Error sensitivity function catalog
Various systematic errors of orbital solutions involving range and range rate radar
Martin, C. F.↗
SEARCH · Engineering Papers
Results for “ERROR FUNCTION”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Using Redundancy To Reduce Errors in Magnetometer Readings
A method of reducing errors in noisy magnetic-field measurements involves exploitation of redundancy in the readings of multiple magnetometers in a cluster. By "redundancy"is meant that the readings are not entirely independent of each other because the relationships among the magnetic-field components that one seeks to measure are governed by the fundamental laws of electromagnetism as expressed by Maxwell's equations. Assuming that the magnetometers are located outside a magnetic material, that the magnetic field is steady or quasi-steady, and that there are no electric currents flowing in or near the magnetometers, the applicable Maxwell 's equations are delta x B = 0 and delta(raised dot) B = 0, where B is the magnetic-flux-density vector. By suitable algebraic manipulation, these equations can be shown to impose three independent constraints on the values of the components of B at the various magnetometer positions. In general, the problem of reducing the errors in noisy measurements is one of finding a set of corrected values that minimize an error function. In the present method, the error function is formulated as (1) the sum of squares of the differences between the corrected and noisy measurement values plus (2) a sum of three terms, each comprising the product of a Lagrange multiplier and one of the three constraints. The partial derivatives of the error function with respect to the corrected magnetic-field component values and the Lagrange multipliers are set equal to zero, leading to a set of equations that can be put into matrix.vector form. The matrix can be inverted to solve for a vector that comprises the corrected magnetic-field component values and the Lagrange multipliers.
Kulikov, Igor↗
An inverse problem strategy based on forward model evaluations: Gradient-based optimization without adjoint solves
This study presents a new nonlinear programming formulation for the solution of inverse problems. First, a general inverse problem formulation based on the compliance error functional is presented. The proposed error functional enables the computation of the Lagrange multipliers, and thus the first order derivative information, at the expense of just one model evaluation. Therefore, the calculation of the Lagrange multipliers does not require the solution of the computationally intensive adjoint problem. This leads to significant speedups for large-scale, gradient-based inverse problems.
42 ENGINEERING↗
SU-E-T-432: A Rapid and Comprehensive Procedure for Daily Proton QA
Purpose: The objective is to develop a rapid and comprehensive daily QA procedure implemented at the S. Lee Kling Proton Therapy Center at Barnes-Jewish Hospital. Methods: A scribed phantom with imbedded fiducials is used for checking lasers accuracy followed by couch isocentricity and for X-ray imaging congruence with isocenter. A Daily QA3 device (Sun Nuclear, FL) was used to check output, range and profiles. Five chambers in the central region possess various build-ups. After converting the thickness of the inherent build-ups into water equivalent thickness (WET) for proton, range of any beam can be checked with additional build-up on the Daily QA3 device. In our procedure, 3 beams from 3 bands (large, small and deep) with nominal range of 20 cm are checked daily. 17cm plastic water with WET of 16.92cm are used as additional build-up so that four chambers sit on the SOBP plateau at various depths and one sit on the distal fall off. Reading from the five chambers are fitted to an error function that has been parameterized to match the SOBP with the same nominal range. Shifting of the error function to maximize the correlation between measurements and the error function is deemed as the range shift from the nominal value. Results: We have found couch isocentricity maintained over 180 degrees. Imaging system exhibits accuracy in regard to imaging and mechanical isocenters. Ranges are within 1mm accuracy from measurements in water tank, and sensitive to change of sub-millimeter. Data acquired since the start of operation show outputs, profiles and range stay within 1% or 1mm from baselines. The whole procedure takes about 40 minutes. Conclusion: Taking advantage of the design of Daily QA3 device turns the device originally designed for photon and electron into a comprehensive and rapid tool for proton daily QA.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Solution of D dimensional Dirac equation for hyperbolic tangent potential using NU method and its application in material properties
