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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Quantum Optimization and Approximation Algorithms

Shor's groundbreaking quantum algorithm for integer factoring provides an exponential speedup over the best-known classical algorithms. In the 20 years since Shor's algorithm was conceived, only a handful of fundamental quantum algorithmic kernels, generally providing modest polynomial speedups over classical algorithms, have been invented. To better understand the potential advantage quantum resources provide over their classical counterparts, one may consider other resources than execution time of algorithms. Quantum Approximation Algorithms direct the power of quantum computing towards optimization problems where quantum resources provide higher-quality solutions instead of faster execution times. We provide a new rigorous analysis of the recent Quantum Approximate Optimization Algorithm, demonstrating that it provably outperforms the best known classical approximation algorithm for special hard cases of the fundamental Maximum Cut graph-partitioning problem. We also develop new types of classical approximation algorithms for finding near-optimal low-energy states of physical systems arising in condensed matter by extending seminal discrete optimization techniques. Our interdisciplinary work seeks to unearth new connections between discrete optimization and quantum information science.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Testing the LSST Difference Image Analysis Pipeline Using Synthetic Source Injection Analysis

Abstract We evaluate the performance of the Legacy Survey of Space and Time Science Pipelines Difference Image Analysis (DIA) on simulated images. By adding synthetic sources to galaxies on images, we trace the recovery of injected synthetic sources to evaluate the pipeline on images from the Dark Energy Science Collaboration Data Challenge 2. The pipeline performs well, with efficiency and flux accuracy consistent with the signal-to-noise ratio of the input images. We explore different spatial degrees of freedom for the Alard–Lupton polynomial-Gaussian image subtraction kernel and analyze for trade-offs in efficiency versus artifact rate. Increasing the kernel spatial degrees of freedom reduces the artifact rate without loss of efficiency. The flux measurements with different kernel spatial degrees of freedom are consistent. We also here provide a set of DIA flags that substantially filter out artifacts from the DIA source table. We explore the morphology and possible origins of the observed remaining subtraction artifacts and suggest that given the complexity of these artifact origins, a convolution kernel with a set of flexible bases with spatial variation may be needed to yield further improvements.

Liu, S. (ORCID:0000000244612143)↗
NASA NTRS1974

Nonparametric maximum likelihood estimation of probability densities by penalty function methods

When it is known a priori exactly to which finite dimensional manifold the probability density function gives rise to a set of samples, the parametric maximum likelihood estimation procedure leads to poor estimates and is unstable; while the nonparametric maximum likelihood procedure is undefined. A very general theory of maximum penalized likelihood estimation which should avoid many of these difficulties is presented. It is demonstrated that each reproducing kernel Hilbert space leads, in a very natural way, to a maximum penalized likelihood estimator and that a well-known class of reproducing kernel Hilbert spaces gives polynomial splines as the nonparametric maximum penalized likelihood estimates.

Demontricher, G. F.↗
DOE OSTI2024

Galaxy bispectrum in the spherical Fourier-Bessel basis

The bispectrum, the three-point correlation in Fourier space, is a crucial statistic for studying many effects targeted by the next-generation galaxy surveys, such as primordial non-Gaussianity (PNG) and general relativistic (GR) effects on large scales. In this work we develop a formalism for the bispectrum in the spherical Fourier-Bessel (SFB) basis—a natural basis for computing correlation functions on the curved sky, as it diagonalizes the Laplacian operator in spherical coordinates. Working in the SFB basis allows for line-of-sight effects such as redshift space distortions and GR to be accounted for exactly, i.e., without having to resort to perturbative expansions to go beyond the plane-parallel approximation. Only analytic results for the SFB bispectrum exist in the literature given the intensive computations needed. We numerically calculate the SFB bispectrum for the first time, enabled by a few techniques: We implement a template decomposition of the redshift-space kernel Z 2 into Legendre polynomials, and separately treat the PNG and velocity-divergence terms. We derive an identity to integrate a product of three spherical harmonics connected by a Dirac delta function as a simple sum and use it to investigate the limit of a homogeneous and isotropic Universe. Furthermore, we present a formalism for convolving the signal with separable window functions and use a toy spherically symmetric window to demonstrate the computation and give insights into the properties of the observed bispectrum signal. While our implementation remains computationally challenging, it is a step toward a feasible full extraction of information on large scales via a SFB bispectrum analysis.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

