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DOE OSTI2024

Downfolding from ab initio to interacting model Hamiltonians: comprehensive analysis and benchmarking of the DFT+cRPA approach

Abstract Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.

Chemistry↗

NASA NTRS1991

Total recall in distributive associative memories

Iterative error correction of asymptotically large associative memories is equivalent to a one-step learning rule. This rule is the inverse of the activation function of the memory. Spectral representations of nonlinear activation functions are used to obtain the inverse in closed form for Sparse Distributed Memory, Selected-Coordinate Design, and Radial Basis Functions.

Danforth, Douglas G.↗

NASA NTRS2000

Robust Inversion and Data Compression in Control Allocation

We present an off-line computational method for control allocation design. The control allocation function delta = F(z)tau = delta (sub 0) (z) mapping commanded body-frame torques to actuator commands is implicitly specified by trim condition delta (sub 0) (z) and by a robust pseudo-inverse problem double vertical line I - G(z) F(z) double vertical line less than epsilon (z) where G(z) is a system Jacobian evaluated at operating point z, z circumflex is an estimate of z, and epsilon (z) less than 1 is a specified error tolerance. The allocation function F(z) = sigma (sub i) psi (z) F (sub i) is computed using a heuristic technique for selecting wavelet basis functions psi and a constrained least-squares criterion for selecting the allocation matrices F (sub i). The method is applied to entry trajectory control allocation for a reusable launch vehicle (X-33).

Hodel, A. Scottedward↗

DOE OSTI2024

Solution of the Schrödinger equation for quasi-one-dimensional materials using helical waves

We formulate and implement a spectral method for solving the Schrödinger equation, as it applies to quasi-one-dimensional materials and structures. This allows for computation of the electronic structure of important technological materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons, chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and systematic manner. Our work is motivated by the observation that one of the most successful methods for carrying out electronic structure calculations of bulk/crystalline systems — the plane-wave method — is a spectral method based on eigenfunction expansion. Our scheme avoids computationally onerous approximations involving periodic supercells often employed in conventional plane-wave calculations of quasi-one-dimensional materials, and also overcomes several limitations of other discretization strategies, e.g., those based on finite differences and atomic orbitals. The basis functions in our method — called helical waves (or twisted waves) — are eigenfunctions of the Laplacian with symmetry adapted boundary conditions, and are expressible in terms of plane waves and Bessel functions in helical coordinates. We describe the setup of fast transforms to carry out discretization of the governing equations using our basis set, and the use of matrix-free iterative diagonalization to obtain the electronic eigenstates. Miscellaneous computational details, including the choice of eigensolvers, use of a preconditioning scheme, evaluation of oscillatory radial integrals and the imposition of a kinetic energy cutoff are discussed. We have implemented these strategies into a computational package called HelicES (Helical Electronic Structure). We demonstrate the utility of our method in carrying out systematic electronic structure calculations of various quasi-one-dimensional materials through numerous examples involving nanotubes, nanoribbons and nanowires. We also explore the convergence properties of our method, and assess its accuracy and computational efficiency by comparison against reference finite difference, transfer matrix method and plane-wave results. We anticipate that our method will find applications in computational nanomechanics and multiscale modeling, for carrying out transport calculations of interest to the field of semiconductor devices, and for the discovery of novel chiral phases of matter that are of relevance to the burgeoning quantum hardware industry.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1984

An urnful of blinding functions

There is a fundamental connection between the mathematical theory of discrete probability distributions and the parametric curves and surfaces of computer-aided geometric design. It is no accident that the blending functions of Bezier curves and surfaces have an obvious probabilistic interpretation, nor is it a coincidence that the normalized uniform B-spline basis functions also model a simple stochastic process. The link between probability and geometry, and how to exploit simple probabilistic arguments to derive many of the classical geometric properties of the parametric curves and surfaces currently in vogue in computer-aided geometric design is discussed. This probabilistic approach is also used to introduce many new types of curves and surfaces, and it is demonstrated how probability theory can be used to simplify, unify, and generalize many well-known results.

