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NASA NTRS1973

A parallel variable metric optimization algorithm

An algorithm, designed to exploit the parallel computing or vector streaming (pipeline) capabilities of computers is presented. When p is the degree of parallelism, then one cycle of the parallel variable metric algorithm is defined as follows: first, the function and its gradient are computed in parallel at p different values of the independent variable; then the metric is modified by p rank-one corrections; and finally, a single univariant minimization is carried out in the Newton-like direction. Several properties of this algorithm are established. The convergence of the iterates to the solution is proved for a quadratic functional on a real separable Hilbert space. For a finite-dimensional space the convergence is in one cycle when p equals the dimension of the space. Results of numerical experiments indicate that the new algorithm will exploit parallel or pipeline computing capabilities to effect faster convergence than serial techniques.

Straeter, T. A.↗

NASA NTRS2014

VIIP Bedrest Analog Roadmap

The objective is to define successive bed rest campaigns leading to a potential VIIP (Vision Impairment and Intracranial Pressure) countermeasure. To determine if the analog is successful, changes need to occur in the following outcome measures (dependent variables): Intracranial pressure; Retinal nerve fiber layer; Choroidal engorgement; Globe flattening; Axial biometry; Optic nerve sheath diameter distention; Cycloplegic refraction; Visual acuity. Study parameters (independent variables) to include: CO2; Sodium; Exercise (resistive & aerobic); Strict tilt angle.

Villarreal, Jennifer D.↗

NASA NTRS2002

Understanding the Long-Term Spectral Variability of Cygnus X-1 from BATSE and ASM Observations

We present a spectral analysis of observations of Cygnus X-1 by the RXTE/ASM (1.5-12 keV) and CGRO/BATSE (20-300 keV), including about 1200 days of simultaneous data. We find a number of correlations between intensities and hardnesses in different energy bands from 1.5 keV to 300 keV. In the hard (low) spectral state, there is a negative correlation between the ASM 1.5-12 keV flux and the hardness at any energy. In the soft (high) spectral state, the ASM flux is positively correlated with the ASM hardness (as previously reported) but uncorrelated with the BATSE hardness. In both spectral states, the BATSE hardness correlates with the flux above 100 keV, while it shows no correlation with the flux in the 20-100 keV range. At the same time, there is clear correlation between the BATSE fluxes below and above 100 keV. In the hard state, most of the variability can be explained by softening the overall spectrum with a pivot at approximately 50 keV. The observations show that there has to be another, independent variability pattern of lower amplitude where the spectral shape does not change when the luminosity changes. In the soft state, the variability is mostly caused by a variable hard (Comptonized) spectral component of a constant shape superimposed on a constant soft blackbody component. These variability patterns are in agreement with the dependence of the rms variability on the photon energy in the two states. We interpret the observed correlations in terms of theoretical Comptonization models. In the hard state, the variability appears to be driven mostly by changing flux in seed photons Comptonized in a hot thermal plasma cloud with an approximately constant power supply. In the soft state, the variability is consistent with flares of hybrid, thermal/nonthermal, plasma with variable power above a stable cold disk. Also, based on broadband pointed observations simultaneous with those of the ASM and BATSE, we find the intrinsic bolometric luminosity increases by a factor of approximately 3-4 from the hard state to the soft one, which supports models of the state transition based on a change of the accretion rate.

Zdziarski, Andrzej A.↗

NASA NTRS1968

Independent doubly truncated gamma variables

Density and distribution functions of the sum of independent variables, each having a truncated gamma density function, were derived for use in the measurement of complex physical phenomena.

