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At least 19 records

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NASA NTRS1988

Controlling basins of attraction in a neural network-based telemetry monitor

The size of the basins of attraction around fixed points in recurrent neural nets (NNs) can be modified by a training process. Controlling these attractive regions by presenting training data with various amount of noise added to the prototype signal vectors is discussed. Application of this technique to signal processing results in a classification system whose sensitivity can be controlled. This new technique is applied to the classification of temporal sequences in telemetry data.

Bell, Benjamin↗

NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - 2

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

Yee, H. C.↗

NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - Part 2

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Yee, H. C.↗

NASA NTRS1994

The Dynamics of Some Iterative Implicit Schemes

Yee, H. C.↗

NASA NTRS1990

On domains of convergence in optimization problems

Numerical optimization algorithms require the knowledge of an initial set of design variables. Starting from an initial design x(sup 0), improved solutions are obtained by updating the design iteratively in a way prescribed by the particular algorithm used. If the algorithm is successful, convergence is achieved to a local optimal solution. Let A denote the iterative procedure that characterizes a typical optimization algorithm, applied to the problem: Find x belonging to R(sup n) that maximizes f(x) subject to x belonging to Omega contained in R(sup n). We are interested in problems with several local maxima (x(sub j))(sup *), j=1, ..., m, in the feasible design space Omega. In general, convergence of the algorithm A to a specific solution (x(sub j))(sup *) is determined by the choice of initial design x(sup 0). The domain of convergence D(sub j) of A associated with a local maximum (x(sub j))(sup *) is a subset of initial designs x(sup 0) in Omega such that the sequence (x(sup k)), k=0,1,2,... defined by x(sup k+1) = A(x(sup k)), k=0,1,... converges to (x(sub j))(sup *). The set D(sub j) is also called the basin of attraction of (x(sub j))(sup *). Cayley first proposed the problem of finding the basin of attraction for Newton's method in 1897. It has been shown that the basin of attraction for Newton's method exhibits chaotic behavior in problems with polynomial objective. This implies that there may be regions in the feasible design space where arbitrarily close starting points will converge to different local optimal solutions. Furthermore, the boundaries of the domains of convergence may have a very complex, even fractal structure. In this paper we show that even simple structural optimization problems solved using standard gradient based (first order) algorithms exhibit similar features.

Diaz, Alejandro R.↗

NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗

NASA NTRS2014

Rewriting the Landform History of One of Africa's Three Largest Basins

The Kalahari Basin in southern Africa - one of the largest basins in Africa, along with the Congo and Chad basins - has attracted attention since David Livingstone traveled through the area in the 1840s. It is a semiarid desert with a large freshwater swampland known as the Okavango Swamp (150 km radius). This prominent megafan (a fan with radii >100 km), with its fingers of dark green forests projecting into the dun colors of the dunes of the Kalahari semi-desert, has been well photographed by astronauts over the years. The study area in the northern Kalahari basin is centered on the Okavango megafan of northwest Botswana, whose swampland has become well known as an African wildlife preserve of importance to biology and tourism alike. The Okavango River is unusual because it has deposited not one but two megafans along its course: the Okavango megafan and the Cubango megafan. The Okavango megafan is one of only three well-known megafans in Africa. Megafans on Earth were once thought to be rare, but recent research has documented 68 in Africa alone. Eleven megafans, plus three more candidates, have been documented in the area immediately surrounding the Okavango feature. These 11 megafans occupy the flattest and smoothest terrains adjacent to the neighboring upland and stand out as the darkest areas in the roughness map of the area. Megafan terrains occupy at least 200,000 sq km of the study area. The roughness map shown is based on an algorithm used first on Mars to quantify topographic roughness. Research of Earth's flattest terrains is just beginning with the aid of such maps, and it appears that these terrains are analogous to the flattest regions of Mars. Implications: 1. The variability in depositional style in each subbasin may apply Africa-wide: rift megafan length is dominated by rift width, whereas Owambo subbasin megafans are probably controlled by upland basin size; Zambezi subbasin megafans appear more like foreland basin types, with the position of the trunk river controlling size. 2. These perspectives were successfully applied to identify the largest megafan in the group (Cubango), a fan that was sufficiently overprinted by dunes and dry lakelets not to be detectable remotely. Such undertsanding can probably be applied on Mars, where Earth experience suggests megafans ought to exist. 3. Sweep angles of rivers on megafans drastically change the hydrology in some subbasins: when the Cubango and Kunene rivers were oriented to the Etosha Pan, it was probably a permanent water body. Now that the rivers are oriented away from the basin, 93 percent of the discharge area from the pan's northerly (main) source area is gone. 4. Biotic contact between major river systems was probably controlled by megafans situated on divides: various fish species that originated in the Congo basin are now found in the Upper Zambezi R., and vice versa, apparently because of river switching behavior on the Cassai megafan that has mediated migrations both to the south and the north.

