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NASA NTRS1992

Clustering of tethered satellite system simulation data by an adaptive neuro-fuzzy algorithm

Recent developments in neuro-fuzzy systems indicate that the concepts of adaptive pattern recognition, when used to identify appropriate control actions corresponding to clusters of patterns representing system states in dynamic nonlinear control systems, may result in innovative designs. A modular, unsupervised neural network architecture, in which fuzzy learning rules have been embedded is used for on-line identification of similar states. The architecture and control rules involved in Adaptive Fuzzy Leader Clustering (AFLC) allow this system to be incorporated in control systems for identification of system states corresponding to specific control actions. We have used this algorithm to cluster the simulation data of Tethered Satellite System (TSS) to estimate the range of delta voltages necessary to maintain the desired length rate of the tether. The AFLC algorithm is capable of on-line estimation of the appropriate control voltages from the corresponding length error and length rate error without a priori knowledge of their membership functions and familarity with the behavior of the Tethered Satellite System.

Mitra, Sunanda↗
NASA NTRS1992

Symplectic maps for the n-body problem - Stability analysis

The stability of new symplectic n-body maps is examined from the point of view of nonlinear dynamics. The resonances responsible for the principal artifacts are identified. These are resonances between the stepsize and the difference of mean motions between pairs of planets. For larger stepsizes resonant perturbations are evident in the variation of the energy of the system corresponding to these stepsize resonances. It is shown that the principal instability of the method can be predicted and corresponds to the overlap of the stepsize resonances. It is noted that the analysis suggests that other artifacts will occur. For example, the overlap of a stepsize resonance with a resonance of the actual system may also give a region of chaotic behavior that is an artifact. It is pointed out that the fact that the principal artifacts corresponds to a particular set of stepsize resonances suggests that it may be possible to perturbatively remove the effect when the stepsize resonances are nonoverlapping.

Wisdom, Jack↗
NASA NTRS1993

Direct calculations of waves in fluid flows using a high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expressions for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations, and the numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the CFL limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about 6 to 8 grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 to 300 time steps. Three numerical examples of waves in compressible flow are included.

Yu, Sheng-Tao↗
NASA NTRS1992

Model based matching using simulated annealing and a minimum representation size criterion

We define the model based matching problem in terms of the correspondence and transformation that relate the model and scene, and the search and evaluation measures needed to find the best correspondence and transformation. Simulated annealing is proposed as a method for search and optimization, and the minimum representation size criterion is used as the evaluation measure in an algorithm that finds the best correspondence. An algorithm based on simulated annealing is presented and evaluated. This algorithm is viewed as a part of an adaptive, hierarchical approach which provides robust results for a variety of model based matching problems.

Ravichandran, B.↗
NASA NTRS1991

On the origin of the streak spacing in turbulent shear flows

It is shown that the ideas of selective amplification and direct resonance, based on linear theory, can not provide an explanation for the well-defined streak spacing of about 100 wall units (referred to as 100(+) hereafter) in wall-bounded turbulent shear flows. In addition, for the direct resonance theory, the streaks are created by the non-linear self-interaction of the vertical velocity rather than of the directly forced vertical vorticity. In view of the failure of these approaches, it is then proposed that the selection mechanism must be inherently non-linear and correspond to a self-sustaining mechanism. The 100(+) value should thus be considered as a critical Reynolds number for that mechanism. Indeed, in the case of Poiseuille flow, this 100(+) criterion for transition to turbulence corresponds to the usually quoted value of 1000 based on the half-width and the centerline velocity. In Couette flow, it corresponds to a critical Reynolds number of about 400 based on the half width and half velocity difference.

Waleffe, Fabian A.↗
NASA NTRS1993

Complex QSO absorbers: High-redshift clusters of galaxies?