The effective number of parameters in kernel density estimation

We devise a new formula for measuring the effective degrees of freedom (EDoF) in kernel density estimation (KDE). Starting from the orthogonal polynomial sequence (OPS) expansion for the ratio of the empirical to the oracle density, we show how convolution with the kernel leads to a new OPS with respect to which one may express the resulting KDE. The expansion coefficients of the two OPS systems can then be related via a kernel sensitivity matrix, which leads to a natural oracle definition of EDoF through the trace operator. Asymptotic properties of the (empirical) plug-in EDoF are worked out through influence functions, and connections with other empirical EDoFs are established. Minimization of Kullback-Leibler divergence is investigated as an alternative to integrated squared error based bandwidth selection rules, yielding a new normal scale rule. The methodology, which arises from a proper oracle formulation and is not restricted to convolution kernels, suggests the possibility of a new bandwidth selection rule based on an information criterion such as AIC.

bandwidth selection↗
DOE OSTI2018

Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind

In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order O(n{sup −r}) in L{sup 2}-norm and order O(n{sup −r+(1/2)}) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n{sup −2r}) in both L{sup 2}-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n{sup −r}) in both L{sup 2}-norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order O(n{sup −3r}) in both infinity and L{sup 2} norm. Numerical examples are given to illustrate the theoretical results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A sparse matrix–vector multiplication based algorithm for accurate density matrix computations on systems of millions of atoms

Here we present an efficient sparse matrix–vector (SpMV) based method to compute the density matrix P from a given Hamiltonian in electronic structure computations. Our method is a hybrid approach based on Chebyshev–Jackson approximation theory and matrix purification methods like the second order spectral projection purification (SP2). Recent methods to compute the density matrix scale as O(N) in the number of floating point operations but are accompanied by large memory and communication overhead, and they are based on iterative use of the sparse matrix–matrix multiplication kernel (SpGEMM), which is known to be computationally irregular. In addition to irregularity in the sparse Hamiltonian H, the nonzero structure of intermediate estimates of P depends on products of H and evolves over the course of computation. On the other hand, an expansion of the density matrix P in terms of Chebyshev polynomials is straightforward and SpMV based; however, the resulting density matrix may not satisfy the required constraints exactly. In this paper, we analyze the strengths and weaknesses of the Chebyshev–Jackson polynomials and the second order spectral projection purification (SP2) method, and propose to combine them so that the accurate density matrix can be computed using the SpMV computational kernel only, and without having to store the density matrix P. Our method accomplishes these objectives by using the Chebyshev polynomial estimate as the initial guess for SP2, which is followed by using sparse matrix–vector multiplications (SpMVs) to replicate the behavior of the SP2 algorithm for purification. We demonstrate the method on a tight-binding model system of an oxide material containing more than 3 million atoms. In addition, we also present the predicted behavior of our method when applied to near-metallic Hamiltonians with a wide energy spectrum.

42 ENGINEERING↗
DOE OSTI2022

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn–Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. In this work, we present a density matrix based method for Kohn–Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann–Feynman forces, and Hellmann–Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw–Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045 K from Kohn–Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

A functional global sensitivity measure and efficient reliability sensitivity analysis with respect to statistical parameters

Sensitivity analysis and reliability assessment are two important aspects of structural and system safety. Epistemic uncertainty with respect to probabilistic model of input parameters due to lack of knowledge is present in many scarce-data applications and complicates the characterization of uncertainty in model response. In this article, we present two importance measures to evaluate the impact of distribution parameters on the probability distribution function (PDF) of the output and the failure probability. The epistemic uncertainty associated with the distribution parameters is modeled as random variables. Additionally, a modified extended polynomial chaos expansion (MEPCE) approach is introduced in which aleatory and epistemic random variables are modeled and propagated simultaneously while allowing the separate assessment for any single epistemic variable. A MEPCE-based kernel density estimation (KDE) construction provides a composite map from each epistemic variable to the response PDF. The functional global sensitivity index of the PDF with respect to the distribution parameters is thus derived, as a function of output, which is both more informative and more efficient than standard scalar sensitivity measures. Reliability sensitivity indices can be readily evaluated by integrating the global sensitivity index function over the failure zone. Three illustrative examples are used to demonstrate the proposed methodology.

