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NASA NTRS2012

Flow Analysis Tool White Paper

Faster networks are continually being built to accommodate larger data transfers. While it is intuitive to think that implementing faster networks will result in higher throughput rates, this is often not the case. There are many elements involved in data transfer, many of which are beyond the scope of the network itself. Although networks may get bigger and support faster technologies, the presence of other legacy components, such as older application software or kernel parameters, can often cause bottlenecks. Engineers must be able to identify when data flows are reaching a bottleneck that is not imposed by the network and then troubleshoot it using the tools available to them. The current best practice is to collect as much information as possible on the network traffic flows so that analysis is quick and easy. Unfortunately, no single method of collecting this information can sufficiently capture the whole endto- end picture. This becomes even more of a hurdle when large, multi-user systems are involved. In order to capture all the necessary information, multiple data sources are required. This paper presents a method for developing a flow analysis tool to effectively collect network flow data from multiple sources and provide that information to engineers in a clear, concise way for analysis. The purpose of this method is to collect enough information to quickly (and automatically) identify poorly performing flows along with the cause of the problem. The method involves the development of a set of database tables that can be populated with flow data from multiple sources, along with an easyto- use, web-based front-end interface to help network engineers access, organize, analyze, and manage all the information.

Boscia, Nichole K.↗
NASA NTRS2011

Testing of the Support Vector Machine for Binary-Class Classification

The Support Vector Machine is a powerful algorithm, useful in classifying data in to species. The Support Vector Machines implemented in this research were used as classifiers for the final stage in a Multistage Autonomous Target Recognition system. A single kernel SVM known as SVMlight, and a modified version known as a Support Vector Machine with K-Means Clustering were used. These SVM algorithms were tested as classifiers under varying conditions. Image noise levels varied, and the orientation of the targets changed. The classifiers were then optimized to demonstrate their maximum potential as classifiers. Results demonstrate the reliability of SMV as a method for classification. From trial to trial, SVM produces consistent results

autonomous target recognition systemr↗
DOE OSTI2015

Model-based versus specific dosimetry in diagnostic context: Comparison of three dosimetric approaches

Purpose: The dosimetric assessment of novel radiotracers represents a legal requirement in most countries. While the techniques for the computation of internal absorbed dose in a therapeutic context have made huge progresses in recent years, in a diagnostic scenario the absorbed dose is usually extracted from model-based lookup tables, most often derived from International Commission on Radiological Protection (ICRP) or Medical Internal Radiation Dose (MIRD) Committee models. The level of approximation introduced by these models may impact the resulting dosimetry. The aim of this work is to establish whether a more refined approach to dosimetry can be implemented in nuclear medicine diagnostics, by analyzing a specific case. Methods: The authors calculated absorbed doses to various organs in six healthy volunteers administered with flutemetamol ({sup 18}F) injection. Each patient underwent from 8 to 10 whole body 3D PET/CT scans. This dataset was analyzed using a Monte Carlo (MC) application developed in-house using the toolkit GATE that is capable to take into account patient-specific anatomy and radiotracer distribution at the voxel level. They compared the absorbed doses obtained with GATE to those calculated with two commercially available software: OLINDA/EXM and STRATOS implementing a dose voxel kernel convolution approach. Results: Absorbed doses calculated with GATE were higher than those calculated with OLINDA. The average ratio between GATE absorbed doses and OLINDA’s was 1.38 ± 0.34 σ (from 0.93 to 2.23). The discrepancy was particularly high for the thyroid, with an average GATE/OLINDA ratio of 1.97 ± 0.83 σ for the six patients. Differences between STRATOS and GATE were found to be higher. The average ratio between GATE and STRATOS absorbed doses was 2.51 ± 1.21 σ (from 1.09 to 6.06). Conclusions: This study demonstrates how the choice of the absorbed dose calculation algorithm may introduce a bias when gamma radiations are of importance, as is the case in nuclear medicine diagnostics.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2020

On decoupling the integrals of cosmological perturbation theory

ABSTRACT Perturbation theory (PT) is often used to model statistical observables capturing the translation and rotation-invariant information in cosmological density fields. PT produces higher order corrections by integration over linear statistics of the density fields weighted by kernels resulting from recursive solution of the fluid equations. These integrals quickly become high dimensional and naively require increasing computational resources the higher the order of the corrections. Here, we show how to decouple the integrands that often produce this issue, enabling PT corrections to be computed as a sum of products of independent 1D integrals. Our approach is related to a commonly used method for calculating multiloop Feynman integrals in quantum field theory, the Gegenbauer Polynomial x-Space Technique. We explicitly reduce the three terms entering the 2-loop power spectrum, formally requiring 9D integrations, to sums over successive 1D radial integrals. These 1D integrals can further be performed as convolutions, rendering the scaling of this method Nglog Ng with Ng the number of grid points used for each fast Fourier transform. This method should be highly enabling for upcoming large-scale structure redshift surveys where model predictions at an enormous number of cosmological parameter combinations will be required by Monte Carlo Markov Chain searches for the best-fitting values.

