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At least 919 records · Page 51

Chromosome-scale shotgun assembly using an in vitro method for long-range linkage

Long-range and highly accurate de novo assembly from short-read data is one of the most pressing challenges in genomics. Recently, it has been shown that read pairs generated by proximity ligation of DNA in chromatin of living tissue can address this problem, dramatically increasing the scaffold contiguity of assemblies. Here, we describe a simpler approach (“Chicago”) based on in vitro reconstituted chromatin. We generated two Chicago data sets with human DNA and developed a statistical model and a new software pipeline (“HiRise”) that can identify poor quality joins and produce accurate, long-range sequence scaffolds. We used these to construct a highly accurate de novo assembly and scaffolding of a human genome with scaffold N50 of 20 Mbp. We also demonstrated the utility of Chicago for improving existing assemblies by reassembling and scaffolding the genome of the American alligator. With a single library and one lane of Illumina HiSeq sequencing, we increased the scaffold N50 of the American alligator from 508 kbp to 10 Mbp.

59 BASIC BIOLOGICAL SCIENCES↗

Replicated Computational Results (RCR) Report for “Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software”

The article by Flegar et al. titled “Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software” presents a novel, practical implementation of an adaptive precision block-Jacobi preconditioner. Performance results using state-of-the-art GPU architectures for the block-Jacobi preconditioner generation and application demonstrate the practical usability of the method, compared to a traditional full-precision block-Jacobi preconditioner. A production-ready implementation is provided in the Ginkgo numerical linear algebra library. In this report, the Ginkgo library is reinstalled and performance results are generated to perform a comparison to the original results when using Ginkgo’s Conjugate Gradient solver with either the full or the adaptive precision block-Jacobi preconditioner for a suite of test problems on an NVIDIA GPU accelerator. After completing this process, the published results are deemed reproducible.

97 MATHEMATICS AND COMPUTING↗

High performance sparse multifrontal solvers on modern GPUs

Here, we have ported the numerical factorization and triangular solve phases of the sparse direct solver STRUMPACK to GPU. STRUMPACK implements sparse LU factorization using the multifrontal algorithm, which performs most of its operations in dense linear algebra operations on so-called frontal matrices of various sizes. Our GPU implementation off-loads these dense linear algebra operations, as well as the sparse scatter–gather operations between frontal matrices. For the larger frontal matrices, our GPU implementation relies on vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs and rocBLAS and rocSOLVER for AMD GPUs. For the smaller frontal matrices we developed custom CUDA and HIP kernels to reduce kernel launch overhead. Overall, high performance is achieved by identifying submatrix factorizations corresponding to sub-trees of the multifrontal assembly tree which fit entirely in GPU memory. The multi-GPU setting uses SLATE (Software for Linear Algebra Targeting Exascale) as a modern GPU-aware replacement for ScaLAPACK. On 4 nodes of SUMMIT the code runs ~10X faster when using all 24 V100 GPUs compared to when it only uses the 168 POWER9 cores. On 8 SUMMIT nodes, using 48 V100 GPUs, the sparse solver reaches over 50TFlop/s. Compared to SuperLU, on a single V100, for a set of 17 matrices our implementation is faster for all but one matrix, and is on average 5X (median 4X) faster

97 MATHEMATICS AND COMPUTING↗

SU-F-I-49: Vendor-Independent, Model-Based Iterative Reconstruction On a Rotating Grid with Coordinate-Descent Optimization for CT Imaging Investigations

