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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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Model-Free Primal-Dual Methods for Network Optimization with Application to Real-Time Optimal Power Flow

This paper examines the problem of real-time optimization of networked systems and develops online algorithms that steer the system towards the optimal trajectory without explicit knowledge of the system model. The problem is modeled as a dynamic optimization problem with time-varying performance objectives and engineering constraints. The design of the algorithms leverages the online zero-order primal-dual projected-gradient method. In particular, the primal step that involves the gradient of the objective function (and hence requires a networked systems model) is replaced by its zero-order approximation with two function evaluations using a deterministic perturbation signal. The evaluations are performed using the measurements of the system output, hence giving rise to a feedback interconnection, with the optimization algorithm serving as a feedback controller. The paper provides some insights on the stability and tracking properties of this interconnection. Finally, the paper applies this methodology to a real-time optimal power flow problem in power systems, and shows its efficacy on the IEEE 37-node distribution test feeder for reference power tracking and voltage regulation.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2023

BUQEYE guide to projection-based emulators in nuclear physics

The BUQEYE collaboration (Bayesian Uncertainty Quantification: Errors in Your effective field theory) presents a pedagogical introduction to projection-based, reduced-order emulators for applications in low-energy nuclear physics. The term emulator refers here to a fast surrogate model capable of reliably approximating high-fidelity models. As the general tools employed by these emulators are not yet well-known in the nuclear physics community, we discuss variational and Galerkin projection methods, emphasize the benefits of offline-online decompositions, and explore how these concepts lead to emulators for bound and scattering systems that enable fast and accurate calculations using many different model parameter sets. We also point to future extensions and applications of these emulators for nuclear physics, guided by the mature field of model (order) reduction. All examples discussed here and more are available as interactive, open-source Python code so that practitioners can readily adapt projection-based emulators for their own work.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Subcellular Feature-Based Classification of α and β Cells Using Soft X-ray Tomography

The dysfunction of α and β cells in pancreatic islets can lead to diabetes. Many questions remain on the subcellular organization of islet cells during the progression of disease. Existing three-dimensional cellular mapping approaches face challenges such as time-intensive sample sectioning and subjective cellular identification. To address these challenges, we have developed a subcellular feature-based classification approach, which allows us to identify α and β cells and quantify their subcellular structural characteristics using soft X-ray tomography (SXT). We observed significant differences in whole-cell morphological and organelle statistics between the two cell types. Additionally, we characterize subtle biophysical differences between individual insulin and glucagon vesicles by analyzing vesicle size and molecular density distributions, which were not previously possible using other methods. These sub-vesicular parameters enable us to predict cell types systematically using supervised machine learning. We also visualize distinct vesicle and cell subtypes using Uniform Manifold Approximation and Projection (UMAP) embeddings, which provides us with an innovative approach to explore structural heterogeneity in islet cells. This methodology presents an innovative approach for tracking biologically meaningful heterogeneity in cells that can be applied to any cellular system.

3D cell mapping↗
NASA NTRS1975

Implementation of the 64-meter-diameter Antennas at the Deep Space Stations in Australia and Spain

The management and construction aspects of the Overseas 64-m Antenna Project in which two 64-m antennas were constructed at the Tidbinbilla Deep Space Communications Complex in Australia, and at the Madrid Deep Space Communications Complex in Spain are described. With the completion of these antennas the Deep Space Network is equipped with three 64-m antennas spaced around the world to maintain continuous coverage of spacecraft operations. These antennas provide approximately a 7-db gain over the capabilities of the existing 26-m antenna nets. The report outlines the project organization and management, resource utilization, fabrication, quality assurance, and construction methods by which the project was successfully completed. Major problems and their solutions are described as well as recommendations for future projects.

Bartos, K. P.↗
NASA NTRS2011

Climate Hazard Assessment for Stakeholder Adaptation Planning in New York City

This paper describes a time-sensitive approach to climate change projections, developed as part of New York City's climate change adaptation process, that has provided decision support to stakeholders from 40 agencies, regional planning associations, and private companies. The approach optimizes production of projections given constraints faced by decision makers as they incorporate climate change into long-term planning and policy. New York City stakeholders, who are well-versed in risk management, helped pre-select the climate variables most likely to impact urban infrastructure, and requested a projection range rather than a single 'most likely' outcome. The climate projections approach is transferable to other regions and consistent with broader efforts to provide climate services, including impact, vulnerability, and adaptation information. The approach uses 16 Global Climate Models (GCMs) and three emissions scenarios to calculate monthly change factors based on 30-year average future time slices relative to a 30- year model baseline. Projecting these model mean changes onto observed station data for New York City yields dramatic changes in the frequency of extreme events such as coastal flooding and dangerous heat events. Based on these methods, the current 1-in-10 year coastal flood is projected to occur more than once every 3 years by the end of the century, and heat events are projected to approximately triple in frequency. These frequency changes are of sufficient magnitude to merit consideration in long-term adaptation planning, even though the precise changes in extreme event frequency are highly uncertain

