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Kernel type density estimators calculated by the method of sieves. Proofs are presented for the characterization theorem: Let x(1), x(2),...x(n) be a random sample from a population with density f(0). Let sigma 0 and consider estimators f of f(0) defined by (1).
The flutter boundaries of six thin highly-swept delta-platform wings have been calculated. Comparisons are made between experimental data and results using several aerodynamic methods. The aerodynamic methods used include a subsonic and supersonic kernel function, second order piston theory, and a transonic small disturbance code. The dynamic equations of motion are solved using analytically calculated mode shapes and frequencies.
Many advances have been made in building technology to help save energy, but influencing the behavior of the occupants is still necessary to achieve low-energy use targets. One of the most practical ways to influence and change occupant behaviors is through incentives. Developing incentives for energy-saving and quantifying the impact of occupant behaviors are both active areas of research. Here, we propose a data analytics framework for detecting changes in occupant behaviors, which will help build an analytics feedback loop from behavior impact to incentive design. The framework has two major parts. The first forecasts energy consumption for each occupant, while the second determines a probability distribution for changes in energy consumption. The parts are interchangeable with other existing machine learning and statistical methods. A specific instantiation of the framework, using kernel ridge-regression for forecasting and k-means to find an empirical behavior distribution, is described in detail. An HVAC use-case with 5 different incentivized behaviors is used as an example to show that the framework can detect behavior changes induced by incentives. Furthermore, we show that some simpler behavior-change detection methods do not work, further justifying the use of advanced analytics.
Time-dependent orbital-free density functional theory (DFT) is an efficient method for calculating the dynamic properties of large-scale quantum systems due to the low computational cost compared to standard time-dependent DFT. In this work, we formalize this method by mapping the real system of interacting fermions onto a fictitious system of noninteracting bosons. The dynamic Pauli potential and associated kernel emerge as key ingredients of time-dependent orbital-free DFT. Using the uniform electron gas as a model system, we derive an approximate frequency-dependent Pauli kernel. Pilot calculations suggest that space nonlocality is a key feature for this kernel. Nonlocal terms arise already in the second-order expansion with respect to unitless frequency and reciprocal space variable ($\frac{ω}{qk_F}$ and $\frac{q}{2k_F}$, respectively). Given the encouraging performance of the proposed kernel, we expect it will lead to more accurate orbital-free DFT simulations of nanoscale systems out of equilibrium. Additionally, the proposed path to formulate nonadiabatic Pauli kernels presents several avenues for further improvements which can be exploited in future work to improve the results.
We introduce an Eulerian Perturbation Theory to study the clustering of tracers for cosmologies in the presence of massive neutrinos. Our approach is based on mapping recently-obtained Lagrangian Perturbation Theory results to the Eulerian framework. We add Effective Field Theory counterterms, IR-resummations and a biasing scheme to compute the one-loop redshift-space power spectrum. To assess our predictions, we compare the power spectrum multipoles against synthetic halo catalogues from the QUIJOTE simulations, finding excellent agreement on scales k ≲ 0.25 h Mpc-1. One can obtain the same fitting accuracy using higher wave-numbers, but then the theory fails to give a correct estimation of the linear bias parameter. We further discuss the implications for the tree-level bispectrum. Finally, calculating loop corrections is computationally costly, hence we derive an accurate approximation wherein we retain only the main features of the kernels, as produced by changes to the growth rate. As a result, we show how FFTLog methods can be used to further accelerate the loop computations with these reduced kernels.
This paper presents a new methodology for automatic knowledge driven image mining based on the theory of Mercer Kernels; which are highly nonlinear symmetric positive definite mappings from the original image space to a very high, possibly infinite dimensional feature space. In that high dimensional feature space, linear clustering, prediction, and classification algorithms can be applied and the results can be mapped back down to the original image space. Thus, highly nonlinear structure in the image can be recovered through the use of well-known linear mathematics in the feature space. This process has a number of advantages over traditional methods in that it allows for nonlinear interactions to be modelled with only a marginal increase in computational costs. In this paper, we present the theory of Mercer Kernels, describe its use in image mining, discuss a new method to generate Mercer Kernels directly from data, and compare the results with existing algorithms on data from the MODIS (Moderate Resolution Spectral Radiometer) instrument taken over the Arctic region. We also discuss the potential application of these methods on the Intelligent Archive, a NASA initiative for developing a tagged image data warehouse for the Earth Sciences.