Purpose: Iterative reconstruction methods show promise for improving image quality and lowering the dose in helical CT. We aim to develop a novel model-based reconstruction method that offers potential for dose reduction with reasonable computation speed and storage requirements for vendor-independent reconstruction from clinical data on a normal desktop computer. Methods: In 2012, Xu proposed reconstructing on rotating slices to exploit helical symmetry and reduce the storage requirements for the CT system matrix. Inspired by this concept, we have developed a novel reconstruction method incorporating the stored-system-matrix approach together with iterative coordinate-descent (ICD) optimization. A penalized-least-squares objective function with a quadratic penalty term is solved analytically voxel-by-voxel, sequentially iterating along the axial direction first, followed by the transaxial direction. 8 in-plane (transaxial) neighbors are used for the ICD algorithm. The forward problem is modeled via a unique approach that combines the principle of Joseph’s method with trilinear B-spline interpolation to enable accurate reconstruction with low storage requirements. Iterations are accelerated with multi-CPU OpenMP libraries. For preliminary evaluations, we reconstructed (1) a simulated 3D ellipse phantom and (2) an ACR accreditation phantom dataset exported from a clinical scanner (Definition AS, Siemens Healthcare). Image quality was evaluated in the resolution module. Results: Image quality was excellent for the ellipse phantom. For the ACR phantom, image quality was comparable to clinical reconstructions and reconstructions using open-source FreeCT-wFBP software. Also, we did not observe any deleterious impact associated with the utilization of rotating slices. The system matrix storage requirement was only 4.5GB, and reconstruction time was 50 seconds per iteration. Conclusion: Our reconstruction method shows potential for furthering research in low-dose helical CT, in particular as part of our ongoing development of an acquisition/reconstruction pipeline for generating images under a wide range of conditions. Our algorithm will be made available open-source as “FreeCT-ICD”. NIH U01 CA181156; Disclosures (McNitt-Gray): Institutional research agreement, Siemens Healthcare; Past recipient, research grant support, Siemens Healthcare; Consultant, Toshiba America Medical Systems; Consultant, Samsung Electronics.

60 APPLIED LIFE SCIENCES↗

Bringing heterogeneity to the CMS software framework

The advent of computing resources with co-processors, for example Graphics Processing Units (GPU) or Field-Programmable Gate Arrays (FPGA), for use cases like the CMS High-Level Trigger (HLT) or data processing at leadership-class supercomputers imposes challenges for the current data processing frameworks. These challenges include developing a model for algorithms to offload their computations on the co-processors as well as keeping the traditional CPU busy doing other work. The CMS data processing framework, CMSSW, implements multithreading using the Intel Threading Building Blocks (TBB) library, that utilizes tasks as concurrent units of work. In this paper we will discuss a generic mechanism to interact effectively with non-CPU resources that has been implemented in CMSSW. In addition, configuring such a heterogeneous system is challenging. In CMSSW an application is configured with a configuration file written in the Python language. The algorithm types are part of the configuration. The challenge therefore is to unify the CPU and co-processor settings while allowing their implementations to be separate. We will explain how we solved these challenges while minimizing the necessary changes to the CMSSW framework. We will also discuss on a concrete example how algorithms would offload work to NVIDIA GPUs using directly the CUDA API.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Shotgun metagenome data of a defined mock community using Oxford Nanopore, PacBio and Illumina technologies

Metagenomic sequence data from defined mock communities is crucial for the assessment of sequencing platform performance and downstream analyses, including assembly, binning and taxonomic assignment. We report a comparison of shotgun metagenome sequencing and assembly metrics of a defined microbial mock community using the Oxford Nanopore Technologies (ONT) MinION, PacBio and Illumina sequencing platforms. Our synthetic microbial community BMock12 consists of 12 bacterial strains with genome sizes spanning 3.2-7.2 Mbp, 40-73% GC content, and 1.5-7.3% repeats. Size selection of both PacBio and ONT sequencing libraries prior to sequencing was essential to yield comparable relative abundances of organisms among all sequencing technologies. While the Illumina-based metagenome assembly yielded good coverage with few misassemblies, contiguity was greatly improved by both, Illumina + ONT and Illumina + PacBio hybrid assemblies but increased misassemblies, most notably in genomes with high sequence similarity to each other. Our resulting datasets allow evaluation and benchmarking of bioinformatics software on Illumina, PacBio and ONT platforms in parallel.

54 ENVIRONMENTAL SCIENCES↗

Replicated computational results (RCR) report for "BLIS: A framework for rapidly instantiating BLAS functionality"

“BLIS: A Framework for Rapidly Instantiating BLAS Functionality” includes single-platform BLIS performance results for both level-2 and level-3 operations that is competitive with OpenBLAS, ATLAS, and Intel MKL. A detailed description of the configuration used to generate the performance results was provided to the reviewer by the authors. All the software components used in the comparison were reinstalled and new performance results were generated and compared to the original results. After completing this process, the published results are deemed replicable by the reviewer.

