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NASA NTRS1989

Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings

A novel design approach is presented for nominally nonpolarizing beamsplitters with very low polarization over an extended wavelength range, with low angular sensitivity. A high degree of intensity matching for s and p polarization in a given wavelength interval, from 500 to 600 nm, has been obtained for multilayer stacks derived from an initial graded-index film design by matching of the characteristic matrices. This graded-index film design maintains a very low polarization ratio for almost the complete visible region, covering a wavelength range about twice that covered by the derived homogeneous multilayer stack.

Zukic, Muamer↗

NASA NTRS1972

On the equinoctial orbit elements.

This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.

Broucke, R. A.↗

DOE OSTI2023

An accurate and efficient fragmentation approach via the generalized many-body expansion for density matrices

With relevant chemical space growing larger and larger by the day, the ability to extend computational tractability over that larger space is of paramount importance in virtually all fields of science. The solution we aim to provide here for this issue is in the form of the generalized many-body expansion for building density matrices (GMBE-DM) based on the set-theoretical derivation with overlapping fragments, through which the energy can be obtained by a single Fock build. In combination with the purification scheme and the truncation at the one-body level, the DM-based GMBE(1)-DM-P approach shows both highly accurate absolute and relative energies for medium-to-large size water clusters with about an order of magnitude better than the corresponding energy-based GMBE(1) scheme. Simultaneously, GMBE(1)-DM-P is about an order of magnitude faster than the previously proposed MBE-DM scheme [F. Ballesteros and K. U. Lao, J. Chem. Theory Comput. 18, 179 (2022)] and is even faster than a supersystem calculation without significant parallelization to rescue the fragmentation method. For even more challenging systems including ion–water and ion–pair clusters, GMBE(1)-DM-P also performs about 3 and 30 times better than the energy-based GMBE(1) approach, respectively. In addition, this work provides the first overlapping fragmentation algorithm with a robust and effective binning scheme implemented internally in a popular quantum chemistry software package. Thus, GMBE(1)-DM-P opens a new door to accurately and efficiently describe noncovalent clusters using quantum mechanics.

Chemistry↗

DOE OSTI2024

Training quantum neural networks using the quantum information bottleneck method

Abstract We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error ε that requires O ( log 2 ⁡ ( 1 / ϵ ) + 1 / δ 2 ) queries to a purification of the input density operator if its spectrum is supported on { 0 } ⋃ [ δ , 1 − δ ] for δ > 0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.

Çatlı, Ahmet Burak (ORCID:0000000152294141)↗

DOE OSTI2023

Integrability Breaking from Backscattering

Herein we analyze the onset of diffusive hydrodynamics in the one-dimensional hard-rod gas subject to stochastic backscattering. While this perturbation breaks integrability and leads to a crossover from ballistic to diffusive transport, it preserves infinitely many conserved quantities corresponding to even moments of the velocity distribution of the gas. In the limit of small noise, we derive the exact expressions for the diffusion and structure factor matrices, and show that they generically have off diagonal components. We find that the particle density structure factor is non-Gaussian and singular near the origin, with a return probability showing logarithmic deviations from diffusion.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1984

Development of a solid hexahedron finite dynamic element

This paper presents the pertinent details of a newly developed solid rectangular hexahedron finite dynamic element, involving the derivation of the higher order stiffness and inertia dynamic correction matrices. Numerical results of a test case are also presented which indicate that adoption of the dynamic elements significantly improves the solution convergence, when compared with the related performance of the corresponding finite elements.

Gupta, K. K.↗

NASA NTRS1988

On the eigenvalue and eigenvector derivatives of a non-defective matrix

A novel approach is introduced to address the problem of existence of differentiable eigenvectors for a nondefective matrix which may have repeated eigenvalues. The existence of eigenvector derivatives for a unique set of continuous eigenvectors corresponding to a repeated eigenvalue is rigorously established for nondefective and analytic matrices. A numerically implementable method is then developed to compute the differentiable eigenvectors associated with repeated eigenvalues. The solutions of eigenvalue and eigenvector derivatives for repeated eigenvalues are then derived. An example is given to illustrate the validity of formulations developed in this paper.