The analytical solution of D-dimensional Dirac equation for hyperbolic tangent potential is investigated using Nikiforov-Uvarov method. In the case of spin symmetry the D dimensional Dirac equation reduces to the D dimensional Schrodinger equation. The D dimensional relativistic energy spectra are obtained from D dimensional relativistic energy eigen value equation by using Mat Lab software. The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi polynomials. The thermodynamically properties of materials are generated from the non-relativistic energy eigen-values in the classical limit. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy. The thermal quantities of the system, partition function and specific heat, are expressed in terms of error function and imaginary error function which are numerically calculated using Mat Lab software.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Using the Diffusion Equation to Predict CST Resin Breakthroughs - 19051
The crystalline silicotitanate, CST, resin manufactured by UOP Honeywell is well-suited for single-use decontamination of caustic liquid wastes at Hanford and Savannah River Sites. There are many measurements of column loading and breakthrough for CST resin decontamination of Cs-137 in various Hanford and SRS waste simulants. Such measurements provide a very good basis for estimating CST resin demand and these more refined predictions make the CST column loading as efficient as possible given the very expensive CST resin. Although there are a variety of tools for predicting chromatographic resin column performance, [1-4], there is a particularly useful method that is suitable as a simple spreadsheet model [4]. This spreadsheet model is an analytical error function for the mass transfer zone (mtz) diffusion width in resin columns and is well-suited for resin breakthrough and column loading. This paper describes just such an error function spreadsheet model and how well the error function agrees with more traditional models and fits the observed small scale breakthrough curves for CST that can then better predict large scale column loading efficiency. The very high sorption of Cs onto CST tends to be slower than that of other lower sorption resins and as a result of the slow CST kinetics, there is also a marked asymmetry in many CST breakthrough curves. The CST breakthrough asymmetry is largely due to the distribution of CST particle sizes because smaller particles sorb much faster than larger particles. It is then the smaller particles that determine the leading mtz at a decontamination factor of 1,000 breakthrough and therefore the decontamination efficiency for a given loading. The larger CST particles correspondingly determine the tailing mtz width and therefore the overall loading efficiency and it is important to model both leading and tailing mtz widths to best optimize CST for a particular Cs-137 decontamination application. (authors)
12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
Nodal variational principle for excited states
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. In conclusion, simple numerical illustrations of the error functions are presented.
74 ATOMIC AND MOLECULAR PHYSICS↗
Kinetic modelling for zinc (II) ions biosorption onto Luffa cylindrica
The biosorption of Zinc (II) ions onto a biomaterial - Luffa cylindrica has been studied. This biomaterial was characterized by elemental analysis, surface area, pore size distribution, scanning electron microscopy, and the biomaterial before and after sorption, was characterized by Fourier Transform Infra Red (FTIR) spectrometer. The kinetic nonlinear models fitted were Pseudo-first order, Pseudo-second order and Intra-particle diffusion. A comparison of non-linear regression method in selecting the kinetic model was made. Four error functions, namely coefficient of determination (R{sup 2}), hybrid fractional error function (HYBRID), average relative error (ARE), and sum of the errors squared (ERRSQ), were used to predict the parameters of the kinetic models. The strength of this study is that a biomaterial with wide distribution particularly in the tropical world and which occurs as waste material could be put into effective utilization as a biosorbent to address a crucial environmental problem.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Halogen Thermochemistry Assessed with Density Functional Theory: Systematic Errors, Swift Corrections and Effects on Electrochemistry
Abstract Despite its sizable errors, density functional theory (DFT) is extensively used to evaluate thermochemical properties of gases, liquids and their interfaces with solids. As numerous halogen‐containing compounds appear as reactants, products and/or electrolytes in electrochemical reactions, and ionic effects are currently an active area of research, it is important to evaluate the accuracy of DFT for halogen thermochemistry. Herein, we assess the formation energies of interhalogens, hydrogen halides, diatomic and atomic halogens and their ions using six widespread functionals at the GGA, meta‐GGA and hybrid levels. We observe that DFT errors with respect to experiments are correlated with the electronegativity of the species and there are systematic trends across functionals, such that swift corrections were devised. Specifically, the average of the mean absolute errors for the six functionals decreased from 0.19 eV before the corrections to 0.08 eV after them. Besides, the overall maximum absolute error (MAX) decreased from 0.76 to 0.44 eV and the average of the MAXs decreased from 0.51 to 0.24 eV. Finally, we illustrate the qualitative and quantitative impact of gas‐phase errors on the predictions of surface Pourbaix diagrams.
