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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Bringing Federated Identity to Grid Computing

The Fermi National Accelerator Laboratory (FNAL) is facing the challenge of providing scientific data access and grid submission to scientific collaborations that span the globe but are hosted at FNAL. Users in these collaborations are currently required to register as an FNAL user and obtain FNAL credentials to access grid resources to perform their scientific computations. These requirements burden researchers with managing additional authentication credentials, and put additional load on FNAL for managing user identities. Our design integrates the existing InCommon federated identity infrastructure, CILogon Basic CA, and MyProxy with the FNAL grid submission system to provide secure access for users from diverse experiments and collab orations without requiring each user to have authentication credentials from FNAL. The design automates the handling of certificates so users do not need to manage them manually. Although the initial implementation is for FNAL's grid submission system, the design and the core of the implementation are general and could be applied to other distributed computing systems.

Teheran, Jeny↗
DOE OSTI2018

Effect of pre-cold rolling on the evolution of microstructure, microtexture, and mechanical properties of the mushy state rolled in-situ Al-4.5Cu-5TiB2 composite

Highlights: • Microstructure-Microtexture evolution study of the mushy state rolled composite (MRC) and MRC with pre-cold rolling (PCMRC). • The Microstructure and microtexture were characterized using SEM, EBSD, and TEM. • Exactly opposite trends in micro-hardness variation across the thickness of MRC and PCMRC were discussed. • The PCMRC exhibited the best combination of hardness, strength, and ductility. • A proper microstructure-microtexture-property correlation was established with experimental evidence. The mushy state rolling has been found as a promising technique for near-net forming of the cast Al-4.5 wt% Cu-5 wt% TiB{sub 2} in-situ composite to obtain microstructural refinement and moderate mechanical properties. Cold rolling with 30% reduction in thickness was carried out before mushy state rolling, for achieving a relatively more homogeneous distribution of CuAl{sub 2}-TiB{sub 2} particles as well as an appropriate microstructure to enhance the mechanical properties. A comparative investigation on the microstructural and microtextural evolution, along with mechanical properties of both the mushy state rolled composite (MRC) and pre-cold rolled mushy-state rolled composite (PCMRC) was performed, and the microstructure-property correlation was established. The PCMRC exhibited the lowest homogenized lengthscale parameter (L{sub H} 29 μm) indicating the maximum uniformity of particle distribution, computed with the help of multi-scalar analysis of area fractions (MSAAF) technique. Electron back-scattered diffraction (EBSD) and transmission electron microscopy (TEM) studies revealed the presence of fine sub-grains within centrally located deformed Al grains due to dynamic recovery, as well as the particle stimulated nucleation of equiaxed fine Al grains by dynamic recrystallization or rapid solidification of liquid near edges for both the investigated composites. The PCMRC displayed a superior combination of hardness (96.82 ± 4.5 HV{sub 5kgf}), 40% higher ultimate tensile strength (301.4 ± 1.8 MPa), and 68% more uniform elongation (30.8% ± 0.5%), compared to the MRC. The improvement in mechanical properties is attributed to the uniform particle distribution and presence of both higher density of low angle boundaries and recrystallized grains adjacent to the particles, obtained from prior cold rolling.