Wilkinson, Justin↗

NASA NTRS1986

Statistical approach to cubic Langmuir turbulence

Previous work on the cubic direct interaction approximation applied to the truncated (in Fourier space) cubically nonlinear Schroedinger equation model of Langmuir turbulence is extended to more modes. In the undriven, undamped case, excellent agreement between the statistical theory and a numerical ensemble of solutions of the dynamic equations is obtained. In the driven, damped case, satisfactory agreement is obtained provided the dynamic ensemble is limited to initial conditions in the basin of attraction.

Sun, G.-Z.↗

NASA NTRS1988

Orthogonal patterns in binary neural networks

A binary neural network that stores only mutually orthogonal patterns is shown to converge, when probed by any pattern, to a pattern in the memory space, i.e., the space spanned by the stored patterns. The latter are shown to be the only members of the memory space under a certain coding condition, which allows maximum storage of M=(2N) sup 0.5 patterns, where N is the number of neurons. The stored patterns are shown to have basins of attraction of radius N/(2M), within which errors are corrected with probability 1 in a single update cycle. When the probe falls outside these regions, the error correction capability can still be increased to 1 by repeatedly running the network with the same probe.

Baram, Yoram↗

NASA NTRS1994

On the Dynamics of Computing a Chemically Relaxed Nonequilibrium Flow

In order to gain insights into the strong dependence of numerical solutions on initial data for finite time steps, a set of nonlinear test problems rich enough to capture the behavior of difference schemes were recently identified and the numerical basins of attraction for these problems were computed using commonly used time discretizations in CFD. Our study revealed a wealth of surprisingly nonlinear behavior of numerical schemes that were not observed before, in particular for the implicit time discretizations that are commonly used in CFD. The goal of this work is to apply these tools to study a practical model from non-equilibrium flowfield relaxation. This type of problem arises in chemically nonequilibrium hypersonic flows such as in a shock tube experiment or an expanding nozzle. Here we consider a reacting mixture of (N2, N) for an inviscid one-dimensional steady model. Preliminary numerical results indicate that, aside from the possibility of spurious numerical solutions being introduced by the time discretizations, limitations on the model for physical or accurate solutions may also play a part in the dynamics observed.

Sweby, P. K.↗

NASA NTRS1995

On the Dynamics of Implicit Linear Multistep Methods

Some new guidelines on the usage of implicit linear multistep methods (LMMs) as time-dependent approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) are explored. The commonly used implicit LMMs in CFD belong to the class of superstable time discretizations. It can be shown that the nonlinear asymptotic behavior in terms of bifurcation diagrams and basins of attractions of these schemes can provide an improved range of initial data and time step over the linearized stability limit.

Yee, H. C.↗

NASA NTRS1988

Nonlinear dynamics in cardiac conduction

Electrical conduction in the heart shows many phenomena familiar from nonlinear dynamics. Among these phenomena are multiple basins of attraction, phase locking, and perhaps period-doubling bifurcations and chaos. We describe a simple cellular-automation model of electrical conduction which simulates normal conduction patterns in the heart as well as a wide range of disturbances of heart rhythm. In addition, we review the application of percolation theory to the analysis of the development of complex, self-sustaining conduction patterns.