Slepian, Zachary↗
DOE OSTI2026

Small-𝑥 behavior in QCD from maximal entanglement and conformal invariance

Recent evidence suggests that, at small Bjorken 𝑥, QCD evolution drives the proton into a state of maximal entanglement. If the evolution kernel is assumed to be conformally invariant—as is the case for the Balitsky-Fadin-Kuraev-Lipatov equation—we can describe it by a conformal field theory. Moreover, the central charge 𝑐 of the corresponding conformal field theory emerges as the key parameter governing the 𝑥 dependence of both the entanglement entropy and the structure function. Here we apply the exact Bethe ansatz methods to the quantum spin chain dual to Lipatov’s high energy effective action to extract the central charge of the theory, and find that 𝑐 = 1. This implies the ∼𝑥 −1/3 small 𝑥 behavior for the structure function—the prediction that can be tested at the forthcoming Electron-Ion Collider.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2012

A statistically defined anthropomorphic software breast phantom

Purpose: Digital anthropomorphic breast phantoms have emerged in the past decade because of recent advances in 3D breast x-ray imaging techniques. Computer phantoms in the literature have incorporated power-law noise to represent glandular tissue and branching structures to represent linear components such as ducts. When power-law noise is added to those phantoms in one piece, the simulated fibroglandular tissue is distributed randomly throughout the breast, resulting in dense tissue placement that may not be observed in a real breast. The authors describe a method for enhancing an existing digital anthropomorphic breast phantom by adding binarized power-law noise to a limited area of the breast. Methods: Phantoms with (0.5 mm){sup 3} voxel size were generated using software developed by Bakic et al. Between 0% and 40% of adipose compartments in each phantom were replaced with binarized power-law noise ({beta} = 3.0) ranging from 0.1 to 0.6 volumetric glandular fraction. The phantoms were compressed to 7.5 cm thickness, then blurred using a 3 Multiplication-Sign 3 boxcar kernel and up-sampled to (0.1 mm){sup 3} voxel size using trilinear interpolation. Following interpolation, the phantoms were adjusted for volumetric glandular fraction using global thresholding. Monoenergetic phantom projections were created, including quantum noise and simulated detector blur. Texture was quantified in the simulated projections using power-spectrum analysis to estimate the power-law exponent {beta} from 25.6 Multiplication-Sign 25.6 mm{sup 2} regions of interest. Results: Phantoms were generated with total volumetric glandular fraction ranging from 3% to 24%. Values for {beta} (averaged per projection view) were found to be between 2.67 and 3.73. Thus, the range of textures of the simulated breasts covers the textures observed in clinical images. Conclusions: Using these new techniques, digital anthropomorphic breast phantoms can be generated with a variety of glandular fractions and patterns. {beta} values for this new phantom are comparable with published values for breast tissue in x-ray projection modalities. The combination of conspicuous linear structures and binarized power-law noise added to a limited area of the phantom qualitatively improves its realism.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2025

Conformal Hierarchical Simulation-Based Inference with Local Validity

Trustworthy and interpretable uncertainty quantification is a long-standing challenge in artificial intelligence. Simulation-based inference (SBI) comprises a broad swath of approaches for estimating latent parameters with uncertainties. Although flexible neural density estimators in SBI can be remark- ably expressive capturing highly structured, high-dimensional posteriors their credible regions can be badly mis-calibrated and are often only accompanied by heuristic coverage checks. We present the first SBI framework that delivers finite-sample local valid coverage guarantees that hold in the neighborhood of each observation. Our framework can couple any off-the-shelf hierarchical SBI engine with a confor- mal Bayesian post-processing step that operates on the posterior predictive density. A kernel-weighted conformity score adapts the conformal quantile to the local geometry of the data, yielding prediction sets that are simultaneously (i) marginally calibrated, (ii) locally valid, and (iii) hierarchical, handling global and observation-specific parameters in a single pass. Through experiments on synthetic data and benchmarks from neuroscience and physics, we show that our approach attains 1 − α coverage, where prior SBI methods under- or over-cover. Our approach also maintains a competitive, credible set size with minimal computational overhead. Finally, our approach can be used to make predictions on real data and give valid credible regions modulo weight-initialization-based model mis-specification.