36 MATERIALS SCIENCE↗
DOE OSTI2012

EMISSION PATTERNS OF SOLAR TYPE III RADIO BURSTS: STEREOSCOPIC OBSERVATIONS

Simultaneous observations of solar type III radio bursts obtained by the STEREO A, B, and WIND spacecraft at low frequencies from different vantage points in the ecliptic plane are used to determine their directivity. The heliolongitudes of the sources of these bursts, estimated at different frequencies by assuming that they are located on the Parker spiral magnetic field lines emerging from the associated active regions into the spherically symmetric solar atmosphere, and the heliolongitudes of the spacecraft are used to estimate the viewing angle, which is the angle between the direction of the magnetic field at the source and the line connecting the source to the spacecraft. The normalized peak intensities at each spacecraft R{sub j} = I{sub j} /{Sigma}I{sub j} (the subscript j corresponds to the spacecraft STEREO A, B, and WIND), which are defined as the directivity factors are determined using the time profiles of the type III bursts. It is shown that the distribution of the viewing angles divides the type III bursts into: (1) bursts emitting into a very narrow cone centered around the tangent to the magnetic field with angular width of {approx}2 Degree-Sign and (2) bursts emitting into a wider cone with angular width spanning from {approx} - 100 Degree-Sign to {approx}100 Degree-Sign . The plots of the directivity factors versus the viewing angles of the sources from all three spacecraft indicate that the type III emissions are very intense along the tangent to the spiral magnetic field lines at the source, and steadily fall as the viewing angles increase to higher values. The comparison of these emission patterns with the computed distributions of the ray trajectories indicate that the intense bursts visible in a narrow range of angles around the magnetic field directions probably are emitted in the fundamental mode, whereas the relatively weaker bursts visible to a wide range of angles are probably emitted in the harmonic mode.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

Continuous Security and Configuration Monitoring of HPC Clusters

Continuous security and configuration monitoring of information systems has been a time consuming and laborious task for system administrators at the High Performance Computing (HPC) center. Prior to this project, system administrators had to manually check the settings of thousands of nodes, which required a significant number of hours rendering the old process ineffective and inefficient. This paper explains the application of Splunk Enterprise, a software agent, and a reporting tool in the development of a user application interface to track and report on critical system updates and security compliance status of HPC Clusters. In conjunction with other configuration management systems, the reporting tool is to provide continuous situational awareness to system administrators of the compliance state of information systems. Our approach consisted of the development, testing, and deployment of an agent to collect any arbitrary information across a massively distributed computing center, and organize that information into a human-readable format. Using Splunk Enterprise, this raw data was then gathered into a central repository and indexed for search, analysis, and correlation. Following acquisition and accumulation, the reporting tool generated and presented actionable information by filtering the data according to command line parameters passed at run time. Preliminary data showed results for over six thousand nodes. Further research and expansion of this tool could lead to the development of a series of agents to gather and report critical system parameters. However, in order to make use of the flexibility and resourcefulness of the reporting tool the agent must conform to specifications set forth in this paper. This project has simplified the way system administrators gather, analyze, and report on the configuration and security state of HPC clusters, maintaining ongoing situational awareness. Rather than querying each cluster independently, compliance checking can be managed from one central location.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

This work discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Co-location Pattern Mining of Geosocial Data to Characterize Urban Functional Spaces

Spatial Co-location Pattern (SCP) mining continues to play a critical role in understanding the morphology of urban functional spaces of world cities. It requires a large amount of fine-granular data and computing efficiency to handle the combinatorial explosion of co-location patterns. To this end, this work has two main contributions - i) We showcase a novel approach to perform SCP mining to characterize intra-city scale structure of urban functionality or co-located activity patterns using geosocial Points-of-Interest (POI) vector data. ii) We present a generalized and optimized parallel/distributed SCP mining algorithm implemented on a Hadoop MapReduce system and demonstrate the utility of our approach using the city of Berlin (Germany) as an example. The SCPs tend to vary across Berlin's municipal boroughs and at different spatial scales. Our findings on Berlin's functional structure conform to existing urban geography models. Such a data-driven exploration of massive urban POIs using distributed computing is first of its kind and can help better understand the changing dynamics of urban functionality, as well as physical, and social network structure around the world.

