistic examples of some common implementation patterns in C, currently-typical C++, and modern C++, and show (when necessary, down to the level of generated assembly language code) the quality of the executable code produced by recent C++ compilers, with the aim of allowing the HEP community to make informed decisions on the costs and benefits of the use of modern C++.

Paterno, M.↗

PipeSight: A High-Performance Computing Platform for Pipeline Integrity Management

The Phase I feasibility study completed as part of this project has led to a number of innovative technologies being developed and has laid the foundation for a successful Phase II effort to commercialize a platform for managing the integrity of pipelines for the damage mechanisms of the new, hybrid-energy based economy. To ground the development efforts and direction of the project, an extensive market research and customer discovery effort was undertaken early in Phase I. Through this effort, a number of pipeline owners and operators were interviewed, and the following key findings were discovered about the pipeline industry: • Small pipeline operators do not have the central engineering groups necessary to perform their own independent analysis of inspection data, but instead rely on summarized tally sheets provided to them by inspection service providers. • The time it takes to go from an inspection to a completed engineering assessment, even for small segments of pipeline, can take anywhere from 30-120 days. During this delay, critical threats can (and have been known to) cause failures. • Uncertainty is often not accounted for in the assessment of pipeline integrity. The tally sheets provided by third-party service providers are almost always deterministic in nature, identifying threats that present a concern only to the current (not the future) integrity of the pipeline. • It is uncommon to apply the latest technologies to perform advanced assessments of damaged pipelines. There is a desire to use more advanced analysis capabilities to assess threats. Many pipeline operators indicated that they would often excavate a pipeline to perform an inspection and find that the damage was not as bad as they anticipated, thus using limited resources unnecessarily. Companies are not consistent in their use of inspection data to determine corrosion rates, and those that do only calculate deterministic corrosion rates. • The industry has prominently relied on time-based inspections but has recently started to transition to risk-based inspections. However, there appears to be no uniform guidance on how to do so while properly accounting for all sources of uncertainty. • Companies are not storing inspection data in a manner that allows for the ready determination of temporal trends. • Predictive maintenance principles and practices are beginning to be used by early adopters • Some pipelines are being re-purposed to transport different process fluids than they were designed for, e.g., H 2 and CO 2 rich process streams to serve the new hybrid-energy based economy, which are presenting new integrity concerns for the existing pipeline network that crisscrosses the United States. As a result of these discoveries, we were able to target the development efforts in Phase I to best serve the needs of the industry. In Phase I, we developed a way to correlate multiple large-scale scans of the pipeline to determine a probabilistic corrosion rate that accounts for all sources of error and uncertainty in the inspection process. This probabilistic corrosion rate can be used to predict the future thickness distribution of the pipe wall. We demonstrate how this analysis may be performed in an analytical fashion and has been implemented in such a manner that it can be readily distributed using GPU computing through integration of the Kokkos programming model. We also make a very novel extension of the analytical corrosion rate model to Bayesian Networks (an explainable AI technique) that can account for non-parametric distributions of corrosion rates. With the predictions made above for the probabilistic corrosion rate and corresponding future distribution of the pipe wall thickness, we can assess the integrity of the pipeline through the use of a probabilistic engineering assessment. We developed a novel screening data analysis approach that can rapidly identify ‘hotspots’ (local thin areas) where the integrity of the pipeline is a concern. Once more, we implemented this screening approach in C++ to leverage GPU computing via the Kokkos programming model. After the critical hotspots are identified, we developed a program that can automatically generate an advanced finite element model of the damaged regions. Since the number of damaged regions that require advanced analysis can number in the thousands, we integrated an open-source container-native workflow engine for orchestrating parallel jobs on the cloud. Initially, these advanced numerical models were only designed to account for loading due to internal pressure. However, in a slight pivot from the initial Phase I proposal, we developed a complete pipe stress analysis program (called Simflex) which can simulate the complete pipeline and its response to thermal expansion, pressure, thermal bowing, weight, wind, earthquake, support displacement, support friction and external forces. This pipe stress analysis program was written generically, to handle any piping system, but contains the features needed to model long pipelines (i.e., it incorporates a model for soil mechanics and can account for the nonlinear boundary conditions necessary to simulate long underground pipelines). This pipe stress analysis program can simulate any segment of the pipeline (simple or complex) under any set of conditions and loads, to determine the supplemental loads (axial forces and bending moments) at the location of damage. This enables the most accurate state of stress to be accounted for in the pipeline, which can prove critical when evaluating the integrity of a damaged region. In the process of developing the technologies to perform the integrity assessment of the pipeline, we also extended one of the industry standard approaches for performing the assessment of local thin areas that extend more in the circumferential direction than the longitudinal direction of the pipeline. This approach was presented to the API 579-1/AS ME FFS-1 steering committee in November 2021 for consideration in the next edition of the industry standard for Fitness-For-Service (expected to be released in 2023). To help pipeline operators make decisions with the results on any integrity assessment, we developed a new approach to the life-cycle management of pipelines which uses a Bayesian Decision Network. The network is designed to help pipeline operators plan and prioritize inspection activities and ultimately make smarter, more cost-effective decisions. The Bayesian approach accounts for all sources of uncertainty and carries them through to the final optimal decisions, providing a probabilistic framework for optimizing inspection intervals. The proof-of-concept networks developed in the feasibility study are complete, verified, and are focused on a subset of the pipeline. To expand this novel approach to the scale necessary for an entire network of pipelines in Phase II, we will leverage the DOE-funded Bengi solver for industrial-scale decision making with Bayesian Networks [22]. Once implemented, we will be able to provide the pipeline industry with a much-needed tool for optimal inspection planning using truly explainable artificial intelligence (XAI). To handle all of these advanced capabilities into a cloud-based platform, the architecture of the Equity Engineering Cloud (EEC) was extended to include Argo Workflows, a framework capable of distributing and managing a massive number of jobs that consume their own resources, such that thousands of serial finite element simulations can be run in parallel. As part of this substantial undertaking, we also integrated Argo Continuous Delivery (CD) into the EEC, to aid with the rapid prototyping and iterations that will be imperative to the success of the PipeSight platform’s Agile development process in Phase II. As part of the pipe stress analysis program, we also developed a custom visualizer that leverages the DOE-funded VTK visualization library. We added custom contouring capabilities and a means for interacting visually with both the inputs and outputs of the pipe stress analysis program. We also developed routines for automating the post-processing of the finite element simulations to determine if any failure criteria are met and to visualize the deformations, stresses and strains in ParaView using the exodus II file format (a subset of netCDF).

