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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 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Photoionization of Atomic Systems Using the Random-Phase Approximation Including Relativistic Interactions

Approximation methods are unavoidable in solving a many-electron problem. One of the most successful approximations is the random-phase approximation (RPA). Miron Amusia showed that it can be used successfully to describe atomic photoionization processes of many-electron atomic systems. In this article, the historical reasons behind the term “random-phase approximation” are revisited. A brief introduction to the relativistic RPA (RRPA) developed by Walter Johnson and colleagues is provided and some of its illustrative applications are presented.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2024

Solving a class of infinite-dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

Here, we examine a method for solving an infinite-dimensional tensor eigenvalue problem Hx = λx, where the infinite-dimensional symmetric matrix H exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e -Ht is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step to ensure accurate and rapid convergence of the power method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Approximate CFTs and random tensor models

Abstract A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS 3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Physics↗
DOE OSTI2025

Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work

Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Johnson, Charles A.↗
DOE OSTI2022

Sensitivity of the electronic and magnetic structures of cuprate superconductors to density functional approximations

Abstract We discuss the crystal, electronic, and magnetic structures of La 2− x Sr x CuO 4 (LSCO) for x = 0.0 and x = 0.25 employing 13 density functional approximations, representing the local, semi-local, and hybrid exchange-correlation approximations within the Perdew–Schmidt hierarchy. The meta-generalized gradient approximation (meta-GGA) class of functionals is found to perform well in capturing the key properties of LSCO, a prototypical high-temperature cuprate superconductor. In contrast, the localspin-density approximation, GGA, and the hybrid density functional fail to capture the metal-insulator transition under doping.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids—A case study in diamond

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for ab initio electronic structure calculations, in practice, the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros). This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced. This approach is illustrated in the case of the carbon diamond using Slater–Jastrow trial wavefunctions including up to one million Slater determinants. Fixed-node DMC energies obtained with such large expansions are much improved, and the fixed-node error is found to decrease monotonically and smoothly as a function of the number of determinants in the trial wavefunction, a property opening the way to a better control of this error. The cohesive energy extrapolated to the thermodynamic limit is in close agreement with the estimated experimental value. Interestingly, this is also the case at the single-determinant level, thus, indicating a very good error cancellation in carbon diamond between the bulk and atomic total fixed-node energies when using single-determinant nodes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Enabling attractive-repulsive potentials in binary-collision-approximation monte-carlo codes for ion-surface interactions

Abstract Binary Collision Approximation (BCA) codes for ion-material interactions, such as SRIM, Tridyn, F-TRIDYN, and SDtrimSP, have historically been limited to screened Coulomb potentials even at low energies due to the difficulty in numerically solving the Distance of Closest Approach (DOCA) problem for attractive-repulsive potentials. Techniques such as direct n-body simulation or modifications to Newton’s method are either prohibitively costly or not guaranteed to work for all potentials. Advanced rootfinding techniques, such as companion matrix solvers, offer a solution. For many attractive-repulsive potentials, however, a companion matrix cannot be used directly, because there is no way to put the associated functions into a monomial basis form. A complementary technique is proxy rootfinding—by finding the best-fit polynomial approximant of a function, the zeros of the approximant can be guaranteed to be close to the zeros of the function. Using the Chebyshev basis and grid offers additional guarantees with regards to the quality of the approximation, the speed of convergence, and the avoidance of Runge’s phenomenon. By finding Chebyshev interpolants and using the Chebyshev-Frobenius companion matrix, the zeros of any real function on a bounded domain can be found. Here we show that using an Adaptive Chebyshev Proxy Rootfinder with Automatic Subdivision (ACPRAS) with appropriate scaling functions, numerical issues presented by attractive-repulsive potentials, including those of scale, can be handled. Using these techniques, we show that it is possible to include any physically reasonable interatomic potential in a BCA code, and to guarantee correctness of the resulting scattering angle calculations.

