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DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

Nonlinear problems of the theory of heterogeneous slightly curved shells

An account if given of the variational method of the solution of physically and geometrically nonlinear problems of the theory of heterogeneous slightly curved shells. Examined are the bending and supercritical behavior of plates and conical and spherical cupolas of variable thickness in a temperature field, taking into account the dependence of the elastic parameters on temperature. The bending, stability in general and load-bearing capacity of flexible isotropic elastic-plastic shells with different criteria of plasticity, taking into account compressibility and hardening. The effect of the plastic heterogeneity caused by heat treatment, surface work hardening and irradiation by fast neutron flux is investigated. Some problems of the dynamic behavior of flexible shells are solved. Calculations are performed in high approximations. Considerable attention is given to the construction of a machine algorithm and to the checking of the convergence of iterative processes.

Kantor, B. Y.↗

NASA NTRS1993

Retrieval of ice thickness from polarimetric SAR data

We describe a potential procedure for retrieving ice thickness from multi-frequency polarimetric SAR data for thin ice. This procedure includes first masking out the thicker ice types with a simple classifier and then deriving the thickness of the remaining pixels using a model-inversion technique. The technique used to derive ice thickness from polarimetric observations is provided by a numerical estimator or neural network. A three-layer perceptron implemented with the backpropagation algorithm is used in this investigation with several improved aspects for a faster convergence rate and a better accuracy of the neural network. These improvements include weight initialization, normalization of the output range, the selection of offset constant, and a heuristic learning algorithm. The performance of the neural network is demonstrated by using training data generated by a theoretical scattering model for sea ice matched to the database of interest. The training data are comprised of the polarimetric backscattering coefficients of thin ice and the corresponding input ice parameters to the scattering model. The retrieved ice thickness from the theoretical backscattering coefficients is compare with the input ice thickness to the scattering model to illustrate the accuracy of the inversion method. Results indicate that the network convergence rate and accuracy are higher when multi-frequency training sets are presented. In addition, the dominant backscattering coefficients in retrieving ice thickness are found by comparing the behavior of the network trained backscattering data at various incidence angels. After the neural network is trained with the theoretical backscattering data at various incidence anges, the interconnection weights between nodes are saved and applied to the experimental data to be investigated. In this paper, we illustrate the effectiveness of this technique using polarimetric SAR data collected by the JPL DC-8 radar over a sea ice scene.

Kwok, R.↗

NASA NTRS1990

Behavior of deep convective clouds in the tropical Pacific deduced from ISCCP radiances

The characteristic features, the diurnal cycle, and the spatial distribution of deep convection over the equatorial Pacific and the relationship of deep convection to SST and surface-wind convergence were examined using a combined visible-IR (VS-IR) threshold method and an IR-only threshold method for diagnosing deep convection clouds (DCCs). Results suggest that deep convection is latitudinally confined to a much smaller spatial scale than that suggested by maps of outgoing long-wave radiation. The results suggested that there are two types of relationships between deep convection, SST, and surface-wind convergence: the west Pacific type and the east Pacific type. The latter relationship is observed in the east Pacific only when SST is not abnormally warm.

Fu, Rong↗

NASA NTRS2000

Descending Layer Variability Over Arecibo

Descending layers of ionization over Arecibo exhibit very diverse behavior from night to night that does not appear to be strongly correlated to geomagnetic activity, solar forcing, or average semidiurnal tidal winds. On some nights, three or more distinct layers are observed to form near 170 km over timescales of approx. 2 hours. Rather than descending smoothly over periods of several hours, these layers stall, abruptly disappear, or even reverse direction in the midst of their descent. The time scales for their disappearance are examined and compared to loss rates arising from diffusion and recombination. Diffusion alone is found to be too slow to account for the observations, but recombination is fast enough provided that the convergent wind shear that forms the layer is relatively weak coincident with their disappearance. The continuity equation is solved in conjunction with a time sequence of radar profiles to estimate the vertical drift and horizontal neutral wind consistent with the observed behavior. The resultant wind field is northward, has an average speed of approx. 80 m/ s, and varies significantly near the altitude where the layers are observed. These inferred winds are consistent with the presence of the observed layers, and their magnitudes as obtained from the classical continuity and momentum equations are reasonable for this altitude range.