Trivedi, Shubhendu [Fermilab]↗
NASA NTRS2013

SpF: Enabling Petascale Performance for Pseudospectral Dynamo Models

Pseudospectral (PS) methods possess a number of characteristics (e.g., efficiency, accuracy, natural boundary conditions) that are extremely desirable for dynamo models. Unfortunately, dynamo models based upon PS methods face a number of daunting challenges, which include exposing additional parallelism, leveraging hardware accelerators, exploiting hybrid parallelism, and improving the scalability of global memory transposes. Although these issues are a concern for most models, solutions for PS methods tend to require far more pervasive changes to underlying data and control structures. Further, improvements in performance in one model are difficult to transfer to other models, resulting in significant duplication of effort across the research community.We have developed an extensible software framework for pseudospectral methods called SpF that is intended to enable extreme scalability and optimal performance. High-level abstractions provided by SpF unburden applications of the responsibility of managing domain decomposition and load balance while reducing the changes in code required to adapt to new computing architectures. The key design concept in SpF is that each phase of the numerical calculation is partitioned into disjoint numerical kernels that can be performed entirely in-processor. The granularity of domain-decomposition provided by SpF is only constrained by the data-locality requirements of these kernels. SpF builds on top of optimized vendor libraries for common numerical operations such as transforms, matrix solvers, etc., but can also be configured to use open source alternatives for portability. SpF includes several alternative schemes for global data redistribution and is expected to serve as an ideal testbed for further research into optimal approaches for different network architectures.In this presentation, we will describe the basic architecture of SpF as well as preliminary performance data and experience with adapting legacy dynamo codes. We will conclude with a discussion of planned extensions to SpF that will provide pseudospectral applications with additional flexibility with regard to time integration, linear solvers, and discretization in the radial direction.

Pseudospectral (PS)↗
DOE OSTI2016

Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2011

An Application of Multivariate Statistical Analysis for Query-Driven Visualization

Driven by the ability to generate ever-larger, increasingly complex data, there is an urgent need in the scientific community for scalable analysis methods that can rapidly identify salient trends in scientific data. Query-Driven Visualization (QDV) strategies are among the small subset of techniques that can address both large and highly complex datasets. This paper extends the utility of QDV strategies with a statistics-based framework that integrates non-parametric distribution estimation techniques with a new segmentation strategy to visually identify statistically significant trends and features within the solution space of a query. In this framework, query distribution estimates help users to interactively explore their query's solution and visually identify the regions where the combined behavior of constrained variables is most important, statistically, to their inquiry. Our new segmentation strategy extends the distribution estimation analysis by visually conveying the individual importance of each variable to these regions of high statistical significance. We demonstrate the analysis benefits these two strategies provide and show how they may be used to facilitate the refinement of constraints over variables expressed in a user's query. We apply our method to datasets from two different scientific domains to demonstrate its broad applicability.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

MO-FG-204-04: How Iterative Reconstruction Algorithms Affect the NPS of CT Images

Purpose: To evaluate how the third generation model based iterative reconstruction (MBIR) compares with filtered back-projection (FBP), adaptive statistical iterative reconstruction (ASiR), and the second generation MBIR based on noise power spectrum (NPS) analysis over a wide range of clinically applicable dose levels. Methods: The Catphan 600 CTP515 module, surrounded by an oval, fat-equivalent ring to mimic patient size/shape, was scanned on a GE HD750 CT scanner at 1, 2, 3, 6, 12 and 19mGy CTDIvol levels with typical patient scan parameters: 120kVp, 0.8s, 40mm beam width, large SFOV, 0.984 pitch and reconstructed thickness 2.5mm (VEO3.0: Abd/Pelvis with Texture and NR05). At each CTDIvol level, 10 repeated scans were acquired for achieving sufficient data sampling. The images were reconstructed using Standard kernel with FBP; 20%, 40% and 70% ASiR; and two versions of MBIR (VEO2.0 and 3.0). For evaluating the effect of the ROI spatial location to the Result of NPS, 4 ROI groups were categorized based on their distances from the center of the phantom. Results: VEO3.0 performed inferiorly comparing to VEO2.0 over all dose levels. On the other hand, at low dose levels (less than 3 mGy), it clearly outperformed ASiR and FBP, in NPS values. Therefore, the lower the dose level, the relative performance of MBIR improves. However, the shapes of the NPS show substantial differences in horizontal and vertical sampling dimensions. These differences may determine the characteristics of the noise/texture features in images, and hence, play an important role in image interpretation. Conclusion: The third generation MBIR did not improve over the second generation MBIR in term of NPS analysis. The overall performance of both versions of MBIR improved as compared to other reconstruction algorithms when dose was reduced. The shapes of the NPS curves provided additional value for future characterization of the image noise/texture features.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2021

Coaction and double-copy properties of configuration-space integrals at genus zero

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension α' or the dimensional-regularization parameter ϵ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in (p, 2) minimal models, which can be normalized to become uniformly transcendental in the p → ∞ limit.