Using the PSPC instrument on the ROSAT satellite, we have obtained deep exposures of the fields centered on the QSO's 1556+335, 1037-271, and 0854+191 in an effort to search for extended X-ray emission corresponding to complex QSO absorption-line systems (QALS's) that could be indicative of clusters of galaxies at high redshift. The QSO 1556+335 is known to have a pair of complicated C IV absorption systems at z equals 1.610 (Delta v = 988 km/s) and z equals 1.650 (Delta v = 1677 km/s) near the z equals 1.641 QSO redshift. It has been suggested that the absorption may be due to two rich clusters, one containing the QSO itself. The QSO 0854+191 (z(sub em) = 1.89) has 6 strong C IV absorption systems in 2 close groups at z equals 1.2954, 1.2973,1.3019 and z equals 1.3522, 1.3543, 1.3558. Assuming q(sub O) equals 0.5 and H(sub O) equals 50 km/s Mpc(exp -l), the redshift range of this absorption-line cluster corresponds to a radial distance of 44 Mpc. Interestingly, there is a nearby (approximately 30') QSO (0856+189) with a projected separation of 16 Mpc whose redshift (z = 1.29) is similar to that of the absorbing material toward 0854+191. In the case of 1037-271 (Z(sub em) = 2.18), common QALSs have been observed at z equals l.90, 1.96, 2.02, 2.08, and 2.14 (a radial range of 66 Mpc) in its spectrum and that of its neighbor Tol 1038-272 (Z(sub em) = 2.32) whose 18' separation corresponds to a linear distance of 9 Mpc at z approximately equals 2. The length scales implied by the absorbers toward 0854+191 and 1037-271 are interesting in that they suggest structures more akin to superclusters than clusters.

Meyer, David M.↗
NASA NTRS1993

Radar backscattering from snow facies of the Greenland ice sheet: Results from the AIRSAR 1991 campaign

In June 1991, the NASA/JPL airborne SAR (AIRSAR) acquired C- (lambda = 5.6cm), L- (lambda = 24cm), and P- (lambda = 68m) band polarimetric SAR data over the Greenland ice sheet. These data are processed using version 3.55 of the AIRSAR processor which provides radiometrically and polarimetrically calibrated images. The internal calibration of the AIRSAR data is cross-checked using the radar response from corner reflectors deployed prior to flight in one of the scenes. In addition, a quantitative assessment of the noise power level at various frequencies and polarizations is made in all the scenes. Synoptic SAR data corresponding to a swath width of about 12 by 50 km in length (compared to the standard 12 x 12 km size of high-resolution scenes) are also processed and calibrated to study transitions in radar backscatter as a function of snow facies at selected frequencies and polarizations. The snow facies on the Greenland ice sheet are traditionally categorized based on differences in melting regime during the summer months. The interior of Greenland corresponds to the dry snow zone where terrain elevation is the highest and no snow melt occurs. The lowest elevation boundary of the dry snow zone is known traditionally as the dry snow line. Beneath it is the percolation zone where melting occurs in the summer and water percolates through the snow freezing at depth to form massive ice lenses and ice pipes. At the downslope margin of this zone is the wet snow line. Below it, the wet snow zone corresponds to the lowest elevations where snow remains at the end of the summer. Ablation produces enough meltwater to create areas of snow saturated with water, together with ponds and lakes. The lowest altitude zone of ablation sees enough summer melt to remove all traces of seasonal snow accumulation, such that the surface comprises bare glacier ice.

Rignot, Eric↗
NASA NTRS1994

A comparison between general circulation model simulations using two sea surface temperature datasets for January 1979

Simulations with the UCLA atmospheric general circulation model (AGCM) using two different global sea surface temperature (SST) datasets for January 1979 are compared. One of these datasets is based on Comprehensive Ocean-Atmosphere Data Set (COADS) (SSTs) at locations where there are ship reports, and climatology elsewhere; the other is derived from measurements by instruments onboard NOAA satellites. In the former dataset (COADS SST), data are concentrated along shipping routes in the Northern Hemisphere; in the latter dataset High Resolution Infrared Sounder (HIRS SST), data cover the global domain. Ensembles of five 30-day mean fields are obtained from integrations performed in the perpetual-January mode. The results are presented as anomalies, that is, departures of each ensemble mean from that produced in a control simulation with climatological SSTs. Large differences are found between the anomalies obtained using COADS and HIRS SSTs, even in the Northern Hemisphere where the datasets are most similar to each other. The internal variability of the circulation in the control simulation and the simulated atmospheric response to anomalous forcings appear to be linked in that the pattern of geopotential height anomalies obtained using COADS SSTs resembles the first empirical orthogonal function (EOF 1) in the control simulation. The corresponding pattern obtained using HIRS SSTs is substantially different and somewhat resembles EOF 2 in the sector from central North America to central Asia. To gain insight into the reasons for these results, three additional simulations are carried out with SST anomalies confined to regions where COADS SSTs are substantially warmer than HIRS SSTs. The regions correspond to warm pools in the northwest and northeast Pacific, and the northwest Atlantic. These warm pools tend to produce positive geopotential height anomalies in the northeastern part of the corresponding oceans. Both warm pools in the Pacific produce large-scale circulation anomalies with a pattern that resembles that obtained using COADS SSTs as well as EOF 1 of the control simulation; the warm pool in the Atlantic does not. These results suggest that the differences obtained with COADS SSTs and HIRS SSTs are mostly due to the differences in the datasets over the northern Pacific. There was a blocking episode near Greenland in late January 1979. Both simulations with warm SST anomalies over the northwest and northeast Pacific show a tendency toward increased incidence of North Atlantic blocking; the simulation with warm SST anomalies over the northwest Atlantic shows a tendency toward decreased incidence. These results suggest that features in both SST datasets that do not have a counterpart in the other dataset contribute signficantly to the differences between the simulated and observed fields. The results of this study imply that uncertainties in current SST distributions for the world oceans can be as important as the SST anomalies themselves in terms of their impact on the atmospheric circulation. Caution should be exercised, therefore, when linking anomalous circulation and SST patterns, especially in long-range prediction.