Masrur, Arif↗
DOE OSTI2021

A Framework for International Collaboration on ITER Using Large-Scale Data Transfer to Enable Near-Real-Time Analysis

The global nature of the ITER project along with its projected ~ petabyte per day data generation presents a unique challenge, but also an opportunity for the fusion community to rethink, optimize, and enhance our scientific discovery process. Recognizing this, collaborative research with computational scientists was undertaken over the past several years to create a framework for large-scale data movement across wide-area networks (WANs), to enable global near-real time analysis of fusion data. This would broaden the available computational resources for analysis/simulation, and increase the number of researchers actively participating in experiments. An official demonstration of this framework for fast, large data transfer and real-time analysis was carried out between the KSTAR tokamak in Daejeon, Korea and PPPL in Princeton, USA. Streaming large data transfer, with near real-time movie creation and analysis of the KSTAR Electron Cyclotron Emission imaging (ECEI) data, was performed using the I/O framework ADIOS, and comparisons made at PPPL with simulation results from the XGC1 code. These demonstrations were made possible utilizing an optimized network configuration at PPPL, which achieved over 8.8 Gbps (88% utilization) in throughput tests from NFRI to PPPL. This demonstration showed the feasibility for large-scale data analysis of KSTAR data, and provides a nascent framework to enable use of globally distributed computational and personnel resources in pursuit of scientific knowledge from the ITER experiment.

43 PARTICLE ACCELERATORS↗
DOE OSTI2023

Quantifying uncertainties due to irreducible three-body forces in deuteron-nucleus reactions

Deuteron-induced nuclear reactions are an essential tool for probing the structure of nuclei as well as astrophysical information such as (n, γ) cross sections. The deuteron-nucleus system is typically described within a Faddeev three-body model consisting of a neutron (n), a proton (p), and the target nucleus (A) interacting through pairwise phenomenological potentials. While Faddeev techniques enable the exact description of the three-body dynamics, their predictive power is limited in part by the omission of irreducible neutron-proton-nucleus three-body force (n–p–A 3BF). Here, our goal is to quantify systematic uncertainties stemming from the reduction of deuteron-nucleus (d + A) dynamics to a picture of three pointlike nuclear clusters interacting via pairwise nucleon-nucleus forces, using as testing grounds d + α scattering and the 6 Li ground state. We particularly focus on quantifying uncertainties arising from the full antisymmetrization of the (A + 2)-body system with the target nucleus fixed in its ground state. We adopt the ab initio no-core shell model coupled with the resonating group method (NCSM/RGM) to compute microscopic n–α and p–α interactions, and use them in a three-body description of the d + α system by means of momentum-space Faddeev-type equations. Simultaneously, we also carry out ab initio calculations of d + α scattering and 6 Li ground state by means of six-body NCSM/RGM calculations to serve as a benchmark for the three-body model predictions given by the Faddeev calculations. By comparing the Faddeev and NCSM/RGM results, we show that the irreducible n–p–α 3BF has a non-negligible effect on bound state and scattering observables alike. Specifically, the Faddeev approach yields a 6 Li ground state that is approximately 600 keV shallower than the one obtained with the NCSM/RGM. Additionally, the Faddeev calculations for d + α scattering yield a 3 + resonance that is located approximately 400 keV higher in energy compared to the NCSM/RGM result. The shape of the d + α angular distributions computed using the two approaches also differ, owing to the discrepancy in the predictions of the 3 + resonance energy. The Faddeev three-body model predictions for d + α scattering and 6 Li using microscopic n–α and p–α potentials differ from those computed microscopically with the NCSM/RGM. These discrepancies are due to the n–p–α 3BF, which arises from two-nucleon exchange terms in the microscopic d–α interaction and are not accounted for in the three-body model Faddeev calculations. This study lays the foundation for future parametrizations of the 3BF due to Pauli exclusion principle effects in improved three-body calculations of deuteron-induced reactions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