Conformal Field Theory↗
DOE OSTI2025

Accurately simulating core-collapse self-interacting dark matter halos

The properties of satellite halos provide a promising probe for dark matter (DM) physics. Observations have motivated current efforts to explain surprisingly compact DM halos. If DM is not collisionless, but has strong self-interactions, halos can undergo gravothermal collapse, leading to higher densities in the central region of the halo. However, it is challenging to model this collapse phase from first principles. To improve on this, we sought to better understand the numerical challenges and convergence properties of self-interacting dark matter (SIDM) N-body simulations in the collapse phase. Especially, our aim was to better understand the evolution of satellite halos. To do so, we ran SIDM N-body simulations of a low-mass halo in isolation and within an external gravitational potential. The simulation set-up was motivated by the perturber of the stellar stream GD-1. We find that the halo evolution is very sensitive to energy conservation errors, and a SIDM kernel size that is too large can artificially speed up the collapse. Moreover, we demonstrate that the King model can describe the density profile at small radii for the late stages that we have simulated. Furthermore, for our most highly resolved simulation (N = 5 × 10 7 ) we have made the data public. It can serve as a benchmark. Overall, we find that the current numerical methods do not suffer from convergence problems in the late collapse phase and provide guidance on how to choose numerical parameters, for example that the energy conservation error is better kept well below 1%. This allows simulations to be run of halos that become concentrated enough to explain observations of GD-1-like stellar streams or strong gravitational lensing systems.

dark matter↗
DOE OSTI2016

Point Spread Function Estimation and Uncertainty Quantification

An important component of analyzing images quantitatively is modeling image blur due to effects from the system for image capture. When the effect of image blur is assumed to be translation invariant and isotropic, it can be generally modeled as convolution with a radially symmetric kernel, called the point spread function (PSF). Standard techniques for estimating the PSF involve imaging a bright point source, but this is not always feasible (e.g. high energy radiography). This work provides a novel non-parametric approach to estimating the PSF from a calibration image of a vertical edge. Moreover, the approach is within a hierarchical Bayesian framework that in addition to providing a method for estimation, also gives a quantification of uncertainty in the estimate by Markov Chain Monte Carlo methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Key results from irradiation and post-irradiation examination of AGR-1 UCO TRISO fuel

The AGR-1 irradiation experiment was performed as the first test of tristructural isotropic (TRISO) fuel in the US Advanced Gas Reactor Fuel Development and Qualification Program. The experiment consisted of 72 right cylinder fuel compacts containing approximately 3 × 105 coated fuel particles with uranium oxide/uranium carbide (UCO) fuel kernels. The fuel was irradiated in the Advanced Test Reactor for a total of 620 effective full power days. Fuel burnup ranged from 11.3 to 19.6% fissions per initial metal atom and time-average, volume-average irradiation temperatures of the individual compacts ranged from 955 to 1136 °C. This paper focuses on key results from the irradiation and post-irradiation examination, which revealed a robust fuel with excellent performance characteristics under the conditions tested and have significantly improved the understanding of UCO coated particle fuel irradiation behavior. The fuel exhibited zero TRISO coating failures (failure of all three dense coating layers) during irradiation and post-irradiation safety testing at temperatures up to 1700 °C. Advanced PIE methods have allowed particles with SiC coating failure that were discovered to be present in a very-low population to be isolated and meticulously examined, which has elucidated the specific causes of SiC failure in these specimens. The level of fission product release from the fuel during irradiation and post-irradiation safety testing has been studied in detail. Results indicated very low release of krypton and cesium through intact SiC and modest release of europium and strontium, while also confirming the potential for significant silver release through the coatings depending on irradiation conditions. Focused study of fission products within the coating layers of irradiated particles down to nanometer length scales has provided new insights into fission product transport through the coating layers and the role various fission products may have on coating integrity. The broader implications of these results and the application of lessons learned from AGR-1 to fuel fabrication and post-irradiation examination for subsequent fuel irradiation experiments as part of the US fuel program are also discussed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2012

Quantitative comparison of noise texture across CT scanners from different manufacturers