Goldman, R. N.↗

DOE OSTI2025

Deep Learning without Global Optimization by Random Fourier Neural Networks

Here we introduce a new training algorithm for deep neural networks that utilize random complex exponential activation functions. Our approach employs a Markov chain Monte Carlo sampling procedure to iteratively train network layers, avoiding global and gradient-based optimization while maintaining error control. It consistently attains the theoretical approximation rate for residual networks with complex exponential activation functions, determined by network complexity. Additionally, it enables efficient learning of multiscale and high-frequency features, producing interpretable parameter distributions. Despite using sinusoidal basis functions, we do not observe Gibbs phenomena in approximating discontinuous target functions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Resolving local structural motifs across the phase evolution of zinc titanates with computational x-ray absorption spectroscopy

Resolving the local structure motifs that characterize phase evolution as a function of composition is a key challenge in structure characterization of complex materials. Here, in this study, we combine first-principles simulations and x-ray absorption near-edge structures (XANES) analysis to gain insights into the structure evolution revealed by measurements across a combinatorial zinc titanate thin film, which was grown with smoothly varying composition over a wide range of the Ti:Zn ratio. Specifically, we propose a cluster blind-signal-separation (cBSS) method for XANES spectral analysis based on a library of the structures and spectra of representative local motifs. In addition to motifs from zinc titanate crystals, two types of Ti-defect models constructed in this study are key to the understanding of the structure characteristics in the Zn-rich region. The cBSS method makes use of both spectral clustering of the simulated site-XANES spectra library and the BSS procedure to construct high-fidelity spectral basis functions from an experimental spectral sequence. The method provides a rigorous measure of the spectral sensitivity and basis completeness. The results of the XANES analysis are corroborated with other experimental modalities, including x-ray diffraction and spectroscopic ellipsometry, to validate the cBSS method. The calculated motif weights resulting from fitting the XANES spectra with the cBSS basis probe the atomic structure characteristics of both crystalline and amorphous phases as a function of the Ti/Zn composition. The insights of the local structure motif evolution are pivotal to the understanding of the nonmonotonic trend in the optical gap, which may lead to potential applications through tuning the optical properties of zinc titanate. The workflow of the XANES spectral analysis developed in this work can be generalized to construct the structure-property relationship in a broad material space.

36 MATERIALS SCIENCE↗

NASA NTRS1995

Expendable launch vehicle studies

Analytical support studies of expendable launch vehicles concentrate on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed and linearized. The information from LeRC about the grids, masses, and modes is incorporated into the model. The eigenvalues of the plant are analyzed for several modeling factors: utilizing diagonal mass matrix, uniform beam assumption, inclusion of aerodynamics, and the interaction between the aerodynamics and the flexible bending motion. Preliminary PID, LQR, and LQG control designs with sensor and actuator dynamics for this system and simulations are also conducted. The initial analysis for comparison of PD (proportional-derivative) and full state feedback LQR Linear quadratic regulator) shows that the split weighted LQR controller has better performance than that of the PD. In order to meet both the performance and robustness requirements, the H(sub infinity) robust controller for the expendable launch vehicle is developed. The simulation indicates that both the performance and robustness of the H(sub infinity) controller are better than that for the PID and LQG controllers. The modelling and analysis support studies team has continued development of methodology, using eigensensitivity analysis, to solve three classes of discrete eigenvalue equations. In the first class, the matrix elements are non-linear functions of the eigenvector. All non-linear periodic motion can be cast in this form. Here the eigenvector is comprised of the coefficients of complete basis functions spanning the response space and the eigenvalue is the frequency. The second class of eigenvalue problems studied is the quadratic eigenvalue problem. Solutions for linear viscously damped structures or viscoelastic structures can be reduced to this form. Particular attention is paid to Maxwell and Kelvin models. The third class of problems consists of linear eigenvalue problems in which the elements of the mass and stiffness matrices are stochastic. dynamic structural response for which the parameters are given by probabilistic distribution functions, rather than deterministic values, can be cast in this form. Solutions for several problems in each class will be presented.