97 MATHEMATICS AND COMPUTING↗

A computational modeling framework for pre-clinical evaluation of cardiac mapping systems

There are a variety of difficulties in evaluating clinical cardiac mapping systems, most notably the inability to record the transmembrane potential throughout the entire heart during patient procedures which prevents the comparison to a relevant “gold standard”. Cardiac mapping systems are comprised of hardware and software elements including sophisticated mathematical algorithms, both of which continue to undergo rapid innovation. The purpose of this study is to develop a computational modeling framework to evaluate the performance of cardiac mapping systems. The framework enables rigorous evaluation of a mapping system’s ability to localize and characterize (i.e., focal or reentrant) arrhythmogenic sources in the heart. The main component of our tool is a library of computer simulations of various dynamic patterns throughout the entire heart in which the type and location of the arrhythmogenic sources are known. Our framework allows for performance evaluation for various electrode configurations, heart geometries, arrhythmias, and electrogram noise levels and involves blind comparison of mapping systems against a “silver standard” comprised of computer simulations in which the precise transmembrane potential patterns throughout the heart are known. A feasibility study was performed using simulations of patterns in the human left atria and three hypothetical virtual catheter electrode arrays. Activation times (AcT) and patterns (AcP) were computed for three virtual electrode arrays: two basket arrays with good and poor contact and one high-resolution grid with uniform spacing. The average root mean squared difference of AcTs of electrograms and those of the nearest endocardial action potential was less than 1 ms and therefore appears to be a poor performance metric. In an effort to standardize performance evaluation of mapping systems a novel performance metric is introduced based on the number of AcPs identified correctly and those considered spurious as well as misclassifications of arrhythmia type; spatial and temporal localization accuracy of correctly identified patterns was also quantified. This approach provides a rigorous quantitative analysis of cardiac mapping system performance. Proof of concept of this computational evaluation framework suggests that it could help safeguard that mapping systems perform as expected as well as provide estimates of system accuracy.

59 BASIC BIOLOGICAL SCIENCES↗

Light Output Fitting Software

Light output response of scintillators is crucial to the utilization of organic scintillators as effective tools in radiation detection and measurement. While the response is a continuous distribution, the light output corresponding to the maximum energy deposition is crucial in effectively understanding and simulating a detector. There are a variety of fits derived in literature that will vary for every detector material. The Light Output Response Fitter, or LORF Program is a python script designed to easily and quickly compute and plot fits for a variety of scintillator light output models. It includes a stopping power library constructed from SRIM including Organic Glass, EJ309, EJ301, Stilbene, EJ276, and their deuterated counterparts by default, with the ability for the user to add custom stopping power libraries. The user is also capable of importing the python package and utilizing its in-built functions as appropriate. Uses for this capability include plotting and computing a model with known parameters.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Modeling liquid organic thin films on substrates

We present the rationale, methods, and results of modeling of thin film organic liquids on various substrates. These liquids may coat surfaces (substrates) either as a result of their production, dispersal via aerosols or spills. Identification of unknown coated surfaces using either reflectance or emittance spectroscopy cannot be accomplished simply through reference to reflectance signature libraries since neither the thickness of the liquid layer nor the substrate type is known beforehand and both contribute to the signature. Liquid spectral libraries offer the complex index of refraction (n,k) as a function of wavelength which by itself is useful only for thick (bulk) liquid layers via computation of reflectance and transmittance coefficients using the Fresnel equations. Thin liquid layers both reflect and refract incident light in combination with reflectance from the substrate. We show modeling of various organic liquids on substrates using commercial thin film design and modeling software, as well as Monte Carlo ray tracing software to demonstrate the variety of potential signatures encountered that depend on the thickness of the liquid layer as well as the characteristics of the substrate (metal or dielectric). These substrates give rise to transflectance behavior, while many dielectric substrates have rich absorption features that provide complex signatures that combine attributes of both the liquid and the substrate. Knowledge of the complex index of refraction of both target liquids and substrates is essential in order to synthesize spectra necessary in the application of target identification algorithms.