This paper presents a new methodology for automatic knowledge driven image mining based on the theory of Mercer Kernels, which are highly nonlinear symmetric positive definite mappings from the original image space to a very high, possibly infinite dimensional feature space. In that high dimensional feature space, linear clustering, prediction, and classification algorithms can be applied and the results can be mapped back down to the original image space. Thus, highly nonlinear structure in the image can be recovered through the use of well-known linear mathematics in the feature space. This process has a number of advantages over traditional methods in that it allows for nonlinear interactions to be modelled with only a marginal increase in computational costs. In this paper we present the theory of Mercer Kernels; describe its use in image mining, discuss a new method to generate Mercer Kernels directly from data, and compare the results with existing algorithms on data from the MODIS (Moderate Resolution Spectral Radiometer) instrument taken over the Arctic region. We also discuss the potential application of these methods on the Intelligent Archive, a NASA initiative for developing a tagged image data warehouse for the Earth Sciences.
In this work, we consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. Furthermore, in order to provide an efficient and reliable approximation of the solution for different values of the parameters, we develop the reduced basis method as a parametric model order reduction approach. Major difficulties arise since the kernels are not affine in the parameters, singular, and discontinuous. Moreover, the spatial regularity of the solutions depends on the varying fractional power $s$. To address this, we derive regularity and differentiability results with respect to $\delta$ and $s$, which are of independent interest for other applications such as optimization and parameter identification. We then use these results to construct affine approximations of the kernels by local polynomials. Finally, we certify the method by providing reliable a posteriori error estimators, which account for all approximation errors, and support the theoretical findings by numerical experiments.
A reduced order modeling approach of the Navier-Stokes equations is presented for the design of a distributed optimal feedback kernel. This approach is based oil a Krylov subspace method where significant modes of the flow are captured in the model This model is then used in all optimal feedback control design where sensing and actuation is performed oil tile entire flow field. This control design approach yields all optimal feedback kernel which provides insight into the placement of sensors and actuators in the flow field. As all evaluation of this approach, a two-dimensional shear layer and driven cavity flow are investigated.
The truncation theory as it pertains to the calculation of geoid undulations based on Stokes' integral, but from limited gravity data, is reexamined. Specifically, the improved procedures of Molodenskii et al. are shown through numerical investigations to yield substantially smaller errors than the conventional method that is often applied in practice. In this improved method, as well as in a simpler alternative to the conventional approach, the Stokes' kernel is suitably modified in order to accelerate the rate of convergence of the error series. These modified methods, however, effect a reduction in the error only if a set of low-degree potential harmonic coefficients is utilized in the computation. Consider, for example, the situation in which gravity anomalies are given in a cap of radius 10 deg and the GEM 9 (20,20) potential field is used. Then, typically, the error in the computed undulation (aside from the spherical approximation and errors in the gravity anomaly data) according to the conventional truncation theory is 1.09 m; with Meissl's modification it reduces to 0.41m, while Molodenskii's improved method gives 0.45 m. A further alteration of Molodenskii's method is developed and yields an RMS error of 0.33 m. These values reflect the effect of the truncation, as well as the errors in the GEM 9 harmonic coefficients. The considerable improvement, suggested by these results, of the modified methods over the conventional procedure is verified with actual gravity anomaly data in two oceanic regions, where the GEOS-3 altimeter geoid serves as the basis for comparison. The optimal method of truncation, investigated by Colombo, is extremely ill-conditioned. It is shown that with no corresponding regularization, this procedure is inapplicable.
An extension of the Chahine relaxation method (1970) for inverting the radiative transfer equation is presented. This method is superior to the original method in that it takes into account in a realistic manner the shape of the kernel function, and its extension to nonlinear systems is much more straightforward. A comparison of the new method with a matrix method due to Twomey (1965), in a problem involving inference of vertical distribution of ozone from spectroscopic measurements in the near ultraviolet, indicates that in this situation this method is stable with errors in the input data up to 4%, whereas the matrix method breaks down at these levels. The problem of non-uniqueness of the solution, which is a property of the system of equations rather than of any particular algorithm for solving them, remains, although it takes on slightly different forms for the two algorithms.