Bainum, Peter M.↗

NASA NTRS1994

Some Aspects of the Implementation of Double Group Symmetry and Electron Correlation in Molecular 4-Component Calculations

The efficient implementation of method for electron correlation in molecular 4-component calculations demands that symmetry be exploited where possible. Algorithms for the construction of matrices and the transformation of integrals over symmetry-adapted basis functions, where the point group is restricted to D(sub 2h) and subgroups, will be presented. The merits of keeping the primitive integrals in the scalar basis will be compared with those of transforming them to the 2-spinor basis.

Dyall, Kenneth G.↗

DOE OSTI2024

Superluminal matter waves

The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1992

Interferometric reconstruction of three-dimensional high-speed aerodynamic flows

Holographic interferometry can be a very useful diagnostic tool in high-speed aerodynamic testing. During this summer research period, various possible approaches for accurately reconstructing three-dimensional flows from limited data were examined. The approach based on the combination of the following three techniques appears to be promising: (1) Continuous Local Basis Function Method - this computational tomographic method has a power to accurately reconstruct continuous regions and is appropriate from well-conditioned to moderately limited data; (2) Variable Basis Method - this computational tomographic method provides accuracy near discontinuities, i.e., shock regions, and is appropriate from moderately-limited to severely-limited data; and (3) Complementary Field Method - this is a general iterative reconstructor that can be coupled with any computational tomographic techniques. Mathematically, it can be shown that this method can provide better accuracy than the direct reconstruction as in a conventional approach. Our numerical simulation of experiments demonstrated improved reconstruction results even when these techniques were individually tested.

Cha, Soyoung Stephen↗

NASA NTRS2004

Planetary Transmission Diagnostics

This report presents a methodology for detecting and diagnosing gear faults in the planetary stage of a helicopter transmission. This diagnostic technique is based on the constrained adaptive lifting algorithm. The lifting scheme, developed by Wim Sweldens of Bell Labs, is a time domain, prediction-error realization of the wavelet transform that allows for greater flexibility in the construction of wavelet bases. Classic lifting analyzes a given signal using wavelets derived from a single fundamental basis function. A number of researchers have proposed techniques for adding adaptivity to the lifting scheme, allowing the transform to choose from a set of fundamental bases the basis that best fits the signal. This characteristic is desirable for gear diagnostics as it allows the technique to tailor itself to a specific transmission by selecting a set of wavelets that best represent vibration signals obtained while the gearbox is operating under healthy-state conditions. However, constraints on certain basis characteristics are necessary to enhance the detection of local wave-form changes caused by certain types of gear damage. The proposed methodology analyzes individual tooth-mesh waveforms from a healthy-state gearbox vibration signal that was generated using the vibration separation (synchronous signal-averaging) algorithm. Each waveform is separated into analysis domains using zeros of its slope and curvature. The bases selected in each analysis domain are chosen to minimize the prediction error, and constrained to have the same-sign local slope and curvature as the original signal. The resulting set of bases is used to analyze future-state vibration signals and the lifting prediction error is inspected. The constraints allow the transform to effectively adapt to global amplitude changes, yielding small prediction errors. However, local wave-form changes associated with certain types of gear damage are poorly adapted, causing a significant change in the prediction error. The constrained adaptive lifting diagnostic algorithm is validated using data collected from the University of Maryland Transmission Test Rig and the results are discussed.

Lewicki, David G.↗

NASA NTRS2007

Dynamic Snap-Through of Thermally Buckled Structures by a Reduced Order Method

The goal of this investigation is to further develop nonlinear modal numerical simulation methods for application to geometrically nonlinear response of structures exposed to combined high intensity random pressure fluctuations and thermal loadings. The study is conducted on a flat aluminum beam, which permits a comparison of results obtained by a reduced-order analysis with those obtained from a numerically intensive simulation in physical degrees-of-freedom. A uniformly distributed thermal loading is first applied to investigate the dynamic instability associated with thermal buckling. A uniformly distributed random loading is added to investigate the combined thermal-acoustic response. In the latter case, three types of response characteristics are considered, namely: (i) small amplitude vibration around one of the two stable buckling equilibrium positions, (ii) intermittent snap-through response between the two equilibrium positions, and (iii) persistent snap-through response between the two equilibrium positions. For the reduced-order analysis, four categories of modal basis functions are identified including those having symmetric transverse, anti-symmetric transverse, symmetric in-plane, and anti-symmetric in-plane displacements. The effect of basis selection on the quality of results is investigated for the dynamic thermal buckling and combined thermal-acoustic response. It is found that despite symmetric geometry, loading, and boundary conditions, the anti-symmetric transverse and symmetric in-plane modes must be included in the basis as they participate in the snap-through behavior.