Ose, Tomoaki↗
NASA NTRS1994

Non-blocking crossbar permutation engine with constant routing latency

The invention is embodied in an N x N crossbar for routing packets from a set of N input ports to a set of N output ports, each packet having a header identifying one of the output ports as its destination, including a plurality of individual links which carry individual packets. Each link has a link input end and a link output end, a plurality of switches. Each of the switches has at least top and bottom switch inputs connected to a corresponding pair of the link output ends and top and bottom switch outputs connected to a corresponding pair of link input ends, whereby each switch is connected to four different links. Each of the switches has an exchange state which routes packets from the top and bottom switch inputs to the bottom and top switch outputs, respectively, and a bypass state which routes packets from the top and bottom switch inputs to the top and bottom switch outputs, respectively. A plurality of individual controller devices governing respective switches for sensing from a header of a packet at each switch input for the identity of the destination output port of the packet and selecting one of the exchange and bypass states in accordance with the identity of the destination output port and with the location of the corresponding switch relative to the destination output port.

Monacos, Steve P.↗
NASA NTRS1994

Venus O2 visible and IR nightglow: Implications for lower thermosphere dynamics and chemistry

The National Center for Atmospheric Research thermospheric general circulation model for the Venus thermosphere is modified to examine two observed night airglow features, both of which serve as sensitive tracers of the thermospheric circulation. New O2 nightglow data from the Pioneer Venus Orbiter (PVO) star tracker (O2 Herzberg II at 400-800 nm) and ground-based telescopes (O2 IR at 1.27 microns) yield additional model constraints for estimating Venus winds over 100-130 km. Atomic oxygen, produced by dayside CO2 photolysis peaking near 110 km, and transported to the nightside by the global wind system, is partially destroyed through three-body recombination, yielding the O2 Herzberg II visible nightglow. This emission is very sensitive to horizontal winds at altitudes between 100 and 130 km. Other trace species catalytic reactions also contribute to the production of the very strong nightside infrared (1.27 microns) emission. This paper examines the dynamical and chemical implications of these new data using the Venus thermospheric general circulation model (VTGCM) as an analysis tool. Three-dimensional calculations are presented for both solar maximum and solar medium conditions, corresponding to early PVO (1979-1981) and PVO entry (mid-1992) time periods. Very distinct periods are identified in which zonal winds are alternately weak and strong in the Venus lower thermosphere. VTGCM sensitivity studies are conducted to assess the impacts of potential changes in thermospheric zonal and day-to-night winds, and eddy diffusion on the corresponding nightglow intensities. It appears that cyclostrophic balance extends above 80 km periodically, owing to a reversal of the upper mesosphere latitudinal temperature gradient, and thereby producing strong zonal winds and correspondingly modified O2 nightglow distributions that are observed.