42 ENGINEERING↗
DOE OSTI2020

Estimation of distributions via multilevel Monte Carlo with stratified sampling

We design and implement a novel algorithm for computing a multilevel Monte Carlo (MLMC) estimator of the joint cumulative distribution function (CDF) of a vector-valued quantity of interest in problems with random input parameters and initial conditions. Our approach combines MLMC with stratified sampling of the input sample space by replacing standard Monte Carlo at each level with stratified Monte Carlo initialized with proportionally allocated samples. We show that the resulting stratified MLMC (sMLMC) algorithm is more efficient than its standard MLMC counterpart due to the additional variance reduction provided by the stratification of the random parameter's domain, especially at the coarsest levels. Additional computational cost savings are obtained by smoothing the indicator function with a Gaussian kernel, which proves to be an efficient and robust alternative to recently developed polynomial-based techniques.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Optimization of the generator coordinate method with machine-learning techniques for nuclear spectra and neutrinoless double- β decay: Ridge regression for nuclei with axial deformation

The generator coordinate method (GCM) is an important tool of choice for modeling large-amplitude collective motion in atomic nuclei. The computational complexity of the GCM increases rapidly with the number of collective coordinates. It imposes a strong restriction on the applicability of the method. In this work, we propose a subspace-reduction algorithm that employs optimal statistical ML models as surrogates for exact quantum-number projection calculations for norm and Hamiltonian kernels. The model space of the original GCM is reduced to a subspace relevant for nuclear low energy spectra and the NME of ground state to ground state 0νββ decay based on the orthogonality condition (OC) and the energy-transition-orthogonality procedure (ENTROP), respectively. For simplicity, the polynomial ridge regression (RR) algorithm is used to learn the norm and Hamiltonian kernels of axially deformed configurations. The efficiency and accuracy of this algorithm are illustrated for 76 Ge and 76 Se by comparing results obtained using the optimal RR models to direct GCM calculations. The low-lying energy spectra of 76 Ge and 76 Se, as well as the 0νββ-decay NME between their ground states, are computed. Furthermore, the results show that the performance of the GCM+OC/ENTROP+RR is more robust than that of the GCM+RR alone, and the former can reproduce the results of the original GCM calculation accurately with a significantly reduced computational cost.

59 ≤ A ≤ 89↗
DOE OSTI2025

Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. In conclusion, we compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

kernel methods↗
DOE OSTI2018

Analytic in a Sector Resolving Families of Operators for Degenerate Evolution Fractional Equations

We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An extended polynomial chaos expansion for PDF characterization and variation with aleatory and epistemic uncertainties

This paper presents an extended polynomial chaos formalism for epistemic uncertainties and a new framework for evaluating sensitivities and variations of output probability density functions (PDF) to uncertainty in probabilistic models of input variables. An ”extended” polynomial chaos expansion (PCE) approach is developed that accounts for both aleatory and epistemic uncertainties, modeled as random variables, thus allowing a unified treatment of both types of uncertainty. We explore in particular epistemic uncertainty associated with the choice of prior probabilistic models for input parameters. A PCE-based Kernel Density (KDE) construction provides a composite map from the PCE coefficients and germ to the PDF of quantities of interest (QoI). Here, the sensitivities of these PDF with respect to the input parameters are then evaluated. Input parameters of the probabilistic models are considered. By sampling over the epistemic random variable, a family of PDFs is generated and the failure probability is itself estimated as a random variable with its own PCE. Integrating epistemic uncertainties within the PCE framework results in a computationally efficient paradigm for propagation and sensitivity evaluation. Two typical illustrative examples are used to demonstrate the proposed approach.

Aleatory uncertainty↗
DOE OSTI2018

Quantizations of the classical time of arrival and their dynamics

The classical time of arrival in the interacting case is quantized by way of quantizing its expansion about the free time of arrival. The quantization is formulated in coordinate representation which represents ordering rules in terms of two variable polynomial functions. This leads to representations of the quantized time of arrival operators as integral operators whose kernels are determined by the chosen ordering rule. The formulation lends itself to generalization which allows construction of time of arrival operators that cannot be obtained by direct quantization using particular ordering rules. Weyl, symmetric and Born–Jordan quantizations are specifically studied. The dynamics of the eigenfunctions of the different time of arrival operators are investigated. The eigenfunctions exhibit unitary arrival at the intended arrival point at their respective eigenvalues.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Exponential concentration in quantum kernel methods

Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model’s parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Compadre Toolkit

The Compadre Toolkit solves a minimization problem, that once solved allows a user to reconstruct a function from sample data collected from a cloud of data sites. The solution to this minimization can also be used to assemble a linear system that can be solved numerically using some other software. The minimization problem the toolkit solves is to find the optimal finite polynomial in a user specified finite dimensional space with regards to minimizing an objective function which measures a linear sampling functional applied to the polynomial, compared to the same sampling functional applied to the source function, weighted by a positive kernel that has compact support.

SciDAC↗