Lavender, D. E.↗

NASA NTRS2007

Uncertainty Analysis of Historical Hurricane Data

An analysis of variance (ANOVA) study was conducted for historical hurricane data dating back to 1851 that was obtained from the U. S. Department of Commerce National Oceanic and Atmospheric Administration (NOAA). The data set was chosen because it is a large, publicly available collection of information, exhibiting great variability which has made the forecasting of future states, from current and previous states, difficult. The availability of substantial, high-fidelity validation data, however, made for an excellent uncertainty assessment study. Several factors (independent variables) were identified from the data set, which could potentially influence the track and intensity of the storms. The values of these factors, along with the values of responses of interest (dependent variables) were extracted from the data base, and provided to a commercial software package for processing via the ANOVA technique. The primary goal of the study was to document the ANOVA modeling uncertainty and predictive errors in making predictions about hurricane location and intensity 24 to 120 hours beyond known conditions, as reported by the data set. A secondary goal was to expose the ANOVA technique to a broader community within NASA. The independent factors considered to have an influence on the hurricane track included the current and starting longitudes and latitudes (measured in degrees), and current and starting maximum sustained wind speeds (measured in knots), and the storm starting date, its current duration from its first appearance, and the current year fraction of each reading, all measured in years. The year fraction and starting date were included in order to attempt to account for long duration cyclic behaviors, such as seasonal weather patterns, and years in which the sea or atmosphere were unusually warm or cold. The effect of short duration weather patterns and ocean conditions could not be examined with the current data set. The responses analyzed were the storm latitude, longitude and intensity, as recorded in the data set, 24 or 120 hours beyond the current state. Several ANOVA modeling schemes were examined. Two forms of validation were used: 1) comparison with official hurricane prediction performance metrics and 2) cases studies conducted on hurricanes from the 2005 season, which were not included within the model construction and ANOVA assessment. In general, the ANOVA technique did not perform as well as the established official prediction performance metrics published by NOAA; still, the technique did remarkably well in this demonstration with a difficult data set and could probably be made to perform better with more knowledge of hurricane development and dynamics applied to the problem. The technique provides a repeatable prediction process that eliminates the need for judgment in the forecast.

Green, Lawrence L.↗

NASA NTRS2017

Automation of POST Cases via External Optimizer and "Artificial p2" Calculation

During early conceptual design of complex systems, speed and accuracy are often at odds with one another. While many characteristics of the design are fluctuating rapidly during this phase there is nonetheless a need to acquire accurate data from which to down-select designs as these decisions will have a large impact upon program life-cycle cost. Therefore enabling the conceptual designer to produce accurate data in a timely manner is tantamount to program viability. For conceptual design of launch vehicles, trajectory analysis and optimization is a large hurdle. Tools such as the industry standard Program to Optimize Simulated Trajectories (POST) have traditionally required an expert in the loop for setting up inputs, running the program, and analyzing the output. The solution space for trajectory analysis is in general non-linear and multi-modal requiring an experienced analyst to weed out sub-optimal designs in pursuit of the global optimum. While an experienced analyst presented with a vehicle similar to one which they have already worked on can likely produce optimal performance figures in a timely manner, as soon as the "experienced" or "similar" adjectives are invalid the process can become lengthy. In addition, an experienced analyst working on a similar vehicle may go into the analysis with preconceived ideas about what the vehicle's trajectory should look like which can result in sub-optimal performance being recorded. Thus, in any case but the ideal either time or accuracy can be sacrificed. In the authors' previous work a tool called multiPOST was created which captures the heuristics of a human analyst over the process of executing trajectory analysis with POST. However without the instincts of a human in the loop, this method relied upon Monte Carlo simulation to find successful trajectories. Overall the method has mixed results, and in the context of optimizing multiple vehicles it is inefficient in comparison to the method presented POST's internal optimizer functions like any other gradient-based optimizer. It has a specified variable to optimize whose value is represented as optval, a set of dependent constraints to meet with associated forms and tolerances whose value is represented as p2, and a set of independent variables known as the u-vector to modify in pursuit of optimality. Each of these quantities are calculated or manipulated at a certain phase within the trajectory. The optimizer is further constrained by the requirement that the input u-vector must result in a trajectory which proceeds through each of the prescribed events in the input file. For example, if the input u-vector causes the vehicle to crash before it can achieve the orbital parameters required for a parking orbit, then the run will fail without engaging the optimizer, and a p2 value of exactly zero is returned. This poses a problem, as this "non-connecting" region of the u-vector space is far larger than the "connecting" region which returns a non-zero value of p2 and can be worked on by the internal optimizer. Finding this connecting region and more specifically the global optimum within this region has traditionally required the use of an expert analyst.

Dees, Patrick D.↗

NASA NTRS1985

Universal Keplerian state transition matrix

A completely general method for computing the Keplerian state transition matrix in terms of Goodyear's universal variables is presented. This includes a new scheme for solving Kepler's problem which is a necessary first step to computing the transition matrix. The Kepler problem is solved in terms of a new independent variable requiring the evaluation of only one transcendental function. Furthermore, this transcendental function may be conveniently evaluated by means of a Gaussian continued fraction.