complex index of refraction↗

The MCDB System for Management and Analysis of Petabyte-Scale Uncertain Data

Analysts working with very large data sets often use statistical models to “guess” at unknown, inaccurate, or missing information associated with the data. For example, a distant object viewed through an optical lens will have its position slightly shifted by imperfections in the lens. Thus, rather than considering the object’s observed position to be absolutely correct, it makes sense to take into account the lens’s imperfections to obtain a probabilistic guess as to the object’s true position. For another example, it might be important to associate some sort of error distribution with each of the individual sensors in an array of magnetometers. This error distribution may be complex and include spatially-driven covariances, because errors in nearby sensors are likely to be correlated (caused, for example, by the presence of some nearby, fixed metal object). This project is concerned with the design and implementation of a prototype data management system called the Monte Carlo Database System, or MCDB for short. MCDB allows an expert-level analyst or statistician to attach arbitrary stochastic models to very large data sets in order to “guess” the values for unknown or inaccurate data, such as the actual position of the observed object in the lens example above. When the resulting data set is analyzed, the underlying stochastic models are used to generate hundreds or thousands of possible data set instances, and each of those possible instances is analyzed separately by MCDB. Thus, MCDB does not just give a single answer to the analysis, but it actually gives an empirical distribution of query results that embody the underlying uncertainty, and can in turn be analyzed using standard statistical techniques. The stochastic models in MCDB are implemented as user-defined, external C++ libraries called Variable Generation functions (VG functions for short). Because the VG function interface is exceedingly general, it allows MCDB to be used in a very wide variety of application domains, in conjunction with virtually any stochastic model. Ultimately, we ended up calling the system that we developed over the course of the project SimSQL, to denote the fact that it allowed for stochastic simulation of data. The software developed during the lifetime of the project is available for download from the SimSQL project website: http://cmj4.web.rice.edu/SimSQL/SimSQL.html.

42 ENGINEERING↗

THE NEW ONLINE METADATA EDITOR FOR GENERATING STRUCTURED METADATA

Nobody is better suited to “describe” data than the scientist who created it. This “description” about a data is called Metadata. In general terms, Metadata represents the who, what, when, where, why and how of the dataset [1]. eXtensible Markup Language (XML) is the preferred output format for metadata, as it makes it portable and, more importantly, suitable for system discoverability. The newly developed ORNL Metadata Editor (OME) is a Web-based tool that allows users to create and maintain XML files containing key information, or metadata, about the research. Metadata include information about the specific projects, parameters, time periods, and locations associated with the data. Such information helps put the research findings in context. In addition, the metadata produced using OME will allow other researchers to find these data via Metadata clearinghouses like Mercury [2][4]. OME is part of ORNL’s Mercury software fleet [2][3]. It was jointly developed to support projects funded by the United States Geological Survey (USGS), U.S. Department of Energy (DOE), National Aeronautics and Space Administration (NASA) and National Oceanic and Atmospheric Administration (NOAA). OME’s architecture provides a customizable interface to support project-specific requirements. Using this new architecture, the ORNL team developed OME instances for USGS’s Core Science Analytics, Synthesis, and Libraries (CSAS&L), DOE’s Next Generation Ecosystem Experiments (NGEE) and Atmospheric Radiation Measurement (ARM) Program, and the international Surface Ocean Carbon Dioxide ATlas (SOCAT).Researchers simply use the ORNL Metadata Editor to enter relevant metadata into a Web-based form. From the information on the form, the Metadata Editor can create an XML file on the server that the editor is installed or to the user’s personal computer. Researchers can also use the ORNL Metadata Editor to modify existing XML metadata files.As an example, an NGEE Arctic scientist use OME to register their datasets to the NGEE data archive and allows the NGEE archive to publish these datasets via a data search portal (http://ngee.ornl.gov/data). These highly descriptive metadata created using OME allows the Archive to enable advanced data search options using keyword, geo-spatial, temporal and ontology filters. Similarly, ARM OME allows scientists or principal investigators (PIs) to submit their data products to the ARM data archive.How would OME help Big Data Centers like the Oak Ridge National Laboratory Distributed Active Archive Center (ORNL DAAC)?The ORNL DAAC is one of NASA’s Earth Observing System Data and Information System (EOSDIS) data centers managed by the Earth Science Data and Information System (ESDIS) Project. The ORNL DAAC archives data produced by NASA's Terrestrial Ecology Program. The DAAC provides data and information relevant to biogeochemical dynamics, ecological data, and environmental processes, critical for understanding the dynamics relating to the biological, geological, and chemical components of the Earth's environment. Typically data produced, archived and analyzed is at a scale of multiple petabytes, which makes the discoverability of the data very challenging. Without proper metadata associated with the data, it is difficult to find the data you are looking for and equally difficult to use and understand the data.OME will allow data centers like the NGEE and ORNL DAAC to produce meaningful, high quality, standards-based, descriptive information about their data products in-turn helping with the data discoverability and interoperability. Useful Links:USGS OME: http://mercury.ornl.gov/OME/NGEE OME: http://ngee-arctic.ornl.gov/ngeemetadata/ARM OME: http://archive2.ornl.gov/armome/Contact: Ranjeet Devarakonda (devarakondar@ornl.gov)References:[1] Federal Geographic Data Committee. Content standard for digital geospatial metadata. Federal Geographic Data Committee, 1998.[2] Devarakonda, Ranjeet, et al. "Mercury: reusable metadata management, data discovery and access system." Earth Science Informatics 3.1-2 (2010): 87-94.[3] Wilson, B. E., Palanisamy, G., Devarakonda, R., Rhyne, B. T., Lindsley, C., & Green, J. (2010). Mercury Toolset for Spatiotemporal Metadata.[4] Pouchard, L. C., Branstetter, M. L., Cook, R. B., Devarakonda, R., Green, J., Palanisamy, G., ... & Noy, N. F. (2013). A Linked Science investigation: enhancing climate change data discovery with semantic technologies. Earth science informatics, 6(3), 175-185.