NASA Discipline Cardiopulmonary↗

NASA NTRS1996

The Lunar Crust: Global Structure and Signature of Major Basins

New lunar gravity and topography data from the Clementine Mission provide a global Bouguer anomaly map corrected for the gravitational attraction of mare fill in mascon basins. Most of the gravity signal remaining after corrections for the attraction of topography and mare fill can be attributed to variations in depth to the lunar Moho and therefore crustal thickness. The large range of global crustal thickness (approx. 20-120 km) is indicative of major spatial variations in melting of the lunar exterior and/or significant impact-related redistribution. The 6l-km average crustal thickness, constrained by a depth-to-Moho measured during the Apollo 12 and 14 missions, is preferentially distributed toward the farside, accounting for much of the offset in center-of-figure from the center-of-mass. While the average farside thickness is 12 km greater than the nearside, the distribution is nonuniform, with dramatic thinning beneath the farside, South Pole-Aitken basin. With the global crustal thickness map as a constraint, regional inversions of gravity and topography resolve the crustal structure of major mascon basins to half wavelengths of 150 km. In order to yield crustal thickness maps with the maximum horizontal resolution permitted by the data, the downward continuation of the Bouguer gravity is stabilized by a three- dimensional, minimum-slope and curvature algorithm. Both mare and non-mare basins are characterized by a central upwarped moho that is surrounded by rings of thickened crust lying mainly within the basin rims. The inferred relief at this density interface suggests a deep structural component to the surficial features of multiring lunar impact basins. For large (greater than 300 km diameter) basins, moho relief appears uncorrelated with diameter, but is negatively correlated with basin age. In several cases, it appears that the multiring structures were out of isostatic equilibrium prior to mare emplacement, suggesting that the lithosphere was strong enough to maintain their state of stress to the present.

Neumann, Gregory A.↗

NASA NTRS2012

Are the Basins of Tui Regio and Hotei Arcus Sites of Former Titanian Seas?

Features observed in the basins of Tui Regio and Hotei Arcus on Titan have attracted the attention of the Cassini-era investigators. At both locations, VIMS observed discrete 5-micron bright approx.500-km wide features described as lobate in shape. Several studies have proposed that these materials are cryo-volcanic flows; in the case of the Hotei Arcus feature this inference was buttressed with SAR RADAR images showing bright and dark patches with lobate margins. We propose an alternative explanation. First we note that all landforms on Titan that are unambiguously identifiable can be explained by exogenic processes (aeolian, fluvial, impact cratering, and mass wasting). Suggestions of endogenically produced cryovolcanic constructs and flows have, without exception, lacked conclusive diagnostic evidence. Recently published topographic profiles across Tui Regio and the lobate feature region north of Hotei Arcus indicate these features appear to occur in large regional basins, at least along the direction of the profiles. SAR images show that the terrains surrounding both 5-micron bright features exhibit fluvial networks that appear to converge and debauch into the probable basins. The 5-micron bright features themselves correspond to fields of discrete radar-bright depressions whose bounding edges are commonly rounded and cumulate in planform in SAR images. These fields of discrete radar-bright depressions strongly resemble fields of features seen at Titan s high latitudes usually attributed to be dry lakes. Thus the combination of (1) the resemblance to high-latitude dry lakes, (2) location in the centers of probable regional depressions, and (3) convergence of fluvial networks are inferred by us to best explain the 5-micron bright regions at Tui Regio and Hotei Arcus as sites of dry seas or at least paleolake clusters. Such equatorial seas, if real, may be evidence of substantially larger inventories of liquid alkanes in Titan s past.

Moore, Jeffrey Morgan↗

NASA NTRS2016

Identifying and Characterizing Impact Melt Outcrops in the Nectaris Basin

The Nectaris Basin is an 820-km diameter, multi-ring impact basin located on the near side of the Moon. Nectaris is a defining stratigraphic horizon based on relationships between ejecta units, giving its name to the Nectarian epoch of lunar history. Lunar basin chronology based on higher resolution LRO imagery and topography, while assigning some important basins like Serenitatis to pre-Nectarian time, were generally consistent with those previously derived. Based on this stratigraphy, at least 11 large basins formed in the time between Nectaris and Imbrium. The absolute age of Nectaris, therefore, is a crucial marker in the lunar time-stratigraphic sequence for understanding the impact flux on the Moon, and by extension, the entire inner solar system. For several decades, workers have attempted to constrain the age of the Nectaris basin through radiometric dating of lunar samples. However, there is little agreement on which samples in our collection represent Nectaris, if any, and what the correct radiometric age of such samples is. The importance of the age of Nectaris goes far beyond assigning a stratigraphic marker to lunar chronology. Several dynamical models use Nectaris as their pin date, so that this date becomes crucial in understanding the time-correlated effects in the rest of the solar system. The importance of the Nectaris basin age, coupled with its nearside, mid-latitude location, make remnants of the impact-melt sheet an attractive target for a future mission, either for in-situ dating or for sample return. We have started exploring this possibility. We have begun a consortium data-analysis effort bringing multiple datasets and analysis methods to bear on these putative impact-melt deposits to characterize their extent, elemental composition and mineralogy, maturity and geologic setting, and to identify potential landing sites that meet both operational safety and science requirements.