Przekop, Adam↗

NASA NTRS1992

Constraints on the age and evolution of the Galaxy from the white dwarf luminosity function

The white dwarf disk luminosity function is explored using observational results of Liebert et al. (1988, 1989) as a template for comparison, and the cooling curves of Wood (1990, 1991) as the input basis functions for the integration. The star formation rate over the history of the Galaxy is found to be constant to within an order of magnitude, and the disk age lies in the range 6-13.5 Gyr, where roughly 40 percent of the uncertainty is due to the observational uncertainties. Using the best current estimates as inputs to the integration, the disk ages range from 7.5 to 11 Gyr, i.e., they are substantially younger than most estimates for the halo globular clusters but in reasonable agreement with those for the disk globular clusters and open clusters. The ages of these differing populations, taken together, are consistent with the pressure-supported collapse models of early spiral Galactic evolution.

Wood, M. A.↗

NASA NTRS1998

The Wavelet Element Method: Realization and Additional Features in 2D and 3D - Part 2

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D and 3D cases. We address additional features such as symmetry, vanishing moments and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.

Canuto, Claudio↗

NASA NTRS2006

Dynamic Snap-Through of Thin-Walled Structures by a Reduced Order Method

The goal of this investigation is to further develop nonlinear modal numerical simulation methods for application to geometrically nonlinear response of structures exposed to combined high intensity random pressure fluctuations and thermal loadings. The study is conducted on a flat aluminum beam, which permits a comparison of results obtained by a reduced-order analysis with those obtained from a numerically intensive simulation in physical degrees-of-freedom. A uniformly distributed thermal loading is first applied to investigate the dynamic instability associated with thermal buckling. A uniformly distributed random loading is added to investigate the combined thermal-acoustic response. In the latter case, three types of response characteristics are considered, namely: (i) small amplitude vibration around one of the two stable buckling equilibrium positions, (ii) intermittent snap-through response between the two equilibrium positions, and (iii) persistent snap-through response between the two equilibrium positions. For the reduced order analysis, four categories of modal basis functions are identified including those having symmetric transverse (ST), anti-symmetric transverse (AT), symmetric in-plane (SI), and anti-symmetric in-plane (AI) displacements. The effect of basis selection on the quality of results is investigated for the dynamic thermal buckling and combined thermal-acoustic response. It is found that despite symmetric geometry, loading, and boundary conditions, the AT and SI modes must be included in the basis as they participate in the snap-through behavior.

Przekop, Adam↗

DOE OSTI2025

The latent variable proximal point algorithm for variational problems with inequality constraints

The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge–Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. Here, in this paper, we describe the LVPP algorithm in a general form and apply it to ten problems from across mathematics.

Inequality constraints↗

NASA NTRS1983

Estimation of a remote sensing system point-spread function from measured imagery

A general approach to identifying the point spread function (PSF) of a remotely sensed scene is demonstrated in terms of a step function for an abrupt change in the gray level along the row or column of the image data. The estimate of the PSF is made in terms of a finite sum of basis functions, employing a sequence of rectangular pulses covering the spatial extent of the PSF. The approximation, if narrow impulses are employed, provides accurate fidelity to the PSF. The method becomes practical when the geometrical structure of the scene elements producing the measured response is known. The field boundary is obtained through consideration of the differing intensities on each side of the boundary, which is a step discontinuity. The mathematical procedure for the technique is provided, together with a sample problem from Landsat-4 Thematic Mapper data. Atmospheric blurring and electronic effects on the overall PSF and the cubic convolution resampling effects are noted.

Mcgillem, C. D.↗