Shepperd, S. W.↗

NASA NTRS1992

Rotary engine performance limits predicted by a zero-dimensional model

A parametric study was performed to determine the performance limits of a rotary combustion engine. This study shows how well increasing the combustion rate, insulating, and turbocharging increase brake power and decrease fuel consumption. Several generalizations can be made from the findings. First, it was shown that the fastest combustion rate is not necessarily the best combustion rate. Second, several engine insulation schemes were employed for a turbocharged engine. Performance improved only for a highly insulated engine. Finally, the variability of turbocompounding and the influence of exhaust port shape were calculated. Rotary engines performance was predicted by an improved zero-dimensional computer model based on a model developed at the Massachusetts Institute of Technology in the 1980's. Independent variables in the study include turbocharging, manifold pressures, wall thermal properties, leakage area, and exhaust port geometry. Additions to the computer programs since its results were last published include turbocharging, manifold modeling, and improved friction power loss calculation. The baseline engine for this study is a single rotor 650 cc direct-injection stratified-charge engine with aluminum housings and a stainless steel rotor. Engine maps are provided for the baseline and turbocharged versions of the engine.

Bartrand, Timothy A.↗

NASA NTRS1974

Optimal control problems with mixed control-phase variable equality and inequality constraints

In this paper, necessary conditions are obtained for optimal control problems containing equality constraints defined in terms of functions of the control and phase variables. The control system is assumed to be characterized by an ordinary differential equation, and more conventional constraints, including phase inequality constraints, are also assumed to be present. Because the first-mentioned equality constraint must be satisfied for all t (the independent variable of the differential equation) belonging to an arbitrary (prescribed) measurable set, this problem gives rise to infinite-dimensional equality constraints. To obtain the necessary conditions, which are in the form of a maximum principle, an implicit-function-type theorem in Banach spaces is derived.

Makowski, K.↗

NASA NTRS1986

Aircrew coordination and decisionmaking: Peer ratings of video tapes made during a full mission simulation

Six professionally active, retired captains rated the coordination and decisionmaking performances of sixteen aircrews while viewing videotapes of a simulated commercial air transport operation. The scenario featured a required diversion and a probable minimum fuel situation. Seven point Likert-type scales were used in rating variables on the basis of a model of crew coordination and decisionmaking. The variables were based on concepts of, for example, decision difficulty, efficiency, and outcome quality; and leader-subordin ate concepts such as person and task-oriented leader behavior, and competency motivation of subordinate crewmembers. Five-front-end variables of the model were in turn dependent variables for a hierarchical regression procedure. The variance in safety performance was explained 46%, by decision efficiency, command reversal, and decision quality. The variance of decision quality, an alternative substantive dependent variable to safety performance, was explained 60% by decision efficiency and the captain's quality of within-crew communications. The variance of decision efficiency, crew coordination, and command reversal were in turn explained 78%, 80%, and 60% by small numbers of preceding independent variables. A principle component, varimax factor analysis supported the model structure suggested by regression analyses.

Murphy, M. R.↗

NASA NTRS1983

Lagrange thermodynamic potential and intrinsic variables for He-3 He-4 dilute solutions

For a two-fluid model of dilute solutions of He-3 in liquid He-4, a thermodynamic potential is constructed that provides a Lagrangian for deriving equations of motion by a variational procedure. This Lagrangian is defined for uniform velocity fields as a (negative) Legendre transform of total internal energy, and its primary independent variables, together with their thermodynamic conjugates, are identified. Here, similarities between relations in classical physics and quantum statistical mechanics serve as a guide for developing an alternate expression for this function that reveals its character as the difference between apparent kinetic energy and intrinsic internal energy. When the He-3 concentration in the mixtures tends to zero, this expression reduces to Zilsel's formula for the Lagrangian for pure liquid He-4. An investigation of properties of the intrinsic internal energy leads to the introduction of intrinsic chemical potentials along with other intrinsic variables for the mixtures. Explicit formulas for these variables are derived for a noninteracting elementary excitation model of the fluid. Using these formulas and others also derived from quantum statistical mechanics, another equivalent expression for the Lagrangian is generated.

Jackson, H. W.↗

NASA NTRS1994

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented; four methods are compared. Each scheme requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent flow example; both geometric-shape and flow-condition design variables are considered. Several methods are tested; results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method; for six independent variables, it is at least two to four times faster than central finite differences and requires only 60 percent more memory than the original code; the performance is expected to improve further in the future.