Horton, Radley M.↗
NASA NTRS2017

On High-Order Upwind Methods for Advection

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom 𝐾 per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 2𝐾−1, which is higher than the expected order of 𝐾.

high-order methods↗
DOE OSTI2023

Extending Parsimonious Bayesian Inference

Parsimonious Bayesian inference is a theoretical framework for efficient data assimilation that seeks to balance increased consistency between predictions and training data against corresponding increases in model complexity. Within this framework, over-training is understood as optimization that encodes excessive information within model parameters while only achieving small improvements between predictions and training data. This project aims to develop practical methods of limiting excess model information during optimization. One key observation is that practical heuristics for parsimonious learning in high-dimensions must balance expressivity, i.e. the ability of the model to capture diverse predictions with only a few non-zero parameters, against discoverability, i.e. the ability to train the model with gradient-based optimization and drive parameters to low information states. As such, we developed logical activation functions that are able to adaptively approximate arbitrary truth tables that define Boolean logic operations within a probabilistic framework. These functions have demonstrated the ability to learn exclusive disjunction (XOR) and conditioned disjunction (if [condition] then [result_if_true] else [result_if_false]) within a single layer of a neural network. To efficiently exploit these activation functions to drive parsimonious learning required several other advances within the domain of variational inference. The most efficient form of complexity suppression is structured sparsification, driving most model parameters to zero while achieving the structural coherence among nonzeros needed for bandwidth reduction. Such models are not only far more efficient at suppressing information-theoretic complexity, they also reduce the other forms of complexity (computations, communication, storage, and the number of dependencies needed to evaluate predictions). Aiming to support enhanced sparsification, this project examined new approaches to high-dimensional variational inference that allow us to calibrate and control parameter uncertainty during optimization. By identifying which parameters can sustain sparsifying perturbations with little impact on prediction quality, we can develop better pruning strategies by framing them as approximate Bayesian inference. These advances also open paths to mitigate concerns with deploying advanced learning methods in resource-constrained environments, such as running models on power-limited or communication-limited devices.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Improved energies and local energies with weighted variational Monte Carlo

Neural network parametrizations have increasingly been used to represent the ground and excited states in variational Monte Carlo (VMC) with promising results. However, traditional VMC methods only optimize the wave function in regions of peak probability. The wave function is uncontrolled in the tails of the probability distribution, which can limit the accuracy of the trained wave function. To improve the approximation accuracy in the probability tails, this paper interprets VMC as a gradient flow in the space of wave functions, followed by a projection step. From this perspective, arbitrary probability distributions can be used in the projection step, allowing the user to prioritize accuracy in different regions of state space. Motivated by this theoretical perspective, the paper tests a weighted VMC method on the antiferromagnetic Heisenberg model for a periodic spin chain. Compared to traditional VMC, weighted VMC reduces the error in the ground state energy by a factor of 2, and it reduces the errors in the local energies away from the mode by large factors of 10 2 –10 4 .

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1982

Analysis of defect structure in silicon. Silicon sheet growth development for the large area silicon sheet task of the Low-Cost Solar array Project

One hundred ninety-three silicon sheet samples, approximately 880 square centimeters, were analyzed for twin boundary density, dislocation pit density, and grain boundary length. One hundred fifteen of these samples were manufactured by a heat exchanger method, thirty-eight by edge defined film fed growth, twenty-three by the silicon on ceramics process, and ten by the dendritic web process. Seven solar cells were also step-etched to determine the internal defect distribution on these samples. Procedures were developed or the quantitative characterization of structural defects such as dislocation pits, precipitates, twin & grain boundaries using a QTM 720 quantitative image analyzing system interfaced with a PDP 11/03 mini computer. Characterization of the grain boundary length per unit area for polycrystalline samples was done by using the intercept method on an Olympus HBM Microscope.