Earle, G. D.↗

NASA NTRS1994

Planetary-scale circulations in the presence of climatological and wave-induced heating

Interaction between the large-scale circulation and the convective pattern is investigated in a coupled system governed by the linearized primitive equations. Convection is represented in terms of two components of heating: A 'climatological component' is prescribed stochastically to represent convection that is maintained by fixed distributions of land and sea and sea surface temperature (SST). An 'induced component' is defined in terms of the column-integrated moisture flux convergence to represent convection that is produced through feedback with the circulation. Each component describes the envelope organizing mesoscale convective activity. As SST on the equator is increased, induced heating amplifies in the gravest zonal wavenumbers at eastward frequencies, where positive feedback offsets dissipation. Under barotropic stratification, a critical SST of 29.5 C results in positive feedback exactly cancelling dissipation in wavenumber 1 for an eastward phase speed of 6 m/s. Sympathetic interaction between the circulation and the induced heating is the basis for 'frictional wave-Conditional Instability of the Second Kind (CISK)', which is distinguished from classical wave-CISK by rendering the gravest zonal dimensions most unstable. Under baroclinic stratification, the coupled system exhibits similar behavior. The critical SST is only 26.5 C for conditions representative of equinox, but in excess of 30 C for conditions representative of solstice. Having the form of an unsteady Walker circulation, the disturbance produced by frictional wave-CISK compares favorably with the observed life cycle of the Madden-Julian oscillation (MJO). SST above the critical value produces an amplifying disturbance in which enhanced convection coincides with upper-tropospheric westerlies and is positively correlated with temperature and surface convergence. Conversely, SST below the critical value produces a decaying disturbance in which enhanced convection coincides with upper-tropospheric easterlies and is nearly in quadrature with temperature and surface convergence. While sharing essential features with the MJO in the Eastern Hemisphere, frictional wave-CISK does not explain observed behavior in the Western Hemisphere, where the convective signal is largely absent. Comprised of Kelvin structure with the same frequency, observed behavior in the Western Hemisphere can be understood as a propagating response that is excited in and radiates away from the fluctuation of convection in the Eastern Hemisphere.

Salby, Murry L↗

NASA NTRS1990

Implicit solution of three-dimensional internal turbulent flows

The scalar form of the approximate factorization method was used to develop a new code for the solution of three-dimensional internal laminar and turbulent compressible flows. The Navier-Stokes equations in their Reynolds-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit and explicit artificial damping schemes that proved to be particularly efficient in speeding up convergence and enhancing the algorithm robustness. A conservative treatment of these terms at domain boundaries is proposed in order to avoid undesired mass and/or momentum artificial fluxes. Turbulence effects are accounted for by the zero-equation Baldwin-Lomax turbulence model and the q-omega two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The flow in a developing S-duct is then solved in the laminar regime at Reynolds number (Re) 790 and in the turbulent regime at Re=40,000 using the Baldwin-Lomax model . The Stanitz elbow is then solved using an inviscid version of the same code at M(sub inlet)=0.4. Grid dependence and convergence rate are investigated showing that for this solver the implicit damping scheme may play a critical role for convergence characteristics. The same flow at Re=2.5x10(exp 6) is solved with the Baldwin-Lomax and the q-omega models. Both approaches showed satisfactory agreement with experiments, although the q-omega model is slightly more accurate.

Michelassi, V.↗

DOE OSTI2021

When and why PINNs fail to train: A neural tangent kernel perspective

Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known about why such models sometimes fail to train at all. Here in this work, we aim to investigate these questions through the lens of the Neural Tangent Kernel (NTK); a kernel that captures the behavior of fully-connected neural networks in the infinite width limit during training via gradient descent. Specifically, we derive the NTK of PINNs and prove that, under appropriate conditions, it converges to a deterministic kernel that stays constant during training in the infinite-width limit. This allows us to analyze the training dynamics of PINNs through the lens of their limiting NTK and find a remarkable discrepancy in the convergence rate of the different loss components contributing to the total training error. To address this fundamental pathology, we propose a novel gradient descent algorithm that utilizes the eigenvalues of the NTK to adaptively calibrate the convergence rate of the total training error. Finally, we perform a series of numerical experiments to verify the correctness of our theory and the practical effectiveness of the proposed algorithms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Space-Split Algorithm for Sensitivity Analysis of Discrete Chaotic Systems With Multidimensional Unstable Manifolds

