Cowell orbit generator error and time analysis program
Computer program for numerical integration of equations of motion for artificial earth satellite orbits
Cigarski, G. A.↗
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A class of optimum digital phase locked loops for the DSN advanced receiver
A class of optimum digital filters for digital phase locked loop of the deep space network advanced receiver is discussed. The filter minimizes a weighted combination of the variance of the random component of the phase error and the sum square of the deterministic dynamic component of phase error at the output of the numerically controlled oscillator (NCO). By varying the weighting coefficient over a suitable range of values, a wide set of filters are obtained such that, for any specified value of the equivalent loop-noise bandwidth, there corresponds a unique filter in this class. This filter thus has the property of having the best transient response over all possible filters of the same bandwidth and type. The optimum filters are also evaluated in terms of their gain margin for stability and their steady-state error performance.
Hurd, W. J.↗
A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements
We describe a variation of the Optimal Estimation (OE) method for greenhouse gas remote sensing retrievals using a singular value decomposition (SVD) and an uninformative prior. The SVD method is capable of discerning vertical information in column integrated absorption measurements. While traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We discuss the method, illustrate its use on an integrated path differential absorption CO2 lidar measurement model, and compare it to traditional optimal estimation using numerical simulations. In the absence of forward model error, the mathematics behind the SVD method guarantee it to be bias-free, which is confirmed by the numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. While the SVD approach can be used for most trace gas retrievals, it is particularly useful for situations where prior knowledge of the trace gas profile is poor. The SVD analysis is also useful in illustrating how vertical information is treated by the traditional OE approach.
Ramanathan, Anand K.↗
A score-based diffusion model approach for adaptive learning of stochastic partial differential equation solutions
In this paper, we propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.
97 MATHEMATICS AND COMPUTING↗
Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
Bayesian approximation error↗
Compact finite difference schemes for the time fractional diffusion equation with nonlocal boundary conditions
Fourth-order compact finite difference schemes for solving the time fractional diffusion equation with nonlocal boundary conditions are considered in this paper. The second-order derivative in one space variable is approximated by a fourth-order compact finite difference. Two different time derivatives, the Caputo derivative and Riemann–Liouville derivative are considered in the equations that we discuss. The Caputo time derivative is discretized by the L1-approximation and a second-order approximation is used for the Riemann–Liouville derivative. Stability analysis using the discrete energy method for Scheme I is given, and error estimation of this scheme is derived by conducting the local truncation error analysis, using the stability result. Numerical results are provided to show that both two fully discrete schemes have fourth-order accuracy in space, and 2−γ order of accuracy in time for Scheme I with γ being the order of the fractional derivative, while Scheme II has second-order accuracy in time when the theoretical solutions are smooth with zero initial values.
97 MATHEMATICS AND COMPUTING↗
Validations of calibration-free measurements of electron temperature using double-pass Thomson scattering diagnostics from theoretical and experimental aspects
This paper evaluates the accuracy of electron temperature measurements and relative transmissivities of double-pass Thomson scattering diagnostics. The electron temperature (T{sub e}) is obtained from the ratio of signals from a double-pass scattering system, then relative transmissivities are calculated from the measured T{sub e} and intensity of the signals. How accurate the values are depends on the electron temperature (T{sub e}) and scattering angle (θ), and therefore the accuracy of the values was evaluated experimentally using the Large Helical Device (LHD) and the Tokyo spherical tokamak-2 (TST-2). Analyzing the data from the TST-2 indicates that a high T{sub e} and a large scattering angle (θ) yield accurate values. Indeed, the errors for scattering angle θ = 135° are approximately half of those for θ = 115°. The method of determining the T{sub e} in a wide T{sub e} range spanning over two orders of magnitude (0.01–1.5 keV) was validated using the experimental results of the LHD and TST-2. A simple method to provide relative transmissivities, which include inputs from collection optics, vacuum window, optical fibers, and polychromators, is also presented. The relative errors were less than approximately 10%. Numerical simulations also indicate that the T{sub e} measurements are valid under harsh radiation conditions. This method to obtain T{sub e} can be considered for the design of Thomson scattering systems where there is high-performance plasma that generates harsh radiation environments.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Error Estimates of the Ares I Computed Turbulent Ascent Longitudinal Aerodynamic Analysis
Numerical predictions of the longitudinal aerodynamic characteristics for the Ares I class of vehicles, along with the associated error estimate derived from an iterative convergence grid refinement, are presented. Computational results are based on an unstructured grid, Reynolds-averaged Navier-Stokes analysis. The validity of the approach to compute the associated error estimates, derived from a base grid to an extrapolated infinite-size grid, was first demonstrated on a sub-scaled wind tunnel model at representative ascent flow conditions for which the experimental data existed. Such analysis at the transonic flow conditions revealed a maximum deviation of about 23% between the computed longitudinal aerodynamic coefficients with the base grid and the measured data across the entire roll angles. This maximum deviation from the wind tunnel data was associated with the computed normal force coefficient at the transonic flow condition and was reduced to approximately 16% based on the infinite-size grid. However, all the computed aerodynamic coefficients with the base grid at the supersonic flow conditions showed a maximum deviation of only about 8% with that level being improved to approximately 5% for the infinite-size grid. The results and the error estimates based on the established procedure are also presented for the flight flow conditions.