Natesh, R.↗
NASA NTRS1989

Performance of FFT methods in local gravity field modelling

Fast Fourier transform (FFT) methods provide a fast and efficient means of processing large amounts of gravity or geoid data in local gravity field modelling. The FFT methods, however, has a number of theoretical and practical limitations, especially the use of flat-earth approximation, and the requirements for gridded data. In spite of this the method often yields excellent results in practice when compared to other more rigorous (and computationally expensive) methods, such as least-squares collocation. The good performance of the FFT methods illustrate that the theoretical approximations are offset by the capability of taking into account more data in larger areas, especially important for geoid predictions. For best results good data gridding algorithms are essential. In practice truncated collocation approaches may be used. For large areas at high latitudes the gridding must be done using suitable map projections such as UTM, to avoid trivial errors caused by the meridian convergence. The FFT methods are compared to ground truth data in New Mexico (xi, eta from delta g), Scandinavia (N from delta g, the geoid fits to 15 cm over 2000 km), and areas of the Atlantic (delta g from satellite altimetry using Wiener filtering). In all cases the FFT methods yields results comparable or superior to other methods.

Forsberg, Rene↗
DOE OSTI2023

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Hunting for Polluted White Dwarfs and Other Treasures with Gaia XP Spectra and Unsupervised Machine Learning

White dwarfs (WDs) polluted by exoplanetary material provide the unprecedented opportunity to directly observe the interiors of exoplanets. However, spectroscopic surveys are often limited by brightness constraints, and WDs tend to be very faint, making detections of large populations of polluted WDs difficult. In this paper, we aim to increase considerably the number of WDs with multiple metals in their atmospheres. Using 96,134 WDs with Gaia DR3 BP/RP (XP) spectra, we constructed a 2D map using an unsupervised machine-learning technique called Uniform Manifold Approximation and Projection (UMAP) to organize the WDs into identifiable spectral regions. The polluted WDs are among the distinct spectral groups identified in our map. We have shown that this selection method could potentially increase the number of known WDs with five or more metal species in their atmospheres by an order of magnitude. Such systems are essential for characterizing exoplanet diversity and geology.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2013

Evaluating Light Rain Drop Size Estimates from Multiwavelength Micropulse Lidar Network Profiling

This paper investigates multiwavelength retrievals of median equivolumetric drop diameter D(sub 0) suitable for drizzle and light rain, through collocated 355-/527-nm Micropulse Lidar Network (MPLNET) observations collected during precipitation occurring 9 May 2012 at the Goddard Space Flight Center (GSFC) project site. By applying a previously developed retrieval technique for infrared bands, the method exploits the differential backscatter by liquid water at 355 and 527 nm for water drops larger than approximately 50 micrometers. In the absence of molecular and aerosol scattering and neglecting any transmission losses, the ratio of the backscattering profiles at the two wavelengths (355 and 527 nm), measured from light rain below the cloud melting layer, can be described as a color ratio, which is directly related to D(sub 0). The uncertainty associated with this method is related to the unknown shape of the drop size spectrum and to the measurement error. Molecular and aerosol scattering contributions and relative transmission losses due to the various atmospheric constituents should be evaluated to derive D(sub 0) from the observed color ratio profiles. This process is responsible for increasing the uncertainty in the retrieval. Multiple scattering, especially for UV lidar, is another source of error, but it exhibits lower overall uncertainty with respect to other identified error sources. It is found that the total error upper limit on D(sub 0) approaches 50%. The impact of this retrieval for long-term MPLNET monitoring and its global data archive is discussed.

light-rain drop size estimates↗
DOE OSTI2022

EERC Final Topical Report – Findings on Subtask 3.2 – Produced Water Management Through Geologic Homogenization, Conditions, and Reuse