Bougher, S. W.↗
NASA NTRS1994

Neodymium, strontium and chromium isotopic studies of the LEW86010 and Angra dos Reis meteorites and the chronology of the angrite parent body

Neodymium, stontium, and chromium isotopic studies of the LEW86010 angrite established its absolute age and the formation interval between its crystallization and condensation of Allende CAIs from the solar nebula. Pyroxene and phosphate were found to contain approximately 8% of its Sm and Nd inventory. A conventional Sm-147-Nd-143 isochron yielded an age of 4.53 +/- 0.04 Ga (2 sigma and Epsilon(sub Nd sup 143)) = 0.45 +/- 1.1. An Sm-146-Nd-142 isochron gives initial Sm-146/Sm-144 = 0.0076 +/- 0.0009 and Epsilon (sub Nd sup 142) = -2.5 +/- 0.4. The Rb-Sr analyses give initial Sr-87/Sr-86 Iota(sub Sr sup 87) = 0.698972 +/- 8 and 0.698970 +/- 18 for LEW and ADOR, respectively, relative to Sr-87/Sr-86 = 0.71025 for NBS987. The difference, Delta Iota(sub Sr Sup 87), between Iota (sub sr sup 87) for the angrites and literature values for Allende CAIs, corresponds to approximately Ma of growth in a solar nebula with a CI chondrite value of Rb-87/Sr-86 = 0.91, or approximately 5 Ma in a nebula with solar photospheric Rb-87/Sr-86 = 1.51. Excess Cr-53 from extinct Mn-53(t(sub 1/2) = 3.7 Ma)in LEW86010 corresponds to initial Mn-53/Mn-55 = 4.4 +/- 1.0 x 10(exp -5) for the inclusions as previously reported by the Paris group (Birck and Allegre, 1988). The Sm-146/Sm-144 value found for LEW86010 corresponds to solar system initial (Sm-146/Sm-144) = 0.0080 +/- 0.0009 for crystallization 8 Ma after Allende, the difference between Pb-Pb ages of angrites and Allende, or 0.0086 +/- 0.0009 for crystallation 18 Ma after Allende, using the Mn-Cr formation interval. The isotopic data are discussed in the context of a model in which an undifferentiated 'chondritic' parent body formed from the solar nebula approximately Ma after Allende CAIs and subsequently underwent differentiation accompanied by loss of volatiles. Parent bodies with Rb/Sr similar to that of CI, CM, or CO chondrites could satisfy the Cr and Sr isotopic systematics. If the angrite parent body had Rb/Sr similar to that of CV meteorites, it would have to form slightly later, approximately 2.6 Ma after the CAIs, to satisfy the Sr and CR isotopic systematics.

Nyquist, L. E.↗
NASA NTRS1994

Landau damping of auroral hiss

Auroral hiss is observed to propagate over distances comparable to an Earth radius from its source in the auroral oval. The role of Landau damping is investigated for upward propagating auroral hiss. By using a ray tracing code and a simplified model of the distribution function, the effect of Landau damping is calculated for auroral hiss propagation through the environment around the auroral oval. Landau damping is found to be the likely mechanism for explaining some of the one-sided auroral hiss funnels observed by Dynamics Explorer 1. It is also found that Landau damping puts a lower limit on the wavelength of auroral hiss. Poleward of the auroral oval, Landau damping is found in a typical case to limit omega/k(sub parallel) to values of 3.4 x 10(exp 4) km/s or greater, corresponding to resonance energies of 3.2 keV or greater and wavelengths of 2 km or greater. For equatorward propagation, omega/k(sub parallel) is limited to values greater than 6.8 x 10(exp 4) km/s, corresponding to resonance energies greater than 13 keV and wavelengths greater than 3 km. Independent estimates based on measured ratios of the magnetic to electric field intensity also show that omega/k(sub parallel) corresponds to resonance energies greater than 1 keV and wavelengths greater than 1 km. These results lead to the difficulty that upgoing electron beams sufficiently energetic to directly generate auroral hiss of the inferred wavelength are not usually observed. A partial transmission mechanism utilizing density discontinuities oblique to the magnetic field is proposed for converting auroral hiss to wavelengths long enough to avoid damping of the wave over long distances. Numerous reflections of the wave in an upwardly flared density cavity could convert waves to significantly increased wavelengths and resonance velocities.