FGDC CSDGM↗

Calculating uncertainty on K-effective with MONK10 - 14583

Criticality safety assessments require a demonstration that a particular configuration of fissile material has an adequate sub-critical margin (k-effective sufficiently below unity) to ensure that the risk of criticality under normal operation and accident conditions is acceptable. The required sub-critical margin depends upon the uncertainty in the estimated value of k-effective. The uncertainty in the calculated value of k-effective arises from a number of sources, including: manufacturing tolerances on input data to the code (affecting geometry, compositions and densities), uncertainty in the nuclear data used by the code, stochastic uncertainty resulting from Monte Carlo simulation and modelling approximations/errors, including the inevitable bugs in the software. The ANSWERS Software Service, in collaboration with industrial partners, is developing a number of techniques to better understand and quantify uncertainty on predicted values of k-effective, using MONK. The SPRUCE utility code has been developed to allow uncertainty to be estimated using sampling methods. This can include the sampling of input parameters (including dimensions, compositions and densities) from statistical distributions. It can also include sampling different nuclear data libraries. A set of nuclear data libraries has been generated for this purpose by sampling from statistical distributions that represent the uncertainties in the published nuclear data evaluated files; a set of libraries has been produced for Latin Hypercube Sampling. By varying the input data and nuclear data, separate and combined uncertainties due to manufacturing tolerances and nuclear data can be derived. By performing least squares fitting on the results it is also possible to estimate the contribution of each of the uncertain inputs and a sensitivity method in MONK can break down the nuclear data uncertainty. (authors)

42 ENGINEERING↗

Performance Debugging and Tuning of Flash-X with Data Analysis Tools

State-of-the-art multiphysics simulations running on large scale leadership computing platforms have many variables contributing to their performance and scaling behavior. We recently encountered an interesting performance anomaly in Flash-X, a multiphysics multicomponent simulation software, when characterizing its performance behavior on several large-scale HPC platforms. The anomaly was tracked down to the interaction between the use of dynamic allocation of scratch data and data locality in the cache hierarchy. In this paper we present the details of unexpected performance variability of Flash-X, its extensive analysis using the performance measurement tool TAU to collect the data and Python data analysis libraries to explore the data, and our insights from this experience. In this process, we discovered and removed or mitigated two additional performance limiting bottlenecks for performance tuning.