Accurate approximations of the change of a system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the linear response of chaotic dynamical systems. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting, which regularizes Ruelle's original expression, together with measure-based parameterization of the expanding subspace. We use these tools to rigorously derive trajectory-following recursive relations that converge exponentially fast, and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics. Thanks to the regularization and lack of simplifying assumptions on the system's behavior, our method is immune to the common problems of other popular methods such as the exploding tangent solutions and unphysical shadowing directions. Here, we provide a ready-to-use algorithm, analyze its complexity, and demonstrate several numerical examples of sensitivity computation using physically-inspired low-dimensional systems.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1985

The atmospheric heat source over the Tibetan plateau May-August 1979

Estimates of the time and space variability of the atmospheric heat source over Tibet are presented for the summer of 1979. These estimates rely on new data from the People's Republic of China allowing a better assessment of the surface heat fluxes, and on new satellite data from Nimbus-7 giving the radiation balance at the top of the atmosphere. The estimates of the atmospheric heat source turned out to be considerably smaller than those provided earlier in the literature, mainly because of different assumptions of the drag coefficient. The atmospheric heat source over Tibet is mainly modulated by the release of latent heat. Over the southeastern and southwestern plateau regions the heat source appears to be in phase with the precipitation yield of the Indian summer monsoon, whereas central Tibet reveals an out-of-phase behavior. Over western Tibet there appears to be hardly any net import of moisture from outside the region, whereas the maintenance of the hydrological cycle over eastern Tibet requires moisture flux convergence from outside the region of up to 40 percent of the mean rainfall, in agreement with what is known about the surface hydrology of Tibet.

Chen, L.↗

NASA NTRS1990

New evidence favoring multilevel decomposition and optimization

A new multilevel optimization technique based on a global sensitivity matrix which uses behavior variables in addition to design variables and constraints to decompose the problem is proposed. The technique simplifies the decomposition of the design problem and the programming task and improves the convergence speed of multilevel optimization compared to conventional optimization. The technique is particularly well suited to problems with large numbers of design variables and with computationally expensive constraints.

Padula, Sharon L.↗

DOE OSTI2020

Impact of isolation and fiducial cuts on qT and N-jettiness subtractions

Kinematic selection cuts and isolation requirements are a necessity in experimental measurements for identifying prompt leptons and photons that originate from the hard-interaction process of interest. We analyze how such cuts affect the application of the q T and N-jettiness subtraction methods for fixed-order calculations. We consider both fixed-cone and smooth-cone isolation methods. We find that kinematic selection and isolation cuts both induce parametrically enhanced power corrections with considerably slower convergence compared to the standard power corrections that are already present in inclusive cross sections without additional cuts. Using analytic arguments at next-to-leading order we derive their general scaling behavior as a function of the subtraction cutoff. We also study their numerical impact for the case of gluon-fusion Higgs production in the H → γγdecay mode and for pp → γγ direct diphoton production. We find that the relative enhancement of the additional cut-induced power corrections tends to be more severe forq T , where it can reach an order of magnitude or more, depending on the choice of parameters and subtraction cutoffs. We discuss how all such cuts can be incorporated without causing additional power corrections by implementing the subtractions differentially rather than through a global slicing method. We also highlight the close relation of this formulation of the subtractions to the projection-to-Born method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2005

Analysis of Slope Limiters on Irregular Grids

This paper examines the behavior of flux and slope limiters on non-uniform grids in multiple dimensions. Many slope limiters in standard use do not preserve linear solutions on irregular grids impacting both accuracy and convergence. We rewrite some well-known limiters to highlight their underlying symmetry, and use this form to examine the proper - ties of both traditional and novel limiter formulations on non-uniform meshes. A consistent method of handling stretched meshes is developed which is both linearity preserving for arbitrary mesh stretchings and reduces to common limiters on uniform meshes. In multiple dimensions we analyze the monotonicity region of the gradient vector and show that the multidimensional limiting problem may be cast as the solution of a linear programming problem. For some special cases we present a new directional limiting formulation that preserves linear solutions in multiple dimensions on irregular grids. Computational results using model problems and complex three-dimensional examples are presented, demonstrating accuracy, monotonicity and robustness.