Materials Science↗
DOE OSTI2021

Solving the Bethe-Salpeter equation on a subspace: Approximations and consequences for low-dimensional materials

It is well known that the ambient environment can dramatically renormalize the quasiparticle gap and exciton binding energies in low-dimensional materials, but the effect of the environment on the energy splitting of the spin-singlet and spin-triplet exciton states is less understood. A prominent effect is the renormalization of the exciton binding energy and optical strength (and hence the optical spectrum) through additional screening of the direct Coulomb term describing the attractive electron-hole interaction in the kernel of the Bethe-Salpeter equation. The repulsive exchange interaction responsible for the singlet-triplet splitting, on the other hand, is unscreened within formal many-body perturbation theory. However, Benedict argued that in practical calculations restricted to a subspace of the full Hilbert space, the exchange interaction should be appropriately screened by states outside of the subspace, the so-called S approximation [L. X. Benedict, Phys. Rev. B 66, 193105 (2002)PRBMDO0163-182910.1103/PhysRevB.66.193105]. Here, we systematically explore the accuracy of the S approximation for different confined systems, including a molecule and heterostructures of semiconducting and metallic layered materials. We show that the S approximation is actually exact in the limit of small exciton binding energies (i.e., small direct term) and can be used to significantly accelerate convergence of the exciton energies with respect to the number of empty states, provided that a particular effective screening consistent with the conventional Tamm-Dancoff approximation is employed. We further find that the singlet-triplet splitting in the energy of the excitons is largely unaffected by the external dielectric environment for most quasi-two-dimensional materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Approximate two-body generating Hamiltonian for the particle-hole Pfaffian wave function

We present two two-body Hamiltonians that approximate the exact particle-hole Pfaffian wave function with their ground states for all the system sizes where this wave function has been numerically constructed to date. The approximate wave functions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimization procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a noninteracting model. They belong to a finite region in the variational space of Hamiltonians where each point approximately generates the particle-hole Pfaffian. Here we diagonalize the identified Hamiltonians for up to 20 electrons and find that for them the particle-hole Pfaffian shift appears energetically more favorable. The possibility to interpret the data in terms of composite fermions is discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2021

State densities of heavy nuclei in the static-path plus random-phase approximation

We report that nuclear state densities are important inputs to statistical models of compound-nucleus reactions. State densities are often calculated with self-consistent mean-field approximations that do not include important correlations and must be augmented with empirical collective enhancement factors. Here, we benchmark the static-path plus random-phase approximation (SPA + RPA) to the state density in a chain of samarium isotopes 148–155 Sm against exact results (up to statistical errors) obtained with the shell-model Monte Carlo (SMMC) method. The SPA + RPA method incorporates all static fluctuations beyond the mean field together with small-amplitude quantal fluctuations around each static fluctuation. Using a pairing plus quadrupole interaction, we show that the SPA + RPA state densities agree well with the exact SMMC densities for both the even- and odd-mass isotopes. For the even-mass isotopes, we also compare our results with mean-field state densities calculated with the finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation. We find that the SPA + RPA repairs the deficiencies of the mean-field approximation associated with broken rotational symmetry in deformed nuclei and with the violation of particle-number conservation in the pairing condensate. In particular, in deformed nuclei the SPA + RPA reproduces the rotational enhancement of the state density relative to the mean-field state density.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Low-depth Clifford circuits approximately solve MaxCut

We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, ADAPT-Clifford, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an N -vertex graph by building a depth O ( N ) Clifford circuit. The algorithm has runtime complexity O ( N 2 ) and O ( N 3 ) for sparse and dense graphs, respectively, and space complexity O ( N 2 ) , with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to ∼ 94 % of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW. Published by the American Physical Society 2024

Muñoz-Arias, Manuel H. (ORCID:000000025711029X)↗
DOE OSTI2020

NISQ+: Boosting quantum computing power by approximating quantum error correction