HIPPO – A Software Platform for Electricity Market Research and Development

The goal of this project is to provide Regional transmission organizations (RTOs) and independent system operators (ISOs) a market design and prototyping software, High-Performance Power-Grid Optimization (HIPPO), that they can evaluate electricity market design options, calculate market planning strategies and operational performance. With the high standards and strict reliability requirements for operating power systems, impacts of new technologies need to be fully investigated prior to any consideration for adoption. A market design and prototyping software tool which can be used to prototype electricity market design options, to calculate market planning strategies and operational performance with high precision, and to investigate the impacts for integrating future power grid technologies will be valuable to RTOs/ISOs who operate power systems, to vendors like GE and ABB who provide the market solvers, and to market participants and researchers who are actively doing market research. HIPPO is a such tool that can be used to improve the current market operations and provide capabilities for rigorous forward-looking design and prototyping of next-generation energy markets. HIPPO has a high-resolution model for the day-ahead SCUC, which was validated with MISO and GE-Grid Solutions. HIPPO is built with parallel and distributed computing capabilities and can be executed in both multi-thread and high-performance computing (HPC) settings. This capability provides fast solution speed necessary to handle the larger and more complex SCUC problems of real-world cases and the potentially growing size and complexity of future scenarios. In addition, HIPPO has a concurrent optimizer (CO) which manages multiple algorithm executions simultaneously and leverages the advantages from different algorithms. This structure provides flexibility to better benchmark competing approaches. Highly accurate market model, fast solution technologies and flexible model and algorithm control are the features which will make HIPPO an extensible platform for developing and testing multiple approaches to meet a wide range of future market needs.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2015

Evolution of CMS Workload Management Towards Multicore Job Support

The successful exploitation of multicore processor architectures is a key element of the LHC distributed computing system in the coming era of the LHC Run 2. High-pileup complex-collision events represent a challenge for the traditional sequential programming in terms of memory and processing time budget. The CMS data production and processing framework is introducing the parallel execution of the reconstruction and simulation algorithms to overcome these limitations. CMS plans to execute multicore jobs while still supporting singlecore processing for other tasks difficult to parallelize, such as user analysis. The CMS strategy for job management thus aims at integrating single and multicore job scheduling across the Grid. This is accomplished by employing multicore pilots with internal dynamic partitioning of the allocated resources, capable of running payloads of various core counts simultaneously. An extensive test programme has been conducted to enable multicore scheduling with the various local batch systems available at CMS sites, with the focus on the Tier-0 and Tier-1s, responsible during 2015 of the prompt data reconstruction. Scale tests have been run to analyse the performance of this scheduling strategy and ensure an efficient use of the distributed resources. This paper presents the evolution of the CMS job management and resource provisioning systems in order to support this hybrid scheduling model, as well as its deployment and performance tests, which will enable CMS to transition to a multicore production model for the second LHC run.

Perez-Calero Yzquierdo, A.↗
DOE OSTI2021

Resilient Information Architecture Platform for Smart Grid (RIAPS)

A number of emerging trends will substantially alter the operation and control of the electric grid over the next several decades. These trends include ensuring resiliency under severe weather events, increasing integration of renewable electricity generation, supporting changing electricity demand patterns, and the improving cost effectiveness of distributed energy resources. To address these challenges, the future “Smart Grid” management will need to transition from centralized to coordinated distributed control paradigm. Reliable operation of the Smart Grid depends on distributed intelligence realized through software applications that run on distributed computing devices attached to the power system to collect data and collaboratively manage resources. However, much of the existing software for Smart Grid-enabled devices is either proprietary or developed with custom solutions, which limits interoperability among the heterogeneous devices and hinders the ability to manage system-level reliability, security, and resiliency requirements. Additionally, this approach makes Smart Grid applications hard to maintain, evolve, verify, and replace; resulting in high development and deployment costs. Further development of the Smart Grid requires a reusable software base-layer to move from hard-coded functionality to a plug-and-play architecture capable of managing system-level objectives and constraints in addition to providing consistent common services across heterogeneous devices and applications. Vanderbilt University, in collaboration with North Carolina State University and Washington State University has developed a foundation ‘software platform’ for developing and deploying robust, reliable, effective and secure software applications for the Smart Grid. The Resilient Information Architecture Platform for the Smart Grid (RIAPS) provides core services for building effective and powerful smart grid applications. It offers unique services for real-time data dissemination, fault tolerance, and coordination across apps distributed over the network.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2016

SU-F-J-123: CT-Based Determination of DIBH Variability and Its Dosimetric Impact On Post-Mastectomy Plus Regional Nodal Radiation Therapy