Nectaris Basin↗

NASA NTRS1992

Geological and geophysical field investigations from a lunar base at Mare Smythii

Mare Smythii, located on the equator and east limb of the Moon, has a great variety of scientific and economic uses as the site for a permanent lunar base. Here a complex could be established that would combine the advantages of a nearside base (for ease of communications with Earth and normal operations) with those of a farside base (for shielding a radio astronomical observatory from the electromagnetic noise of Earth). The Mare Smythii region displays virtually the entire known range of geological processes and materials found on the Moon; from this site, a series of field traverses and investigations could be conducted that would provide data on and answers to fundamental questions in lunar geoscience. This endowment of geological materials also makes the Smythii region attractive for the mining of resources for use both on the Moon and in Earth-Moon space. We suggest that the main base complex be located at 0, 90 deg E, within the mare basalts of the Smythii basin; two additional outposts would be required, one at 0, 81 deg E to maintain constant communications with Earth, and and the other, at 0, 101 deg E on the lunar farside, to serve as a radio astronomical observatory. The bulk of lunar surface activities could be conducted by robotic teleoperations under the direct control of the human inhabitants of the base.

Spudis, Paul D.↗

NASA NTRS1996

MASTER: An Orbiter for the Detailed Study of Vesta

MASTER (Mainbelt Asteroid Exploration/Rendezvous), a Discovery-class orbiter, will carry out a global geological and geochemical survey of Vesta. Primary goals include determining Vesta's mean density and interior structure through detailed mapping of the gravity field, imaging surface morphology at 3-m resolution, mapping mineralogy between 0.4 and 2.5 microns, and determining abundances of key elements through X-ray and gamma-ray spectroscopy. Spectroscopic evidence indicates the presence of basaltic lava flows on Vesta and suggests that impact basins may have exposed mantle materials. These possibilities, combined with the likelihood that Vesta is the ultimate source of HED meteorites, makes this asteroid an important target for a comprehensive orbiter mission. MASTER's global survey, combined with ongoing studies of HED meteorites, will be a major step in understanding the chemical, thermal, and geological evolution of Vesta. A particularly attractive opportunity involves a launch in June 2003 with arrival at Vesta in November 2009. The orbital phase of the mission is scheduled to last one year. A flyby of another mainbelt asteroid enroute to Vesta may be possible.

Veverka, J.↗

NASA NTRS2017

Gravity-Assist Trajectories to the Ice Giants: An Automated Method to Catalog Mass- Or Time-Optimal Solutions

This work presents an automated method of calculating mass (or time) optimal gravity-assist trajectories without a priori knowledge of the flyby-body combination. Since gravity assists are particularly crucial for reaching the outer Solar System, we use the Ice Giants, Uranus and Neptune, as example destinations for this work. Catalogs are also provided that list the most attractive trajectories found over launch dates ranging from 2024 to 2038. The tool developed to implement this method, called the Python EMTG Automated Trade Study Application (PEATSA), iteratively runs the Evolutionary Mission Trajectory Generator (EMTG), a NASA Goddard Space Flight Center in-house trajectory optimization tool. EMTG finds gravity-assist trajectories with impulsive maneuvers using a multiple-shooting structure along with stochastic methods (such as monotonic basin hopping) and may be run with or without an initial guess provided. PEATSA runs instances of EMTG in parallel over a grid of launch dates. After each set of runs completes, the best results within a neighborhood of launch dates are used to seed all other cases in that neighborhood-allowing the solutions across the range of launch dates to improve over each iteration. The results here are compared against trajectories found using a grid-search technique, and PEATSA is found to outperform the grid-search results for most launch years considered.

Hughes, Kyle M.↗