The Energy & Environmental Research Center (EERC) was awarded a contract by the North Dakota Industrial Commission (NDIC) Oil and Gas Research Program (NDIC No. G-051-101) to conduct a study on the recycling of water used in oil and gas operations, also known as produced water, from oil- and gas-producing regions of North Dakota as directed by Section 19 of North Dakota House Bill 1014. This final report provides a compilation of results of the study, which include regulatory, scientific, technological, and feasibility methods and considerations associated with North Dakota produced water management. The report provides a synopsis of this project’s previously submitted produced water assessment report entitled “Produced Water Management and Recycling Options in North Dakota” (Energy & Environmental Research Center, 2020), with updated values provided as appropriate. The report provides the results from the investigation of a novel produced water management strategy, referred to as geologic homogenization, conditioning, and reuse (GHCR), which aims to use a subsurface geologic formation as a natural medium for managing produced water recycling and reuse. Water management is a significant technical and economic challenge for sustainable oil and gas production, and water volumes are intrinsically linked to oil production volumes. North Dakota oil production rose to over 1.5 million barrels (MMbbl)/day in 2019, and despite a downturn in oil price in early 2020, North Dakota oil production has recovered to 1.1 MMbbl/day as of August 2021. Bakken petroleum system development between 2008 and 2020 has resulted in a nearly fourfold increase in produced water volumes to 642 MMbbl/yr in 2020 after peaking at 740 MMbbl/yr in 2019 and a fivefold increase in saltwater disposal (SWD) volumes to 565 MMbbl/yr in 2020 after a peak of 682 MMbbl/yr in 2019. Produced water and SWD volumes are forecasted to double by 2030. SWD is the primary method of produced water management used in North Dakota, with approximately 95% of the SWD volume occurring through subsurface injection into sandstones of the Dakota Group (Dakota). Localized pressurization of the Dakota resulting from SWD and projected increases in produced water volumes could impact the economics of North Dakota oil production. As a result, there is an emerging need to pursue alternative produced water management approaches, including recycling and reuse. While produced water recycling is not yet widespread, commercial operators are making strides in overcoming the technical challenges of using high salinity produced water in completion operations (Marathon Oil, 2020). As water management continues to be a key focal point in companies’ environmental, social, and governance (ESG) initiatives, focus on water management, including recycling, will likely continue to increase. Laboratory column testing, field sample collection, geologic modeling and numerical simulation, and techno-economic analysis all indicate that GHCR could feasibly be implemented as a potential water management option. Laboratory column testing and field sample collection indicate that the Inyan Kara sandstone and native formation fluid are capable of homogenizing with the Bakken produced water to a point where the fluid composition appears to stabilize. Extracting that stabilized fluid could be considered homogeneous and capable of providing individual batches of hydraulic fracturing fluid. Numerical simulation results indicate that extraction of fluids from the Inyan Kara in a GHCR implementation scenario is capable of reducing formation pressure, which would help ease localized pressurization of the Inyan Kara and extend the available capacity for nearby existing SWD wells. Economic analysis indicates that there are scenarios where GHCR implementation can be a competitive or even lower-cost option than a conventional water management approach. Site-specific conditions will dictate the economic potential of GHCR, but potentially attractive sites for GHCR implementation will be those that are located above a pressurized zone of the Inyan Kara, need six or more Bakken infill wells, and face high costs for conventional SWD and/or freshwater. Based on the regulatory review, drilling into the Inyan Kara for SWD and to harness as a source water for industrial use have precedent in North Dakota, and a workable regulatory solution for GHCR seems likely. However, restrictions in the state regarding surface storage and transport of produced fluids may limit some activities, which will affect how GHCR could ultimately be implemented. In summary, this study reveals pursuing GHCR can be a viable approach to water management in North Dakota. The GHCR concept addresses some of the challenges that hinder the more traditional approaches to recycling in the industry. Furthermore, an assessment of the current landscape of water management within the state reveals the ongoing trend of increasing volumes of produced water and SWD. Projections reveal that the volumes of produced water that need to be managed are expected to double over the next decade (Energy & Environmental Research Center, 2020). With the continued development of the Bakken and continuing driving factors related to ESG initiatives, implementing a practice such as GHCR is a feasible approach to adding recycling of produced water to industry within the state. This subtask was cofunded through the EERC–U.S. Department of Energy Joint Program on Research and Development for Fossil Energy-Related Resources Cooperative Agreement No. DE-FE0024233. Nonfederal funding was provided by the North Dakota Industrial Commission Oil and Gas Research Program. References Energy & Environmental Research Center, 2020, Produced water management and recycling options in North Dakota: Final Report for North Dakota Legislative Management Energy Development and Transmission Committee and North Dakota Industrial Commission. Marathon Oil, 2020, Sustainability report: https://cdn.sanity.io/files/ghcnw9z2/website/ 91744eb6ef8fbe59505a911c6b8d2e8dd9a537fa.pdf?dl (accessed November 2021).