Quantum computers are growing in size, and design decisions are being made now that attempt to squeeze more computation out of these machines. In this spirit, we design a method to boost the computational power of near-term quantum computers by adapting protocols used in quantum error correction to implement "Approximate Quantum Error Correction (AQEC)." By approximating fully-fledged error correction mechanisms, we can increase the compute volume (qubits × gates, or "Simple Quantum Volume (SQV)") of near-term machines. The crux of our design is a fast hardware decoder that can approximately decode detected error syndromes rapidly. Specifically, we demonstrate a proof-of-concept that approximate error decoding can be accomplished online in near-term quantum systems by designing and implementing a novel algorithm in Single-Flux Quantum (SFQ) superconducting logic technology. This avoids a critical decoding backlog, hidden in all offline decoding schemes, that leads to idle time exponential in the number of T gates in a program. Our design utilizes one SFQ processing module per physical qubit. Employing state-of-the-art SFQ synthesis tools, we show that the circuit area, power, and latency are within the constraints of contemporary quantum system designs. Under pure dephasing error models, the proposed accelerator and AQEC solution is able to expand SQV by factors between 3,402 and 11,163 on expected near-term machines. The decoder achieves a 5% accuracy-threshold and pseudo-thresholds of ~ 5%,4.75%,4.5%, and 3.5% physical error-rates for code distances 3,5,7, and 9. Decoding solutions are achieved in a maximum of ~20 nanoseconds on the largest code distances studied. By avoiding the exponential idle time in offline decoders, we achieve a 10x reduction in required code distances to achieve the same logical performance as alternative designs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

An under-approximation of entropy for elemental multiconfigurational ground state electronic structures

A combinatorial approach has been applied to the allowable permutations of quantum electronic configurations under the constraints of Hund's rule for established ground state configurations toward an under-approximation of electronic structure entropy. Combined with a previously reported over-approximation, the approximations are used in conjunction in an attempt to bracket the upper and lower entropy limits for multiconfigurational ground state electronic structure entropy and compared to known standard molar entropies for the elements. This formality has been used for the application of a classical statistical mechanics methodology to be applied to the discrete sets of quantum mechanical states of Pu in order to calculate orbital occupancies in Pu's multiconfigurational ground state. Without consideration of the relative energies of various possible electronic configurations contributing to the multiconfigurational ground state, the calculations are performed under a general energy degeneracy assumption weighted to the number of permutations for specific configurations. The number of configurations assumed to significantly contribute is gradually constrained in order to approach a low-order approximation of orbital occupancies in Pu that are then compared to experimental and other calculated results from the literature.

Beaux, II, Miles F. (ORCID:000000032192626X)↗
DOE OSTI2022

Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Hybrid quantum-classical algorithms for approximate graph coloring

We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX- k -CUT, the problem of finding an approximate k -vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level p : the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level- 1 QAOA and level- 1 RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level- 1 QAOA and RQAOA with up to 300 qutrits applied to ensembles of randomly generated 3 -colorable constant-degree graphs. We find that level- 1 RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1972

Constrained Chebyshev approximations to some elementary functions suitable for evaluation with floating point arithmetic

Approximations which can be evaluated with precision using floating-point arithmetic are presented. The particular set of approximations thus far developed are for the function TAN and the functions of USASI FORTRAN excepting SQRT and EXPONENTIATION. These approximations are, furthermore, specialized to particular forms which are especially suited to a computer with a small memory, in that all of the approximations can share one general purpose subroutine for the evaluation of a polynomial in the square of the working argument.

Manos, P.↗
NASA NTRS1972

Approximation method for the kinetic Boltzmann equation

The further development of a method for approximating the Boltzmann equation is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann equation to a series of approximating equations.

Shakhov, Y. M.↗
NASA NTRS1973

Approximations in the minimum time-to-climb problem

The minimum time-to-climb problem is formulated as a third order system and three approximate solutions based on reduced order systems are presented. The first of these is the often used energy state, the second is the less frequently used two state and the third is a slightly altered form of the second, herein called the modified two state. These three approximations are discussed and compared both qualitatively and, by using a numerical example, quantitatively. The numerical example is also solved by the steepest descent method to provide a basis for comparison. It is concluded that the modified two state approximation is significantly better than the other two. This approximation is used to assess the sensitivity of climb performance to various vehicle parameters and it is found that, as expected, thrust and weight influence the time-to-climb most strongly.

Ardema, M. D.↗