24 POWER TRANSMISSION AND DISTRIBUTION↗

Data Intensive Computing on Amazon Web Services

The Geophysical Monitoring Program (GMP) has spent the past few years building up the capability to perform data intensive computing using what have been referred to as “big data” tools. These big data tools would be used against massive archives of seismic signals (>300 TB) to conduct research not previously possible. Examples of such tools include Hadoop (HDFS, MapReduce), HBase, Hive, Storm, Spark, Solr, and many more by the day. These tools are useful for performing data analytics on datasets that exceed the resources of traditional analytic approaches. To this end, a research big data cluster (“Cluster A”) was set up as a collaboration between GMP and Livermore Computing (LC).

97 MATHEMATICS AND COMPUTING↗

Measuring the capabilities of quantum computers

Quantum computers can now run interesting programs, but each processor’s capability—the set of programs that it can run successfully—is limited by hardware errors. These errors can be complicated, making it difficult to accurately predict a processor’s capability. Benchmarks can be used to measure capability directly, but current benchmarks have limited flexibility and scale poorly to many-qubit processors. We show how to construct scalable, efficiently verifiable benchmarks based on any program by using a technique that we call circuit mirroring. With it, we construct two flexible, scalable volumetric benchmarks based on randomized and periodically ordered programs. We use these benchmarks to map out the capabilities of twelve publicly available processors, and to measure the impact of program structure on each one. We find that standard error metrics are poor predictors of whether a program will run successfully on today’s hardware, and that current processors vary widely in their sensitivity to program structure.

97 MATHEMATICS AND COMPUTING↗

Advanced Shuttle Strategies for Parallel QCCD Architectures

Trapped ions (TIs) are at the forefront of quantum computing implementation, offering unparalleled coherence, fidelity, and connectivity. However, the scalability of TI systems is hampered by the limited capacity of individual ion traps, necessitating intricate ion shuttling for advanced computational tasks. The quantum charge-coupled device (QCCD) framework has emerged as a promising solution, facilitating ion mobility for universal quantum computation. Current QCCD architectures predominantly feature a linear topology, which is increasingly recognized as inefficient for complex quantum operations. Anticipating the shift toward more efficacious designs, this article introduces an innovative quantum scheduling strategy optimized for parallel QCCD topologies. Our strategy proposes a probabilistic formula for ion movement, alongside ingenious methods for local layer generation and layer compression, yielding a significant reduction in ion shuttle times. Through simulations, we demonstrate that our strategy not only substantially outstrips the linear model but also exhibits better performance over other parallel strategies that employ greedy algorithms. This is achieved through our nuanced resolution of complexities, such as traffic blocks and trap capacity limitations. The consequent reduction in shuttle operations leads to lower energy consumption and an enhancement in the quantum computer's fidelity, ultimately accelerating program execution times.