Abdol-Hamid, Khaled S.↗
Reducing Truncation Error In Integer Processing
Improved method of rounding off (truncation of least-significant bits) in integer processing of data devised. Provides for reduction, to extremely low value, of numerical bias otherwise generated by accumulation of truncation errors from many arithmetic operations. Devised for use in integer signal processing, in which rescaling and truncation usually performed to reduce number of bits, which typically builds up in sequence of operations. Essence of method to alternate direction of roundoff (plus, then minus) on alternate occurrences of truncated values contributing to bias.
Thomas, J. Brooks↗
Real-time orbit estimation for ATS-6 from redundant attitude sensors
A program installed in the ATSOCC on-line computer operates with attitude sensor data to produce a smoothed real-time orbit estimate. This estimate is obtained from a Kalman filter which enables the estimate to be maintained in the absence of T/M data. The results are described of analytical and numerical investigations into the sensitivity of Control Center output to the position errors resulting from the real-time estimation. The results of the numerical investigation, which used several segments of ATS-6 data gathered during the Sensor Data Acquisition run on August 19, 1974, show that the implemented system can achieve absolute position determination with an error of about 100 km, implying pointing errors of less than 0.2 deg in latitude and longitude. This compares very favorably with ATS-6 specifications of approximately 0.5 deg in latitude-longitude.
Englar, T. S., Jr.↗
Algorithm-based fault tolerance for spaceborne computing: basis and implementations
We describe and test the mathematical background for using checksum methods to validate results returned by a numerical subroutine operating in a fault- prone environment that causes unpredictable errors in data.
Algorithms numerical tolerances environment↗
Newton versus the machine: solving the chaotic three-body problem using deep neural networks
ABSTRACT Since its formulation by Sir Isaac Newton, the problem of solving the equations of motion for three bodies under their own gravitational force has remained practically unsolved. Currently, the solution for a given initialization can only be found by performing laborious iterative calculations that have unpredictable and potentially infinite computational cost, due to the system’s chaotic nature. We show that an ensemble of converged solutions for the planar chaotic three-body problem obtained using an arbitrarily precise numerical integrator can be used to train a deep artificial neural network (ANN) that, over a bounded time interval, provides accurate solutions at a fixed computational cost and up to 100 million times faster than the numerical integrator. In addition, we demonstrate the importance of training an ANN using converged solutions from an arbitrary precise integrator, relative to solutions computed by a conventional fixed precision integrator, which can introduce errors in the training data, due to numerical round-off and time discretization, that are learned by the ANN. Our results provide evidence that, for computationally challenging regions of phase space, a trained ANN can replace existing numerical solvers, enabling fast and scalable simulations of many-body systems to shed light on outstanding phenomena such as the formation of black hole binary systems or the origin of the core collapse in dense star clusters.
Breen, Philip G.↗
Accuracy of relative oceanic geoid computations
An error analysis based on a model which requires potential coefficients and gravity anomaly information for the computation of oceanic geoidal undulations is developed. The rigorous models are presented and abandoned owing to the numerical difficulties associated with them. An approximate error model is developed in the frequency domain through which an idea is obtained as to the size and behavior of the errors involved. The use of the method is demonstrated by obtaining data requirements for the realization of a 10-cm relative oceanic geoid. Various data sets could result in the desired accuracy. One such set, indicative of the strictness of the requirements, involves gravity profile spacings of approximately 3 km with observational noise not exceeding 0.5 mGal inside detailed data caps of 30 deg and potential coefficients of prescribed accuracy available to degree and order 70.