02 PETROLEUM↗
DOE OSTI2025

Semi-analytical covariance matrices for two-point correlation function for DESI 2024 data

We present an optimized way of producing the fast semi-analytical covariance matrices for the Legendre moments of the two-point correlation function, taking into account survey geometry and mimicking the non-Gaussian effects. We validate the approach on simulated (mock) catalogs for different galaxy types, representative of the Dark Energy Spectroscopic Instrument (DESI) Data Release 1, used in 2024 analyses. We find only a few percent differences between the mock sample covariance matrix and our results, which can be expected given the approximate nature of the mocks, although we do identify discrepancies between the shot-noise properties of the DESI fiber assignment algorithm and the faster approximation (emulator) used in the mocks. Importantly, we find a close agreement (≤ 8% relative differences) in the projected errorbars for distance scale parameters for the baryon acoustic oscillation measurements. This confirms our method as an attractive alternative to simulation-based covariance matrices, especially for non-standard models or galaxy sample selections, making it particularly relevant to the broad current and future analyses of DESI data.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Exploring Continuous Seismic Data at an Industry Facility Using Unsupervised Machine Learning

Seismic data recorded at industrial sites contain valuable information on anthropogenic activities. With advances in machine learning and computing power, new opportunities have emerged to explore the seismic wavefield in these complex environments. We applied two unsupervised machine learning algorithms to analyze continuous seismic data collected from an industrial facility in Texas, United States. The Uniform Manifold Approximation and Projection for Dimension Reduction algorithm was used to reduce the dimensionality of the data and generate 2D embeddings. Then, the Hierarchical Density-Based Spatial Clustering of Applications with Noise method was employed to automatically group these embeddings into distinct signal clusters. Our analysis of over 1400 hr (around 59 days) of continuous seismic data revealed five and seven signal clusters at two separate stations. At both stations, we identified clusters associated with background noise and vehicle traffic, with the latter’s temporal patterns aligning closely with the facility’s work schedule. Furthermore, the algorithms detected signal clusters from unknown sources and underline the ability of unsupervised machine learning for uncovering previously unrecognized patterns. Our analysis demonstrates the effectiveness of unsupervised approaches in examining continuous seismic data without requiring prior knowledge or pre-existing labels.

58 GEOSCIENCES↗
DOE OSTI2023

Spatiotemporal Super-Resolution with Generative Machine Learning for Creating Renewable Energy Resource Data Under Climate Change Scenarios

As we plan for a future with higher penetrations of renewables and increasing electrification, it becomes more important to understand how the electricity grid will operate under a variety of weather events. We must also consider that the weather our future grid will experience will be different and possibly more extreme than the historical weather that we have extensive data for. We can use data from global climate models (GCMs) to help understand how our climate may change over the next several decades, but there is often a significant gap between the low-resolution GCM data and the high-resolution weather data required to study power systems under specific weather events. Therefore, our objective in this work is to develop tools that can bridge this gap by using low-resolution GCM data to create realistic high-resolution weather datasets that can be used to study renewable energy generation and electricity demand. To accomplish this objective, we have developed a set of generative machine learning models that can rapidly downscale GCM daily average output data at an approximate grid resolution of 100km to hourly data at an approximate 4 km grid resolution. The models can be used to create high resolution data from nearly any GCM included in the Coupled Model Intercomparison Project (CMIP) Phase 5 or 6. Our methods include all datasets regularly used to study the integration of wind and solar power plants as well as changes in electricity demand due to heating and cooling loads. These models and datasets enable power systems modelers to study climate change-influenced weather events and their impact on the grid. We have downscaled and validated wind, solar, temperature, and humidity data with very promising results. The generative machine learning methods are computationally efficient and produce data that has similar statistical characteristics to current state-of-the-art historical datasets. We have trained initial generative models and produced an initial dataset collectively referred to as Sup3rCC: Super-Resolved Renewable Energy Resource Data with Climate Change Impacts. The data covers a (mostly) historical period from 2015-2025 and a future period from 2050-2059. We have also taken hypothetical high-electrification load data and scaled the heating and cooling loads with respect to the 2050-2059 high-resolution Sup3rCC meteorology. The results show how future levels of renewable energy generation and electrified load may be impacted by climate change, setting the stage for capacity expansion models to consider a dynamic climate through model years.

climate change↗
DOE OSTI2022

Development of the uncertainty quantification toolkit's python interface and surrogate construction tutorial

The uncertainty quantification toolkit (UQTk) is a collection of c++ libraries that assess the confidence of numerical models. Surrogate approximations, often polynomial chaos expansions (PCEs), lessen the computational cost of these assessments. I developed a Python interface in UQTk for regression and Bayesian compressive sensing to add to the existing Galerkin projection method. These methods receive an object containing the polynomial basis information and NumPy arrays of sample points, call c++ methods, and return the PCE coefficients in a NumPy array. To demonstrate these methods, I wrote a tutorial in which I use them to construct surrogates for Genz functions and calculate the resulting error.

97 MATHEMATICS AND COMPUTING↗