Morgan, D. D.↗
NASA NTRS1994

The roots of coronal structure in the Sun's surface

We have compared the structures seen on X-ray images obtained by a flight of the NIXT sounding rocket payload on July 11, 1991 with near-simultaneous photospheric and chromospheric structures and magnetic fields observed at Big Bear. The X-ray images reflect emission of both Mg X and Fe XVI, formed at 1 x 10(exp 6) K and 3 x 10(exp 6) K, respectively. The brightest H(alpha) sources correspond to a dying sub-flare and other active region components, all of which reveal coronal enhancements situated spatially well above the H(alpha) emission. The arches appear to lie in a small range of angle in the meridian plane connecting their footpoints. Sunspots are dark on the surface and in the corona. For the first time we see an emerging flux region in X-rays and find the emission extends twice as high as the H(alpha) arches. Many features which we believe to correspond to `X-ray bright points' (XBPs) were observed. Whether by resolution or spectral band, the number detected greatly exceeds that from previous work. All of the brighter XBPs correspond to bipolar H(alpha) features, while unipolar H(alpha) bright points are the base of more diffuse comet-like coronal arches, generally vertical. These diverge from individual features by less than 30 deg, and give a good measure of what the `canopies' must do. The H(alpha) data shows that all the H(alpha) features were present present the entire day, so they are not clearly disappearing or reappearing. We find a new class of XBPs which we call `satellite points', elements of opposite polarity linked to nearby umbrae by invisible field lines. The satellite points change rapidly in X-ray brightness during the flight. An M1.9 flare occurred four hours after the flight; examination of the pre-flare structures reveals nothing unusual.

Golub, Leon↗
NASA NTRS1994

Measurement of diurnal and semidiurnal rotational variations and tidal parameters of Earth

We discuss the determination of diurnal and semidiurnal variations in the rotation rate and the direction of rotation axis of Earth from the analysis of 8 years of very long baseline interferometry (VLBI) data. This analysis clearly show that these variations are largely periodic and tidally driven; that is, the periods of the variations correspond to the periods of the largest lunar and solar tides. For rotation rate variations, expressed in terms of changes in universal time (UT), the tidal lines with the largest observed signals are O1 (amplitude 23.5 microseconds in time (microseconds), period 25.82 solar hours); KL (18.9 microseconds, 23.93 hours); M2 (17.9 microseconds, 12.54 hours); and S2 (8.6 microseconds, 12.00 hours). For variations in the direction of the rotation axis (polar motion), significant signals exist in the retrograde semidiurnal band at the M2 and S2 tides (amplitudes 265 and 119 microarc seconds (microarc seconds, respectively); the prograde diurnal band at the O1, K1, and P1 tides (amplitudes 199, 152, and 60 microarc seconds, respectively); and the prograde semidiurnal band at the M2 and K2 tides (amplitudes 58 and 39 microarc seconds, respectively). Variations in the retrograde diurnal band are represented by corrections with previous estimates except that a previously noted discrepancy in the 13.66-day nutation (corresponding to the O1 tide) is largely removed in this new analysis. We estimate that the standard deviations of these estimates are 1.0 microseconds for the UT1 variations and 14-16 microarc seconds for the polar motion terms. These uncertainties correspond to surface displacements of approximately 0.5 mm. From the analysis of atmospheric angular momentum data we conclude that variations in UT1 excited by the atmosphere with subdaily periods are small (approximately 1 microsecond). We find that the average radial tidal displacements of the VLBI sites in the diurnal band are largely consistent with known deficiencies in current tidal models, i.e., deficiencies of up to 0.9 mm in the treatment of the free core nutation resonance. In the semidiurnal band, our analysis yields estimates of the second-degree harmonic radial Love number h(sub 2) at the M2 tide of 0.604 + i0.005 +/- 0.002. The most likely explanation for the rotational variations are the effects of ocean tides, but there may also be some contributions from atmospheric tides, the effects of triaxiality of Earth, and the equatorial second-degree-harmonic components of the core- mantle boundary.

Herring, Thomas A.↗
NASA NTRS1995

Neutron-capture Cl-36, Ca-41, Ar-36, and Sm-150 in large chondrites: Evidence for high fluences of thermalized neutrons