Chemistry↗
Real-Gas Effects on Binary Mixing Layers
This paper presents a computational study of real-gas effects on the mean flow and temporal stability of heptane/nitrogen and oxygen/hydrogen mixing layers at supercritical pressures. These layers consist of two counterflowing free streams of different composition, temperature, and density. As in related prior studies reported in NASA Tech Briefs, the governing conservation equations were the Navier-Stokes equations of compressible flow plus equations for the conservation of total energy and of chemical- species masses. In these equations, the expressions for heat fluxes and chemical-species mass fluxes were derived from fluctuation-dissipation theory and incorporate Soret and Dufour effects. Similarity equations for the streamwise velocity, temperature, and mass fractions were derived as approximations to the governing equations. Similarity profiles showed important real-gas, non-ideal-mixture effects, particularly for temperature, in departing from the error-function profile, which is the similarity solution for incompressible flow. The temperature behavior was attributed to real-gas thermodynamics and variations in Schmidt and Prandtl numbers. Temporal linear inviscid stability analyses were performed using the similarity and error-function profiles as the mean flow. For the similarity profiles, the growth rates were found to be larger and the wavelengths of highest instability shorter, relative to those of the errorfunction profiles and to those obtained from incompressible-flow stability analysis. The range of unstable wavelengths was found to be larger for the similarity profiles than for the error-function profiles
Okong'o, Nora↗
A new computational approach for the solutions of generalized pantograph-delay differential equations
In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods.
97 MATHEMATICS AND COMPUTING↗
Density Profiles of Liquid/Vapor Interfaces Away from Their Critical Points
Abstract not provided
Bu, Wei↗
Solution of D dimensional Dirac equation for coulombic potential using NU method and its thermodynamics properties
The analytical solution of Ddimensional Dirac equation for Coulombic potential is investigated using Nikiforov-Uvarov method. The D dimensional relativistic energy spectra are obtained from relativistic energy eigenvalue equation by using Mat Lab software.The corresponding D dimensional radial wave functions are formulated in the form of generalized Jacobi and Laguerre Polynomials. In the non-relativistic limit, the relativistic energy equation reduces to the non-relativistic energy which will be applied to determine some thermodynamical properties of the system. The thermodynamical properties of the system are expressed in terms of error function and imaginary error function.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Development of a video image-based QA system for the positional accuracy of dynamic tumor tracking irradiation in the Vero4DRT system
Purpose: To develop and evaluate a new video image-based QA system, including in-house software, that can display a tracking state visually and quantify the positional accuracy of dynamic tumor tracking irradiation in the Vero4DRT system. Methods: Sixteen trajectories in six patients with pulmonary cancer were obtained with the ExacTrac in the Vero4DRT system. Motion data in the cranio–caudal direction (Y direction) were used as the input for a programmable motion table (Quasar). A target phantom was placed on the motion table, which was placed on the 2D ionization chamber array (MatriXX). Then, the 4D modeling procedure was performed on the target phantom during a reproduction of the patient’s tumor motion. A substitute target with the patient’s tumor motion was irradiated with 6-MV x-rays under the surrogate infrared system. The 2D dose images obtained from the MatriXX (33 frames/s; 40 s) were exported to in-house video-image analyzing software. The absolute differences in the Y direction between the center of the exposed target and the center of the exposed field were calculated. Positional errors were observed. The authors’ QA results were compared to 4D modeling function errors and gimbal motion errors obtained from log analyses in the ExacTrac to verify the accuracy of their QA system. The patients’ tumor motions were evaluated in the wave forms, and the peak-to-peak distances were also measured to verify their reproducibility. Results: Thirteen of sixteen trajectories (81.3%) were successfully reproduced with Quasar. The peak-to-peak distances ranged from 2.7 to 29.0 mm. Three trajectories (18.7%) were not successfully reproduced due to the limited motions of the Quasar. Thus, 13 of 16 trajectories were summarized. The mean number of video images used for analysis was 1156. The positional errors (absolute mean difference + 2 standard deviation) ranged from 0.54 to 1.55 mm. The error values differed by less than 1 mm from 4D modeling function errors and gimbal motion errors in the ExacTrac log analyses (n = 13). Conclusions: The newly developed video image-based QA system, including in-house software, can analyze more than a thousand images (33 frames/s). Positional errors are approximately equivalent to those in ExacTrac log analyses. This system is useful for the visual illustration of the progress of the tracking state and for the quantification of positional accuracy during dynamic tumor tracking irradiation in the Vero4DRT system.