Christodoulidis, D. C.↗
A stochastic-dynamic model for global atmospheric mass-field statistics
Global atmospheric mass field error correlations based on satellite observations and on numerical forecasts show strong and systematic latitude dependence. A model for the latitude dependent spatial correlation structure of mass field forecast errors is derived from dynamical considerations. Three methods of solution were tested. In the first method, the equation was solved by expansion in spherical harmonics, and the correlation function was computed analytically using the expansion coefficients. In the second method, the finite difference equivalent of the equation was solved using a fast poisson solver. The correlation function was computed using stratified sampling of the individual realizations. In the third method, a higher order equation was derived, and solved directly in finite differences by two successive applications of the fast poisson solver. The three methods were compared for accuracy and efficiency, and the third method was chosen as clearly superior.
Ghil, M.↗
Evaluation of the Rapid Refresh Numerical Weather Prediction Model over Arctic Alaska
Abstract Despite a need for accurate weather forecasts for societal and economic interests in the U.S. Arctic, thorough evaluations of operational numerical weather prediction in the region have been limited. In particular, the Rapid Refresh Model (RAP), which plays a key role in short-term forecasting and decision-making, has seen very limited assessment in northern Alaska, with most evaluation efforts focused on lower latitudes. In the present study, we verify forecasts from version 4 of the RAP against radiosonde, surface meteorological, and radiative flux observations from two Arctic sites on the northern Alaskan coastline, with a focus on boundary layer thermodynamic and dynamic biases, model representation of surface inversions, and cloud characteristics. We find persistent seasonal thermodynamic biases near the surface that vary with wind direction, and may be related to the RAP’s handling of sea ice and ocean interactions. These biases seem to have diminished in the latest version of the RAP (version 5), which includes refined handling of sea ice, among other improvements. In addition, we find that despite capturing boundary layer temperature profiles well overall, the RAP struggles to consistently represent strong, shallow surface inversions. Further, while the RAP seems to forecast the presence of clouds accurately in most cases, there are errors in the simulated characteristics of these clouds, which we hypothesize may be related to the RAP’s treatment of mixed-phase clouds.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well
A new type weighted reverse Poincare inequality is established for a difference of two continuous weak subsolutions of a linear second order uniformly elliptic partial differential equation in the ball.This result is the key to deriving the error estimate for the gradient of the analytically unknown value function of the optimal stochastic control problem from the uniform error of the value function itself in the related numerical approximation problems.
97 MATHEMATICS AND COMPUTING↗
Kaon mixing beyond the standard model with physical masses
We present nonperturbative results for beyond the standard model kaon mixing matrix elements in the isospin symmetric limit ( m u = m d ) of QCD, including a complete estimate of all dominant sources of systematic error. Our results are obtained from numerical simulations of lattice QCD with N f = 2 + 1 flavors of dynamical domain wall fermions. For the first time, these quantities are simulated directly at the physical pion mass m π ∼ 139 MeV for two different lattice spacings. We include data at three lattice spacings in the range a = 0.11 – 0.07 fm and with pion masses ranging from the physical value up to 450 MeV. Compared to our earlier work, we have added both direct calculations at physical quark masses and a third lattice spacing making the removal of discretization effects significantly more precise and eliminating the need for any significant mass extrapolation beyond the range of simulated data. We renormalize the lattice operators nonperturbatively using RI-SMOM off-shell schemes. These schemes eliminate the need to model and subtract nonperturbative pion poles that arises in the RI-MOM scheme and, since the calculations are performed with domain wall fermions, the unphysical mixing between chirality sectors is suppressed. Our results for the bag parameters in the MS ¯ scheme at 3 GeV are B K ≡ B 1 = 0.5240 ( 17 ) ( 54 ) , B 2 = 0.4794 ( 25 ) ( 35 ) , B 3 = 0.746 ( 13 ) ( 17 ) , B 4 = 0.897 ( 02 ) ( 10 ) and B 5 = 0.6882 ( 78 ) ( 94 ) , where the first error is from lattice uncertainties and the second is the uncertainty due to the perturbative matching to MS ¯ . Published by the American Physical Society 2024
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Numerical Modeling of the Effects of Coating Plates on Terminal Ballistic Performance
This report deals with the development and evaluation of a numerical model to examine applied coating to a metal substrate subjected to a ballistic impact. The numerical model will be used to examine the benefit of the coating in resisting penetration due to the impact. For a detailed examination the Retch-Ipson curve is used as a metric. The numerical data is plotted and then fit to the Retch-Ipson curve and error calculations are used to compare the difference between the numerical output and the experimental data. This initial study is an examination of a few shortcomings of the standard material models used, and demonstrate the future work that is needed to understand the ballistic behavior of materials.
36 MATERIALS SCIENCE↗