We have measured significant concentrations of Cl-36, Ca-41, Ar-36 from decay of Cl-36, and Sm-150 produced from the capture of thermalized neutrons in the large Chico L6 chondrite. Activities of Cl-36 and Ca-41, corrected for a high-energy spallogenic component and a terrestrial age of approximately 50 ka, give average neutron-capture production rates of 208 atoms/min/g-Cl and 1525 atoms/min/kg-Ca, which correspond to thermal neutron (n) fluxes of 6.2 n/sq cm/s and 4.3 n/sq cm/s, respectively. If sustained for the approximately 65 Ma single-stage, cosmic ray exposure age of Chico, these values correspond to thermal neutron fluences of approximately 1.3 x 10(exp 16) and 0.8 x 10(exp 16) n/sq cm for Cl-36 and Ca-41, respectively. Stepwise temperature extraction of Ar in Chico impact melt shows Ar-36/Ar-38 ratios as large as approximately 9. The correlation of high Ar-36/Ar-38 with high Cl/Ca phases in neutron-irradiated Chico indicates that the excess Ar-36 above that expected from spallation is due to decay of neutron-produced Cl-36. Excess Ar-36 in Chico requires a thermal neutron fluence of 0.9-1.7 x 10(exp 16) n/sq cm. Decreases in Sm-149/Sm-152 due to neutron-capture by Sm-149 correlate with increases in Sm-150/Sm-152 for three samples of Chico, and one of the Torino H-chondrite. The 0.08% decrease in Sm-149 shown by Chico corresponds to a neutron fluence of 1.23 x 10(exp 16) n/sq cm. This fluence derived from Sm considers capture of epithermal neutrons and effects of chemical composition on the neutron energy distribution. Excess Ar-36 identified in the Arapahoe, Bruderheim, and Torino chondrites and the Shallowater aubrite suggest exposure to neutron fluences of approximately 0.2-0.2 x 10(exp 16) n/sq cm. Depletion of Sm-149 in Torino and the LEW86010 angrite suggest neutron fluences of 0.8 x 10(exp 16) n/sq cm and 0.25 x 10(exp 16) n/sq cm, respectively. Neutron fluences of approximately 10(exp 16) n/sq cm in Chico are almost as large as those previously observed for some lunar soils. Consideration of exposure ages suggests that the neutron flux in Chico may have been greater than that in many lunar soils.

Bogard, D. D.↗
NASA NTRS1994

Direct calculations of waves in fluid flows using high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expression for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations. The numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the Courant-Friedrichs-Lewy (CFL) limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about six - eight grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 - 300 time steps. Three numerical examples of waves in compressible flow are included: (1) sound propagation in a duct with linear shear, (2) linear wave growth in a compressible free shear layer, and (3) vortex pairing in a compressible free shear layer perturbed at two frequencies.

Yu, Sheng-Tao↗
NASA NTRS1995

Fast temporal neural learning using teacher forcing

A neural network is trained to output a time dependent target vector defined over a predetermined time interval in response to a time dependent input vector defined over the same time interval by applying corresponding elements of the error vector, or difference between the target vector and the actual neuron output vector, to the inputs of corresponding output neurons of the network as corrective feedback. This feedback decreases the error and quickens the learning process, so that a much smaller number of training cycles are required to complete the learning process. A conventional gradient descent algorithm is employed to update the neural network parameters at the end of the predetermined time interval. The foregoing process is repeated in repetitive cycles until the actual output vector corresponds to the target vector. In the preferred embodiment, as the overall error of the neural network output decreasing during successive training cycles, the portion of the error fed back to the output neurons is decreased accordingly, allowing the network to learn with greater freedom from teacher forcing as the network parameters converge to their optimum values. The invention may also be used to train a neural network with stationary training and target vectors.

Toomarian, Nikzad↗
NASA NTRS1996

Multiple Coarse Grid Multigrid Methods for Solving Elliptic Problems

In this paper we describe some classes of multigrid methods for solving large linear systems arising in the solution by finite difference methods of certain boundary value problems involving Poisson's equation on rectangular regions. If parallel computing systems are used, then with standard multigrid methods many of the processors will be idle when one is working at the coarsest grid levels. We describe the use of Multiple Coarse Grid MultiGrid (MCGMG) methods. Here one first constructs a periodic set of equations corresponding to the given system. One then constructs a set of coarse grids such that for each grid corresponding to the grid size h there are four grids corresponding to the grid size 2*h. Multigrid operations such as restriction of residuals and interpolation of corrections are done in parallel at each grid level. For suitable choices of the multigrid operators the MCGMG method is equivalent to the Parallel Superconvergent MultiGrid (PSMG) method of Frederickson and McBryan. The convergence properties of MCGMG methods can be accurately analyzed using spectral methods.

Xiao, Shengyou↗