60 APPLIED LIFE SCIENCES↗
SU-F-T-548: An Extraction Method for Correction of Detector Volume Effect in Small Field Profile Measurement
Purpose: To investigate the appropriateness of Gaussian detector response function in SRS profile measurement. Methods: In-air profile data for a 6MV beam were collected by using ionization chambers and radiochromic film. Two ionization chambers (FC65-G(IBA) and PinPoint(PTWTW31009)) were used to scan the beam profile. Film was used to measure the beam penumbra. A half-blocked beam was acquired with stepping size of 0.2mm and gradient of the measured profile represents the detector response function. The field widths of 2cm and 3cm were measured at SAD. Two detector response functions were evaluated, i.e. a traditional single Gaussian and a three-Gaussian symmetric function. Two corresponding models were used to fit measured data. One was error functions with one shaping parameter. The other model was a symmetrical superposition of three sets of error functions with fixed separation and two different shaping parameters. In both fitting, resultant Gaussian kernels and corresponding σs could be calculated analytically. SNR of the fitted curves were obtained and compared. Results: Our measured data indicated that the ionization chambers had a three-peak response function. From the first modeling, SNR for FC65-G were 36.0dB and 38.3dB for 2cm and 3cm field respectively. The corresponding SNR for the PinPoint were 43.6dB and 42.5dB. From the second modeling, SNR for FC65-G were 42.5dB and 47.4dB for 2cm and 3cm field respectively. The corresponding SNR for the PinPoint were 44.0dB and 44.5 dB. Three-gaussian model had an observable improvement for FC65-G chamber. Film measured penumbra was 3–5 times smaller than that of ionization-chambers. Conclusion: Superposition of three Gaussian kernels resembles the detector response due to central electrode and chamber wall and results in a better fitting to the measurement data than a single Gaussian kernel. Our study indicated that the Three-Gaussian model was more suitable for the detector with sigma comparable to penumbra width.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes
In this paper, we analyze the effects of Dufour number and fractional-order derivative on unsteady natural convection flow of a viscous and incompressible fluid over an infinite vertical plate with constant heat and mass fluxes. The fractional constitutive model is obtained using fractional calculus approach. The Caputo fractional derivative operator is used in this problem. The dimensionless system of equations has been solved by employing Laplace transformation technique. Closed form solutions for concentration, temperature and velocity are presented in the form of Wright function and complementary error function. Effects of fractional and physical parameters on temperature and velocity profiles are illustrated graphically.
97 MATHEMATICS AND COMPUTING↗
Using Density-Corrected DFT to Understand Density-Driven and Functional-Dependent Errors in Ab Initio Simulations of the Hydrated Electron
The hydrated electron, an excess electron in liquid water, plays a crucial role in a plethora of chemical processes, motivating extensive research efforts to characterize its structure, dynamics, and reactivity in solution. Recent theoretical approaches to understanding this intriguing object have involved ab initio simulations based on density functional theory (DFT). Although DFT allows for the study of hydrated electron reactivity and quantum mechanical behavior, it is well-known that anionic systems can suffer from significant density-driven errors (DDEs). Density-corrected DFT (DC-DFT) provides a framework to mitigate such errors; the method reduces DDEs by replacing the self-consistent (SC) density associated with a given density functional with the Hartree–Fock (HF) density. Since HF densities tend to be more localized than DFT SC densities, the DC-DFT scheme significantly improves errors in calculations where the SC density is spuriously delocalized. Here, we investigate how the use of density correction affects the calculated properties of the DFT-simulated (PBEh) hydrated electron, a particularly challenging diffuse anionic system to simulate. First, we analyze charge delocalization in a system consisting of a model octahedral hydrated electron water cluster (the so-called Kevan structure) along with a spatially separated sulfur atom. We show that the use of density correction indeed reduces DDEs in comparison to a standard DFT global hybrid functional. We then propagate molecular dynamics trajectories of the hydrated electron using DC-DFT, where we find that DC further localizes electron density in the cavity region, a signature of reduced charge delocalization. Unfortunately, the decreased radius of gyration of the spin density and corresponding tightening of the local solvation structure from density correction causes predicted observables to deviate further from experimental measurements than when density correction is not employed. Here, we argue that DC’s worse agreement with experiment results from the removal of a fortuitous cancellation of errors that is intrinsic to the PBEh functional. This indicates that the difficulties with DFT to simulate hydrated electrons are primarily due to the inherent approximations in DFT rather than to density-driven errors.
Density functional theory↗
Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion
There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1D, 2D, and 3D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, t d0 . In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, t d0 . The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2D/3D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.
42 ENGINEERING↗