Purpose: Breast cancer radiotherapy delivered using voluntary deep inspiration breath-hold (DIBH) requires reproducible breath holds, particularly when matching supraclavicular fields to tangential fields. We studied the impact of variation in DIBHs on CTV and OAR dose metrics by comparing the dose distribution computed on two DIBH CT scans taken at the time of simulation. Methods: Ten patients receiving 50Gy in 25 fractions to the left chestwall and regional lymph nodes were studied. Two simulation CT scans were taken during separate DIBHs along with a free-breathing (FB) scan. The treatment was planned using one DIBH CT. The dose was recomputed on the other two scans using adaptive planning (Pinnacle 9.10) in which the scans are registered using a cross-correlation algorithm. The chestwall, lymph nodes and OARs were contoured on the scans following the RTOG consensus guidelines. The overall translational and rotational variation between the DIBH scans was used to estimate positional variation between breath-holds. Dose metrics between plans were compared using paired t-tests (p < 0.05) and means and standard deviations were reported. Results: The registration parameters were sub-millimeter and sub-degree. Although DIBH significantly reduced mean heart dose by 2.4Gy compared to FB (p < 0.01), no significant changes in dose were observed for targets or OARs between the two DIBH scans. Nodal coverage as assessed by V90% was 90%±8% and 89%±8% for supraclavicular and 99%±2% and 97%±22% for IM nodes. Though a significant decrease (10.5%±12.4%) in lung volume in the second DIBH CT was observed, the lung V20Gy was unchanged (14±2% and 14±3%) between the two DIBH scans. Conclusion: While the lung volume often varied between DIBHs, the CTV and OAR dose metrics were largely unchanged. This indicates that manual DIBH has the potential to provide consistent dose delivery to the chestwall and regional nodes targets when using matched fields.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2022

Proximity Portability and in Transit , M-to-N Data Partitioning and Movement in SENSEI [Book Chapter]

In high-performance parallel in situ processing, the term in transit processing refers to those configurations where data must move from a producer to a consumer that runs on separate resources. In the context of parallel and distributed computing on an HPC platform one of the central challenges is to determine a mapping of data from producer ranks to consumer ranks. This problem is complicated by the heterogeneity that arises in producer-consumer pairs, such as when producer and consumer codes have different levels of concurrency, different scaling characteristics, or different data models. The resulting mapping and movement of data from M producer to N consumer ranks can have a significant impact on aggregate application performance, particularly when the data consumer requires only a subset of the overall data for its task. This chapter focuses on the design considerations that underlie SENSEI’s implementation to this challenging problem. These design considerations extend the core SENSEI architecture and include ideas like the need to accommodate flexibility in the choice of different partitioning methods, the ability for a data consumer to request and receive only the subset of data needed for its particular operation, and the ability to leverage any of several different data transport tools. The idea of proximity portability, being able to use different data transport methods as part of an in transit workflow, is illustrated through the use of three different transport layers where switching from one transport tool to another is accomplished with only a configuration file change. Here, the chapter also includes a performance analysis summary showing the performance gains that are possible in terms of multiple metrics, such as memory footprint, time to solution, and amount of data moved, when using optimized partitioners in an in transit setting, gains that are made possible by the implementation shaped by specific design considerations.

Bethel, E. Wes↗
DOE OSTI2016

HPC Node Performance and Energy Modeling with the Co-Location of Applications

Multicore processors have become an integral part of modern large-scale and high-performance parallel and distributed computing systems. This advance became necessary as the demand for increased system performance has exceeded the limits that single core processors can provide. Unfortunately, applications co-located on multicore processors can suffer from decreased performance and increased dynamic energy use as a result of access to shared resources, such as memory. Consequently, it is increasingly important to characterize the performance of applications that execute on these architectures. This work investigates some of the disadvantages of co-location, and presents a methodology for building models capable of utilizing varying amounts of information about a target application and its co-located applications to make predictions about the target application’s execution time and the system’s energy use under arbitrary co-locations of a wide range of application types. The proposed methodology is validated on three different server class Intel Xeon multicore processors using eleven applications from two scientific benchmark suites. The model’s utility for scheduling is also demonstrated in a simulated large-scale high-performance computing environment through the creation of a co-location aware scheduling heuristic. This heuristic demonstrates that scheduling using information generated with the proposed modeling methodology is capable of making significant improvements over a scheduling heuristic that is naive to co-location interference.