Huck, Kevin↗

Geant4 based positron beam source (GPos) v1.0

GPos is a software that was created to determine the properties of positron beams resulting from the interaction of the LBNL BELLA center PetaWatt laser-driven plasma-capillary accelerated electron beam and the atoms of a thin solid target. GPos is written in C++, easily compiled with cmake and the spack package manager, which allows for multi-thread and MPI parallel computing. Its functions expand on the Geant4 toolkit library and allow for propagation of the modelled particles through vacuum drift distances with a focusing element (thin lens approximation). Users can change beam-foil-drift-lens parameters - to adapt GPos to other particle sources and infrastructures - in a simple input file. The code particle data output format, openPMD, which is compatible, for example, with the input of the ECP WarpX project code used to explore the physics of particle acceleration in plasmas. Using GPos in conjunction to WarpX allowed us to test various configurations for designing a high-quality and high-energy positron source at BELLA -required for us to address positron acceleration challenges in the development of future linear colliders. GPos can also be advantageous when tackling the physics of muon sources for future muon colliders as well as for the investigation of positron sources in lower energy regimes for applications like annihilation spectroscopy and astrophysical gamma-ray-bursts.

Pinto de Almeida Amorim, Ligia↗

Data Analyst - A Device for Continuous Repeating Real-Time Assay of HPGe, CZT, or Scintillation Gamma Spectra - 18574

There are many situations where a continuous repeating series of quantitative gamma spectral assays is desirable. These include measurements of fluid going through a pipe, of items on a conveyor passing under a detector, a detector aimed at the ground on a moving platform, or a detector in free air as a plume of radioactivity drifts by. While Canberra and many others have built and deployed many of these devices, they have all been custom built for that particular application, and rather expensive. The Data Analyst [DA] is a general purpose device that is designed to fulfill many of these applications without modification, but can be fairly-easily modified if needed. The DA is a small [12 x 18 x 5 cm], low-power [3 W] box that accepts the input from external MCAs, controls the acquisition cycle, analyzes the results, and transmits them to the outside world when needed. The detector-MCAs supported are HPGe with Lynx MCA, NaI or CeBr or LaBr with Osprey MCA, and CZT or CsI with internal MCAs. The DA internal software used for analysis is the full standard Genie analysis software, which is very widely used and well accepted in the industry. This allows exported spectral files to be reviewed and confirmed by Experts, and reanalyzed if needed. The DA hardware is totally autonomous. As soon as power is applied, it starts the pre-programmed sequences of acquisition and analysis. Multiple analysis sequences [called Workflows] can be in operation simultaneously, e.g. with different count times and different nuclide libraries. Only the detector with MCA, and the DA are required in the field. A PC can be connected to the DA locally or remotely via WiFi, USB, or Ethernet. The PC is needed to setup the DA, and to readout the results. The DA also supports [with application-specific programming] external Inputs/Outputs either for recording with the analysis results or for use in the computation of the result, e.g. temperature, flow rate, pressure, ... ]. The DA also has an internal GPS receiver with external antenna. Applications of the DA that are discussed include stack gas effluent activity where a portion of the effluent stream was extracted to flow through a shield with a HPGe detector and Lynx MCA, and measurement of reactor coolant activity where a shielded CZT detector and MCA were aimed at various sections of reactor primary coolant during a maintenance outage. Potential other applications are also described. (author)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

SIDIS-RC EvGen: A Monte-Carlo event generator of semi-inclusive deep inelastic scattering with the lowest-order QED radiative corrections

SIDIS-RC EvGen is a C++ standalone Monte-Carlo event generator for studies of semi-inclusive deep inelastic scattering (SIDIS) processes at medium to high lepton beam energies. In particular, the generator contains binary and library components for generating SIDIS events and calculating cross sections for unpolarized or longitudinally polarized beam and unpolarized, longitudinally or transversely polarized target. The structure of the generator incorporates transverse momentum-dependent parton distribution and fragmentation functions, whereby we obtain multi-dimensional binned simulation results, which will facilitate the extraction of important information about the three-dimensional nucleon structure from SIDIS measurements. In order to build this software, we have used recent elaborate QED calculations of the lowest-order radiative effects, applied to the leading order Born cross section in SIDIS. Here, in this paper, we provide details on the theoretical formalism as well as the construction and operation of SIDIS-RC EvGen, e.g., how we handle the event generation process and perform multi-dimensional integration. We also provide example programs, flowcharts, and numerical results on azimuthal transverse single-spin asymmetries.

97 MATHEMATICS AND COMPUTING↗