43 PARTICLE ACCELERATORS↗

SharP: Towards Programming Extreme-Scale Systems with Hierarchical Heterogeneous Memory

The pre-exascale systems are expected to have a significant amount of hierarchical and heterogeneous on-node memory, and this trend of system architecture in extreme-scale systems is expected to continue into the exascale era. Along with hierarchical-heterogeneous memory, the system typically has a high-performing network and a compute accelerator. This system architecture is not only effective for running traditional High Performance Computing (HPC) applications (Big-Compute), but also running data-intensive HPC applications and Big-Data applications. As a consequence, there is a growing desire to have a single system serve the needs of both Big-Compute and Big-Data applications. Though the system architecture supports the convergence of the Big-Compute and Big-Data, the programming models have yet to evolve to support either hierarchical-heterogeneous memory systems or the convergence. In this work, we propose and develop the programming abstraction called SHARed data-structure centric Programming abstraction (SharP) to address both of these goals, i.e., provide (1) a simple, usable, and portable abstraction for hierarchical-heterogeneous memory and (2) a unified programming abstraction for Big-Compute and Big-Data applications. To evaluate SharP, we implement a Stencil benchmark using SharP, port QMCPack, a petascale-capable application, and adapt Memcached ecosystem, a popular Big-Data framework, to use SharP, and quantify the performance and productivity advantages. Additionally, we demonstrate the simplicity of using SharP on different memories including DRAM, High-bandwidth Memory (HBM), and non-volatile random access memory (NVRAM).

97 MATHEMATICS AND COMPUTING↗

Sensors and methods for monitoring flying objects

Described herein are sensing methods, sensor systems, and non-transitory, computer-readable, storage media having programs for long-duration, continuous monitoring of flying objects during the day or the night and regardless of weather conditions. The methods and systems are computationally efficient and can provide compact, three-dimensional representations of motion from the observed object. A 3D track of the flying object can be generated from a point-matched pair of stereo composite motion track images and not directly from the videos, wherein each composite motion track image is based on a composite of a plurality of video frames composited in part according to video frame numbers.

Matzner, Shari↗

Kokkos Programming Model and Library

High performance computing is in the midst of a many-core revolution driven by rapidly evolving microchips with diverse and heterogeneous architectures, increasing concurrency (thread count), and decreasing memory per thread. Our HPC applications face a major challenge to effectively use these microchips with a single code base that is performance portable across these evolving architectures. Kokkos was developed to enable on-chip concurrency and manage architecture-dependent constraints on memory access patterns. The goal is to maximize the amount of application code that is portable across diverse architectures and still obtain the same (or nearly) performance as a variant of that code written specifically for that architecture. The Kokkos team has ongoing R&D co-design collaborations with Sandia's and vendors' microchip and system engineers to rapidly incorporate evolving computer architecture features into Kokkos. The team is also providing leadership for DOE laboratories' participation on the ISO/C++ language standard committee to subsuming Kokkos features into a future standard.

97 MATHEMATICS AND COMPUTING↗

Advanced Simulation and Computing (FY22 Implementation Plan Rev 0)

The DOE National Nuclear Security Administration (NNSA) Stockpile Stewardship Program (SSP) is an integrated technical program for maintaining the safety, surety, and reliability of the U.S. nuclear stockpile. The SSP incorporates nuclear test data, computational modeling and simulation, and experimental facilities to advance understanding of nuclear weapons. The suite of data analyzed comes from activities including stockpile surveillance, experimental research, and development and engineering programs. This integrated national program requires the continued use of experimental facilities and the computational capabilities to support the SSP missions. These component parts, in addition to an appropriately scaled production capability, enable NNSA to support stockpile requirements. The ultimate goal of the SSP, and thus of the Advanced Simulation and Computing (ASC) Program, is to ensure that the U.S. maintains a safe, secure, and effective strategic deterrent. Specific work activities and scope contained in this Implementation Plan (IP) represent the full-year annual operating plan for FY22. The Initial IP, effective , should be consistent with the Department’s Base Table when operating under a Continuing Resolution (CR). The final IP, effective date TBD, is consistent with the final, enacted appropriation.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗

High performance graphics processor based computed tomography reconstruction algorithms for nuclear and other large scale applications.

The goal of this work is to develop a fast computed tomography (CT) reconstruction algorithm based on graphics processing units (GPU) that achieves significant improvement over traditional central processing unit (CPU) based implementations. The main challenge in developing a CT algorithm that is capable of handling very large datasets is parallelizing the algorithm in such a way that data transfer does not hinder performance of the reconstruction algorithm. General Purpose Graphics Processing (GPGPU) is a new technology that the Science and Technology (S&T) community is starting to adopt in many fields where CPU-based computing is the norm. GPGPU programming requires a new approach to algorithm development that utilizes massively multi-threaded environments. Multi-threaded algorithms in general are difficult to optimize since performance bottlenecks occur that are non-existent in single-threaded algorithms such as memory latencies. If an efficient GPU-based CT reconstruction algorithm can be developed; computational times could be improved by a factor of 20. Additionally, cost benefits will be realized as commodity graphics hardware could potentially replace expensive supercomputers and high-end workstations. This project will take advantage of the CUDA programming environment and attempt to parallelize the task in such a way that multiple slices of the reconstruction volume are computed simultaneously. This work will also take advantage of the GPU memory by utilizing asynchronous memory transfers, GPU texture memory, and (when possible) pinned host memory so that the memory transfer bottleneck inherent to GPGPU is amortized. Additionally, this work will take advantage of GPU-specific hardware (i.e. fast texture memory, pixel-pipelines, hardware interpolators, and varying memory hierarchy) that will allow for additional performance improvements.

97 MATHEMATICS AND COMPUTING↗