A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turbulence-model and chemistry-model diffusion terms, and the corresponding Jacobian contributions. Initially, the EBV method had been implemented for tetrahedral grids and demonstrated multifold acceleration of all viscous-kernel computations. This paper presents an extension of the EBV method for mixed-element grids. In addition to the primal edges of a given mixed-element grid, virtual edges are introduced to connect cell nodes that are not connected by a primal edge. The EBV method uses an efficient loop over all (primal and virtual) edges and features a compact discretization stencil based on the nearest neighbors. This study verifies the EBV method and assesses its efficiency on mixed-element grids by comparing the EBV solution accuracy and iterative convergence with those of well-established solutions obtained using a cell-based viscous (CBV) discretization method. The EBV solver’s memory footprint is optimized and often smaller than the memory footprint of the CBV solver. A multifold speedup is demonstrated for all viscous-kernel computations resulting in significant reduction of the time to solutions for several benchmark mixed-element-grid computations, including simulations of a flow around NASA’s juncture-flow model and a hypersonic, chemically reacting flow around a blunt body.
A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turbulence-model and chemistry-model diffusion terms, and the corresponding Jacobian contributions. Initially, the EBV method had been implemented for tetrahedral grids and demonstrated multifold acceleration of all viscous-kernel computations. This paper presents an extension of the EBV method for mixed-element grids. In addition to the primal edges of a given mixed-element grid, virtual edges are introduced to connect cell nodes that are not connected by a primal edge. The EBV method uses an efficient loop over all (primal and virtual) edges and features a compact discretization stencil based on the nearest neighbors. This study verifies the EBV method and assesses its efficiency on mixed-element grids by comparing the EBV solution accuracy and iterative convergence with those of well-established solutions obtained using a cell-based viscous (CBV) discretization method. The EBV solver’s memory footprint is optimized and often smaller than the memory footprint of the CBV solver. A multifold speedup is demonstrated for all viscous-kernel computations resulting in significant reduction of the time to solutions for several benchmark mixed-element-grid computations, including simulations of a flow around NASA’s juncture-flow model and a hypersonic, chemically reacting flow around a blunt body.
he CMS detector at the High-Luminosity Large Hadron Collider (HL-LHC) will operate in challenging conditions with expected pile-up of up to 200 collisions per bunch crossing, necessitating the development of a more resilient primary vertex (PV) reconstruction method to ensure the integrity of data analysis and the efficiency of the CMS triggering system. This contribution describes preliminary studies on a new ML based PV-Finder method for PV identification. The method is based on a model trained using Kernel Density Estimations (KDEs) derived from the positions of reconstructed tracks at the beamline, incorporating uncertainties from track parameters. It also utilizes target histograms, modeled as Gaussian distributions centered on the actual ground truth values of specific primary vertices.
Streaks caused by satellites are a persistent problem in optical images, and their occurrence will likely increase in the coming years. Many Hyper Suprime-Cam images are already affected by streaks, and the same is expected for Rubin Observatory data. In the LSST Data Release Production (DRP), most streaks and other artifacts are already detected and masked by the algorithm CompareWarpAssembleCoadd. However, some streaks are not caught at this stage and can contaminate the final coadds and detection catalogs. To find and mask these, we adopt a morphologically-based method for detecting streaks, which uses the Kernel-Based Hough Transform to detect straight lines. Once the lines are detected, the streak profile is fit and the affected portion of the image is masked out. This algorithm, maskStreaks, is included in the meas_algorithms package and is used in CompareWarpAssembleCoadd to remove streaks from coadds. Similar implementation in the Alert Production Pipeline is possible but has not been implemented.
KAP-5 computer program for design and analysis of radiation shields in nuclear rocket systems
Data from all current JPL missions are stored in files called SPICE kernels. At present, animators who want to use data from these kernels have to either read through the kernels looking for the desired data, or write programs themselves to retrieve information about all the needed objects for their animations. In this project, methods of automating the process of importing the data from the SPICE kernels were researched. In particular, tools were developed for creating basic scenes in Maya, a 3D computer graphics software package, from SPICE kernels.
PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Itô equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.