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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.↗

DOE OSTI2025

Basin-Size Mapping: Prediction of Metastable Polymorph Synthesizability Across TaC–TaN Alloys

The sizes of the basins of attraction on the potential energy surface are helpful indicators in determining the experimental synthesizability of metastable phases. In principle, these basins can be controlled with changes in thermodynamic conditions such as composition, pressure, and surface energy. Herein, we use random structure sampling to computationally study how alloying smoothly perturbs basin of attraction sizes. The TaC 1-x N x pseudobinary is an ideal test system given the structural and polymorphic contrast of its parent compounds and their technological relevance as epitaxial substrates for Al 1-x Ga x N. While we find limited thermodynamic stability across all computationally observed phases, random structure sampling shows a significant composition region where the rocksalt basin dominates. As such, we predict the potential for the nonequilibrium synthesis of metastable rocksalt TaC 1-x N x alloys as substrates for Al 1-x Ga x N. At higher nitrogen concentrations, other low-energy metastable polymorphs emerge that continue to retain the hexagonal close packing suitable for III-N growth. Confidence in these trends was established through uncertainty quantification of the basin sizes and energy distributions; such analysis utilized the Beta and Dirichlet distributions. In conclusion, we also find (a) polymorph basin sizes can be rationalized in terms of energetic preferences for different coordination environments; and (b) basin sizes universally shrink with increasing nitrogen content, making the system more prone to amorphous growth.

36 MATERIALS SCIENCE↗

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↗

DOE OSTI2023

Consequences and benefits of utilizing continuous vibro-impact representations in constrained pipeline conveying fluid systems

Here, the effectiveness of continuous vibro-impact forcing representations for the cantilevered pipe that conveys fluid is explored and analyzed. The previously accepted forcing model utilizing a smoothened trilinear spring is estimated using three continuous forcing representations, namely, polynomial, rational polynomial, and hyperbolic tangent. The accuracy of the estimated forcing functions is investigated and analyzed by calculating the root mean square error, and bifurcation diagrams are generated and compared to the nominal system. Additionally, the dynamic response of the system is further characterized using Poincare maps, power spectra, and basins of attraction. Once all continuous forcing representations are analyzed and compared to the nominal system, the computational cost of each method is examined, and further limitations of the hyperbolic tangent method are discovered. It is proved that the hyperbolic tangent forcing representation most accurately captures the dynamic response of the pipeline, and the least accurate representation is the rational polynomial representation. Additionally, considerable computational cost is saved when employing the hyperbolic tangent representation compared to the discontinuous representation.

42 ENGINEERING↗

DOE OSTI2023

Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities

Freeplay is a common type of piecewise-smooth nonlinearity in dynamical systems, and it can cause discontinuity-induced bifurcations and other behaviors that may bring about undesirable and potentially damaging responses. Prior research has focused on piecewise-smooth systems with two or three distinct regions, but less attention is devoted to systems with more regions (i.e., multi-segmented systems). In this work, numerical analysis is performed on a dynamical system with multi-segmented freeplay, in which there are four stiffness transitions and five distinct regions in the phase space. Here, the effects of the multi-segmented parameters are studied through bifurcation diagram evolution along with induced multi-stable behavior and different bifurcations. These phenomena are interrogated through various tools, such as harmonic balance, basins of attraction, phase planes, and Poincaré section analysis. Results show that among the three multi-segmented parameters, the asymmetry has the strongest effect on the response of the system.

42 ENGINEERING↗

DOE OSTI2023

Dynamical responses of constrained pipe conveying fluids and its dependence on the modeling of the contact force

Accurately modeling the impact force used in the analysis of loosely constrained cantilevered pipes conveying fluid is imperative. If little information is known of the motion-limiting constraints used in experiments, the analysis of the system may yield inaccurate predictions. Here in this work, multiple forcing representations of the impact force are defined and analyzed for a cantilevered pipe that conveys fluid. Depending on the representation of the impact force, the dynamics of the pipe can vary greatly when only the stiffness of the constraints is known from experiments. Three gap sizes of the constraints are analyzed, and the representation of the impact force used to analyze the system is found to significantly affect the response of the pipe at each gap size. An investigation on the effects of the vibro-impact force representation is performed through using basin of attraction analysis and nonlinear characterization of the system’s response.

42 ENGINEERING↗

DOE OSTI2024

New horizons in the holographic conformal phase transition

Abstract We describe 5D dynamical cosmological solutions of the stabilized holographic dilaton and their role in completion of the conformal phase transition. This analysis corresponds, via the AdS/CFT dictionary, to a study of out-of-equilibrium dynamics where trajectories of the dilaton do not depend solely on thermodynamic quantities in the early universe, but have sensitivity also to initial conditions. Unlike the well-studied thermal transition, which requires quantum tunneling of an infrared brane through the surface of an AdS-Schwarzschild horizon, our approach instead invokes an early epoch in which the cosmology is fully 5-dimensional, with highly relativistic brane motion and with Rindler horizons obscuring the infrared brane at early times. In this context, we demonstrate the existence of a large class of natural initial conditions that seed trajectories where the brane simply passes through the Rindler horizon and into the basin of attraction of the stabilized dilaton potential. This corresponds to successful completion of the phase transition without sacrificing perturbativity of the 5D theory.

Physics↗

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↗

DOE OSTI2026

Distributed Quantum-Enhanced Optimization: A Topographical Preconditioning Approach for High-Dimensional Search

Optimization problems become fundamentally challenging as the number of variables increases. Because the volume of the search space grows exponentially, classical algorithms frequently fail to locate the global minimum of non-convex functions. While quantum optimization offers a potential alternative, mapping continuous problems onto near-term quantum hardware introduces severe scaling limits and barren plateaus. To bridge this gap, we propose the Distributed Quantum-Enhanced Optimization (D-QEO) framework. Instead of forcing the quantum processor to find the exact minimum, we use it simply as a topographical preconditioner. The QPU maps the landscape to locate the most promising basin of attraction, generating high-quality seed points for a classical GPU-accelerated solver to refine. To make this approach viable for utility-scale problems, we exploit the mathematical structure of separable functions. This allows us to cut a 50-qubit (i.e., $2^{50}$) global search space into independent and manageable sub-spaces using 5-qubit subcircuits. By executing these fragments concurrently with CUDA-Q, we completely bypass the overhead of cross-register entanglement and classical tensor knitting for separable functions. Benchmarks on the 10-dimensional Rastrigin and Ackley functions show that D-QEO prevents the exponential failure rates observed in purely classical algorithms. Furthermore, this quantum warm-start significantly reduces the number of classical BFGS iterations required to converge, providing a highly practical blueprint for utilizing near-term quantum resources in complex global search.

Soos, Dominik [Old Dominion U.]↗