Purpose: To quantitatively compare noise texture across computed tomography (CT) scanners from different manufacturers using the noise power spectrum (NPS). Methods: The American College of Radiology CT accreditation phantom (Gammex 464, Gammex, Inc., Middleton, WI) was imaged on two scanners: Discovery CT 750HD (GE Healthcare, Waukesha, WI), and SOMATOM Definition Flash (Siemens Healthcare, Germany), using a consistent acquisition protocol (120 kVp, 0.625/0.6 mm slice thickness, 250 mAs, and 22 cm field of view). Images were reconstructed using filtered backprojection and a wide selection of reconstruction kernels. For each image set, the 2D NPS were estimated from the uniform section of the phantom. The 2D spectra were normalized by their integral value, radially averaged, and filtered by the human visual response function. A systematic kernel-by-kernel comparison across manufacturers was performed by computing the root mean square difference (RMSD) and the peak frequency difference (PFD) between the NPS from different kernels. GE and Siemens kernels were compared and kernel pairs that minimized the RMSD and |PFD| were identified. Results: The RMSD (|PFD|) values between the NPS of GE and Siemens kernels varied from 0.01 mm{sup 2} (0.002 mm{sup -1}) to 0.29 mm{sup 2} (0.74 mm{sup -1}). The GE kernels 'Soft,''Standard,''Chest,' and 'Lung' closely matched the Siemens kernels 'B35f,''B43f,''B41f,' and 'B80f' (RMSD < 0.05 mm{sup 2}, |PFD| < 0.02 mm{sup -1}, respectively). The GE 'Bone,''Bone+,' and 'Edge' kernels all matched most closely with Siemens 'B75f' kernel but with sizeable RMSD and |PFD| values up to 0.18 mm{sup 2} and 0.41 mm{sup -1}, respectively. These sizeable RMSD and |PFD| values corresponded to visually perceivable differences in the noise texture of the images. Conclusions: It is possible to use the NPS to quantitatively compare noise texture across CT systems. The degree to which similar texture across scanners could be achieved varies and is limited by the kernels available on each scanner.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2014

Grating-based phase contrast tomosynthesis imaging: Proof-of-concept experimental studies

Purpose: This paper concerns the feasibility of x-ray differential phase contrast (DPC) tomosynthesis imaging using a grating-based DPC benchtop experimental system, which is equipped with a commercial digital flat-panel detector and a medical-grade rotating-anode x-ray tube. An extensive system characterization was performed to quantify its imaging performance. Methods: The major components of the benchtop system include a diagnostic x-ray tube with a 1.0 mm nominal focal spot size, a flat-panel detector with 96 μm pixel pitch, a sample stage that rotates within a limited angular span of ±30°, and a Talbot-Lau interferometer with three x-ray gratings. A total of 21 projection views acquired with 3° increments were used to reconstruct three sets of tomosynthetic image volumes, including the conventional absorption contrast tomosynthesis image volume (AC-tomo) reconstructed using the filtered-backprojection (FBP) algorithm with the ramp kernel, the phase contrast tomosynthesis image volume (PC-tomo) reconstructed using FBP with a Hilbert kernel, and the differential phase contrast tomosynthesis image volume (DPC-tomo) reconstructed using the shift-and-add algorithm. Three inhouse physical phantoms containing tissue-surrogate materials were used to characterize the signal linearity, the signal difference-to-noise ratio (SDNR), the three-dimensional noise power spectrum (3D NPS), and the through-plane artifact spread function (ASF). Results: While DPC-tomo highlights edges and interfaces in the image object, PC-tomo removes the differential nature of the DPC projection data and its pixel values are linearly related to the decrement of the real part of the x-ray refractive index. The SDNR values of polyoxymethylene in water and polystyrene in oil are 1.5 and 1.0, respectively, in AC-tomo, and the values were improved to 3.0 and 2.0, respectively, in PC-tomo. PC-tomo and AC-tomo demonstrate equivalent ASF, but their noise characteristics quantified by the 3D NPS were found to be different due to the difference in the tomosynthesis image reconstruction algorithms. Conclusions: It is feasible to simultaneously generate x-ray differential phase contrast, phase contrast, and absorption contrast tomosynthesis images using a grating-based data acquisition setup. The method shows promise in improving the visibility of several low-density materials and therefore merits further investigation.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1991

Note on unit tangent vector computation for homotopy curve tracking on a hypercube

Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map and subsequent tracking of some smooth curve in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vector at different points along the zero curve, which amounts to calculating the kernel of the n x (n + 1) Jacobian matrix. While computing the tangent vector is just one part of the curve tracking algorithm, it can require a significant percentage of the total tracking time. This note presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms for the tangent vector computation on a hypercube.

Chakraborty, A.↗