Berger, Marsha↗

DOE OSTI2020

Scaling Analysis of Two–Phase Flow in Fractal Permeability Fields

Fluid mixing in permeable media is essential in many practical applications. The mixing process is a consequence of velocity fluctuations owing to geological heterogeneities and mobility contrast of fluids. Heterogeneities in natural rocks are often spatially correlated, and their properties, such as permeability, may be described using fractal distributions. This work models the fractal characteristics of such permeability fields in which the covariance function is expressed as a power-law function. A generalized scaling relation is derived relating various fractal permeability fields using the magnitude of their fluctuations. Here, this relation reveals the self-similar behavior of two-phase flow in such permeable media. To that end, a recently developed, high-resolution numerical simulator is employed to validate the analytically derived scaling relations. Two flow problems are considered in which flow is governed by 1) a linear, and 2) a nonlinear transport equation. Due to the probabilistic representation of the fractal permeability fields, a sensitivity study is conducted for each flow scenario to determine the number of realizations required for statistical convergence. Scaling analysis is performed using ensemble averages of simulated saturation profiles and their mixing lengths. Results support the validity of the developed scaling relation across the range of investigated flow conditions at intermediate times. The dynamics of linear flow in the asymptotic regime is affected by the correlation structure of heterogeneity. In nonlinear flow, scaling behavior appears to be dominated by the degree of nonlinearity.

58 GEOSCIENCES↗

DOE OSTI2025

A Bayesian approach to time-domain photonic Doppler velocimetry analysis

Photonic Doppler velocimetry (PDV) is an established technique for measuring the velocities of fast-moving surfaces in high-energy-density experiments. In the standard approach to PDV analysis, the short-time Fourier transform (STFT) is used to generate a spectrogram from which the velocity history of the target is inferred. The user chooses the form, duration, and separation of the window function. Here, in this study, we present a Bayesian approach to infer the velocity directly from the PDV oscilloscope trace, without using the spectrogram for analysis. This is clearly a difficult inference problem due to the highly periodic nature of the data, but we find that with carefully chosen prior distributions for the model parameters, we can accurately recover the injected velocity from synthetic data. We validate this method using PDV data collected at the STAR two-stage light gas gun at Sandia National Laboratories, recovering shock-front velocities in quartz that are consistent with those inferred using the STFT-based approach and are interpolated across regions of low signal-to-noise data. Although this method does not rely on the same user choices as the STFT, we caution that it can be prone to misspecification if the chosen model is not sufficient to capture the velocity behavior. Analysis using posterior predictive checks can be used to establish whether a better model is required, although more complex models come with additional computational cost, often taking more than several hours to converge when sampling the Bayesian posterior. We, therefore, recommend it be viewed as a complementary method to that of the STFT-based approach.

Allison, James R. [First Light Fusion Ltd., Yarnto↗

NASA NTRS2026

Development and Experimental Validation of a Path-Dependent Spin Forming Finite Element Model

Spin forming is an advanced manufacturing process widely used in the aerospace and defense sectors to produce lightweight, high-strength cylindrical components with tight dimensional tolerances. This study explores the applicability of the path-dependent Mechanical Threshold Stress (MTS) constitutive model by simulating the evolution of geometry, machining forces, and plastic deformation during the spin forming of a 10-mm thick 6061-O aluminum cylinder. While numerical modeling of spin forming has advanced substantially over the past decade, systematic verification and experimental validation of material models remain limited, particularly in predicting through-thickness process evolution. The MTS model, incorporating a Voce hardening rule, is employed for its ability to represent cyclic loading, rapidly varying temperature fields, and strain rates characteristic of spin forming. Numerical convergence analysis indicates discretization uncertainties between 0.3% and 9.2% for key quantities of interest. Experimental validation demonstrates that the MTS model, when implemented with a verified mesh, accurately reproduces both elastic and plastic behavior of 6061-O aluminum, predicting peak roller loads within 11–18% of measurements, geometric tolerances within 3%, and plastic strain distributions within 10% of experimental values. Collectively, these results establish a validated computational framework for predictive spin-forming simulations with quantified confidence, providing a foundation for extension to other alloys, geometries, and forming conditions.

Spin forming↗

NASA NTRS1984

An investigation of the transient thermal analysis of spur gears

A finite element computer program is developed for evaluating the transient behavior of surface temperature in high performance spur gears. The time dimension is implemented using two and three point finite difference schemes. The different schemes are provided for the purpose of numerical stability and convergence studies. A detailed explanation of the gear cooling process leading to the establishment of a modified Blok model is also included. Other conventional models for approximating the heat transfer coefficients are available for comparison. Preliminary results are given showing snap shots of gear temperature contours at the initial stages of tooth engagement.

El-Bayoumy, L. E.↗

NASA NTRS1985

An investigation of the transient thermal analysis of spur gears

El-Bayoumy, L. E.↗