Taylor, Arthur C., III↗

NASA NTRS2017

A Time-Regularized Multiple Gravity-Assist Low-Thrust Bounded-Impulse Model for Trajectory Optimization

The multiple gravity assist low-thrust (MGALT) trajectory model combines the medium-fidelity Sims-Flanagan bounded-impulse transcription with a patched-conics flyby model and is an important tool for preliminary trajectory design. While this model features fast state propagation via Kepler's equation and provides a pleasingly accurate estimation of the total mass budget for the eventual flight-suitable integrated trajectory it does suffer from one major drawback, namely its temporal spacing of the control nodes. We introduce a variant of the MGALT transcription that utilizes the generalized anomaly from the universal formulation of Kepler's equation as a decision variable in addition to the trajectory phase propagation time. This results in two improvements over the traditional model. The first is that the maneuver locations are equally spaced in generalized anomaly about the orbit rather than time. The second is that the Kepler propagator now has the generalized anomaly as its independent variable instead of time and thus becomes an iteration-free propagation method. The new algorithm is outlined, including the impact that this has on the computation of Jacobian entries for numerical optimization, and a motivating application problem is presented that illustrates the improvements that this model has over the traditional MGALT transcription.

Optimization↗

NASA NTRS2017

A Time-Regularized, Multiple Gravity-Assist Low-Thrust, Bounded-Impulse Model for Trajectory Optimization

The multiple gravity assist low-thrust (MGALT) trajectory model combines the medium-fidelity Sims-Flanagan bounded-impulse transcription with a patched-conics flyby model and is an important tool for preliminary trajectory design. While this model features fast state propagation via Keplers equation and provides a pleasingly accurate estimation of the total mass budget for the eventual flight suitable integrated trajectory it does suffer from one major drawback, namely its temporal spacing of the control nodes. We introduce a variant of the MGALT transcription that utilizes the generalized anomaly from the universal formulation of Keplers equation as a decision variable in addition to the trajectory phase propagation time. This results in two improvements over the traditional model. The first is that the maneuver locations are equally spaced in generalized anomaly about the orbit rather than time. The second is that the Kepler propagator now has the generalized anomaly as its independent variable instead of time and thus becomes an iteration-free propagation method. The new algorithm is outlined, including the impact that this has on the computation of Jacobian entries for numerical optimization, and a motivating application problem is presented that illustrates the improvements that this model has over the traditional MGALT transcription.

Optimization↗

NASA NTRS1955

Optimum flight paths of turbojet aircraft

The climb of turbojet aircraft is analyzed and discussed including the accelerations. Three particular flight performances are examined: minimum time of climb, climb with minimum fuel consumption, and steepest climb. The theoretical results obtained from a previous study are put in a form that is suitable for application on the following simplifying assumptions: the Mach number is considered an independent variable instead of the velocity; the variations of the airplane mass due to fuel consumption are disregarded; the airplane polar is assumed to be parabolic; the path curvatures and the squares of the path angles are disregarded in the projection of the equation of motion on the normal to the path; lastly, an ideal turbojet with performance independent of the velocity is involved. The optimum Mach number for each flight condition is obtained from the solution of a sixth order equation in which the coefficients are functions of two fundamental parameters: the ratio of minimum drag in level flight to the thrust and the Mach number which represents the flight at constant altitude and maximum lift-drag ratio.

CLIMBING SPEEDS - EFFECT OF ALTITUDE↗

NASA NTRS1977

Variational energy principle for compressible, baroclinic flow. 2: Free-energy form of Hamilton's principle

The first and second variations are calculated for the irreducible form of Hamilton's Principle that involves the minimum number of dependent variables necessary to describe the kinetmatics and thermodynamics of inviscid, compressible, baroclinic flow in a specified gravitational field. The form of the second variation shows that, in the neighborhood of a stationary point that corresponds to physically stable flow, the action integral is a complex saddle surface in parameter space. There exists a form of Hamilton's Principle for which a direct solution of a flow problem is possible. This second form is related to the first by a Friedrichs transformation of the thermodynamic variables. This introduces an extra dependent variable, but the first and second variations are shown to have direct physical significance, namely they are equal to the free energy of fluctuations about the equilibrium flow that satisfies the equations of motion. If this equilibrium flow is physically stable, and if a very weak second order integral constraint on the correlation between the fluctuations of otherwise independent variables is satisfied, then the second variation of the action integral for this free energy form of Hamilton's Principle is positive-definite, so the action integral is a minimum, and can serve as the basis for a direct trail and error solution. The second order integral constraint states that the unavailable energy must be maximum at equilibrium, i.e. the fluctuations must be so correlated as to produce a second order decrease in the total unavailable energy.