performance modeling, energy modeling, resource ma↗
DOE OSTI2018

Fractal Dimension Calculation for Big Data Using Box Locality Index

The box-counting approach for fractal dimension calculation is scaled up for big data using a data structure named box locality index (BLI). The BLI is constructed as key-value pair style with the key (an integer array) indexing the location of a “box” (i.e., a gird cell on the multi-dimensional space embedding a given dataset) and the value counting the number of data points inside the box (called “box occupancy”). Compared with the traditionally used E-dim tree, the BLI avoids complex hierarchical structure and encodes only necessary information required by box-counting approach for fractal dimension calculation. Moreover, the key-value pair nature of BLI, together with the fact that the box occupancy is aggregatable, grants box-counting approach the needed scalability for fractal dimension calculation of big data using distributed computing techniques (e.g., MapReduce and Spark). Taking the advantage of the BLI, MapReduce and Spark methods for fractal dimension calculation of big data are developed, which conduct box-counting for each grid level as a cascade of MapReduce/Spark jobs in a bottom-up fashion. In an empirical validation, the MapReduce and Spark methods demonstrated good effectiveness and efficiency in fractal calculation of a big synthetic dataset. In summary, this work provides an efficient solution for the estimation of intrinsic dimension of big data that is essential for many machine learning methods and data analytics.

Fractal dimension, Intrinsic dimension, Box-counti↗
DOE OSTI2024

Parallel quantum computing simulations via quantum accelerator platform virtualization

Quantum circuit execution is a central task in quantum computation. Due to inherent quantum-mechanical constraints, quantum computing workflows often involve a considerable number of independent measurements over a large set of slightly different quantum circuits. Here we discuss a simple model for parallelizing such quantum circuit executions that is based on introducing a large array of virtual quantum processing units (mapped to HPC nodes in our case) as a parallel quantum computing platform. Implemented within the XACC framework, the model can readily take advantage of its backend-agnostic features, enabling parallel quantum computing/simulation over any target backend supported by XACC. We illustrate the performance of this approach by demonstrating strong scaling in two pertinent domain science problems, namely in computing the gradients for the multi-contracted variational quantum eigensolver and in data-driven quantum circuit learning, where we vary the number of qubits and the number of circuit layers. Here, the latter simulation leverages the cuQuantum library to run efficiently on GPU-accelerated HPC platforms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Mesh-free data transfer algorithms for partitioned multiphysics problems: Conservation, accuracy, and parallelism

In this study we analyze and extend mesh-free algorithms for three-dimensional data transfer problems in partitioned multiphysics simulations. We first provide a direct comparison between a mesh-based weighted residual method using the common-refinement scheme and two mesh-free algorithms leveraging compactly supported radial basis functions: one using a spline interpolation and one using a moving least square reconstruction. Through the comparison we assess both the conservation and accuracy of the data transfer obtained from each of the methods. We do so for a varying set of geometries with and without curvature and sharp features and for functions with and without smoothness and with varying gradients. Our results show that the mesh-based and mesh-free algorithms are complementary with cases where each was demonstrated to perform better than the other. We then focus on the mesh-free methods by developing a set of algorithms to parallelize them based on sparse linear algebra techniques. This includes a discussion of fast parallel radius searching in point clouds and restructuring the interpolation algorithms to leverage data structures and linear algebra services designed for large distributed computing environments. The scalability of our new algorithms is demonstrated on a leadership class computing facility using a set of basic scaling studies. Finally, these scaling studies show that for problems with reasonable load balance, our new algorithms for both spline interpolation and moving least square reconstruction demonstrate both strong and weak scalability using more than 100,000 MPI processes with billions of degrees of freedom in the data transfer operation.

97 MATHEMATICS AND COMPUTING↗