Schmid, L. A.↗

NASA NTRS2011

Can Functional Cardiac Age be Predicted from ECG in a Normal Healthy Population

In a normal healthy population, we desired to determine the most age-dependent conventional and advanced ECG parameters. We hypothesized that changes in several ECG parameters might correlate with age and together reliably characterize the functional age of the heart. Methods: An initial study population of 313 apparently healthy subjects was ultimately reduced to 148 subjects (74 men, 84 women, in the range from 10 to 75 years of age) after exclusion criteria. In all subjects, ECG recordings (resting 5-minute 12-lead high frequency ECG) were evaluated via custom software programs to calculate up to 85 different conventional and advanced ECG parameters including beat-to-beat QT and RR variability, waveform complexity, and signal-averaged, high-frequency and spatial/spatiotemporal ECG parameters. The prediction of functional age was evaluated by multiple linear regression analysis using the best 5 univariate predictors. Results: Ignoring what were ultimately small differences between males and females, the functional age was found to be predicted (R2= 0.69, P < 0.001) from a linear combination of 5 independent variables: QRS elevation in the frontal plane (p<0.001), a new repolarization parameter QTcorr (p<0.001), mean high frequency QRS amplitude (p=0.009), the variability parameter % VLF of RRV (p=0.021) and the P-wave width (p=0.10). Here, QTcorr represents the correlation between the calculated QT and the measured QT signal. Conclusions: In apparently healthy subjects with normal conventional ECGs, functional cardiac age can be estimated by multiple linear regression analysis of mostly advanced ECG results. Because some parameters in the regression formula, such as QTcorr, high frequency QRS amplitude and P-wave width also change with disease in the same direction as with increased age, increased functional age of the heart may reflect subtle age-related pathologies in cardiac electrical function that are usually hidden on conventional ECG.

Schlegel, Todd↗

NASA NTRS2017

Biomass Retrieval from L-Band Polarimetric UAVSAR Backscatter and PRISM Stereo Imagery

The forest above-ground biomass (AGB) and spatial distribution of vegetation elements have profound effects on the productivity and biodiversity of terrestrial ecosystems. In this paper, we evaluated biomass estimation from L-band Synthetic Aperture Radar (SAR) data acquired by National Aeronautics and Space Administration (NASA) Uninhabited Aerial Vehicle SAR (UAVSAR) and the improvement of accuracy by adding canopy height information derived from stereo imagery acquired by Japan Aerospace Exploration Agency (JAXA) Panchromatic Remote Sensing Instrument for Stereo Mapping (PRISM) on-board the Advanced Land Observing Satellite (ALOS). Various models for prediction of forest biomass from UAVSAR data were investigated at pixel sizes of 1/4 ha (50 m x 50 m) and 1 ha. The variance inflation factor (VIF) was calculated for each of the explanatory variables in multivariable regression models to assess the multi-collinearity between explanatory variables. In addition, the t-and p-values were used to interpret the significance of the coefficients of each explanatory variables. The R(exp. 2), Root Mean Square Error (RMSE), bias and Akaike information criterion (AIC), and leave-one-out cross-validation (LOOCV) and bootstrapping were used to validate models. At 1/4-ha scale, the R(exp. 2) and RMSE of biomass estimation from a model using a single track of polarimetric UAVSAR data were 0.59 and 52.08 Mg/ha. With canopy height from PRISM as additional independent variable, R(exp. 2) increased to 0.76 and RMSE decreased to 39.74 Mg/ha (28.24%). At 1-ha scale, the RMSE of biomass estimation based on UAVSAR data of a single track was 39.42 Mg/ha with a R(exp. 2) of 0.77. With the canopy height from PRISM, R(exp. 2) increased to 0.86 and RMSE decreased to 29.47 Mg/ha (20.18%). The models using UAVSAR data alone underestimated biomass at levels above approximately 150 Mg/ha showing the saturation phenomenon. Adding canopy height from PRISM stereo imagery significantly improved the biomass estimation and elevated the saturation level in estimating biomass. Combined use of UAVSAR data acquired from opposite directions (odd and even tracks) slightly improved the biomass estimation.Combined use of UAVSAR data acquired from opposite directions (odd and even tracks) slightly improved the biomass estimation at 1/4-ha scale, R(exp. 2) increased from 0.59 to 0.66 and RMSE reduced from 52.08 to 48.57 Mg/ha. Averaging multiple acquisitions of UAVSAR data from the same look azimuth direction did not improve biomass estimation. A biomass map derived from NASA's LVIS (Laser Vegetation Imaging System) wave-form data was used as a reference for evaluation of the biomass maps from these models. The study has also shown that the errors decreased when deciduous, evergreen, and mixed forests were modeled separately but the improvement was not significant

Zhang, Zhiyu↗