Juang, Jer-Nan↗

DOE OSTI2022

Algebraic Bethe Circuits

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-$\frac{1}{2}$ XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Determining the Ensemble N -Representability of Reduced Density Matrices

The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy that embeds an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both simple models as well as molecular systems at finite temperature, demonstrating its robustness.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1992

Interval prediction in structural dynamic analysis

Methods for assessing the predictive accuracy of structural dynamic models are examined with attention given to the effects of modal mass, stiffness, and damping uncertainties. The methods are based on a nondeterministic analysis called 'interval prediction' in which interval variables are used to describe parameters and responses that are unknown. Statistical databases for generic modeling uncertainties are derived from experimental data and incorporated analytically to evaluate responses. Covariance matrices of modal mass, stiffness, and damping parameters are propagated numerically in models of large space structures by means of three methods. The test data tend to fall within the predicted intervals of uncertainty determined by the statistical databases. The present findings demonstrate the suitability of using data from previously analyzed and tested space structures for assessing the predictive accuracy of an analytical model.

Hasselman, Timothy K.↗

NASA NTRS1992

Probabilistic evaluation of on-line checks in fault-tolerant multiprocessor systems

The analysis of fault-tolerant multiprocessor systems that use concurrent error detection (CED) schemes is much more difficult than the analysis of conventional fault-tolerant architectures. Various analytical techniques have been proposed to evaluate CED schemes deterministically. However, these approaches are based on worst-case assumptions related to the failure of system components. Often, the evaluation results do not reflect the actual fault tolerance capabilities of the system. A probabilistic approach to evaluate the fault detecting and locating capabilities of on-line checks in a system is developed. The various probabilities associated with the checking schemes are identified and used in the framework of the matrix-based model. Based on these probabilistic matrices, estimates for the fault tolerance capabilities of various systems are derived analytically.

Nair, V. S. S.↗

NASA NTRS1992

Adaptation of a matrix parameterization of the middle atmospheric radiative cooling for an arbitrary vertical coordinate grid

Matrices corresponding to a revised vertical grid are shown to be derived from the original grid by integrating the radiative cooling rate over model layers. The method is applied to temperature profiles that represent average or extreme atmospheric conditions and a range of vertical resolutions. The example makes use of the 15-micron CO2 band parameterization proposed by Akmaev and Shved (1982) for the LTE region of the middle atmosphere. Cooling rate errors provided by the present method are not more than 0.11 K/day for a vertical grid spacing of less than 5 km. The matrix transformation method is shown to be valuable for adapting effective matrix parameterizations of middle atmosphere cooling to atmospheric models with arbitrary vertical resolutions.

Akmaev, R. A.↗

NASA NTRS2015

Theoretical Studies of Spectroscopic Line Mixing in Remote Sensing Applications

The phenomenon of collisional transfer of intensity due to line mixing has an increasing importance for atmospheric monitoring. From a theoretical point of view, all relevant information about the collisional processes is contained in the relaxation matrix where the diagonal elements give half-widths and shifts, and the off-diagonal elements correspond to line interferences. For simple systems such as those consisting of diatom-atom or diatom-diatom, accurate fully quantum calculations based on interaction potentials are feasible. However, fully quantum calculations become unrealistic for more complex systems. On the other hand, the semi-classical Robert-Bonamy (RB) formalism, which has been widely used to calculate half-widths and shifts for decades, fails in calculating the off-diagonal matrix elements. As a result, in order to simulate atmospheric spectra where the effects from line mixing are important, semi-empirical fitting or scaling laws such as the ECS (Energy-Corrected Sudden) and IOS (Infinite-Order Sudden) models are commonly used. Recently, while scrutinizing the development of the RB formalism, we have found that these authors applied the isolated line approximation in their evaluating matrix elements of the Liouville scattering operator given in exponential form. Since the criterion of this assumption is so stringent, it is not valid for many systems of interest in atmospheric applications. Furthermore, it is this assumption that blocks the possibility to calculate the whole relaxation matrix at all. By eliminating this unjustified application, and accurately evaluating matrix elements of the exponential operators, we have developed a more capable formalism. With this new formalism, we are now able not only to reduce uncertainties for calculated half-widths and shifts, but also to remove a once insurmountable obstacle to calculate the whole relaxation matrix. This implies that we can address the line mixing with the semi-classical theory based on interaction potentials between molecular absorber and molecular perturber. We have applied this formalism to address the line mixing for Raman and infrared spectra of molecules such as N2, C2H2, CO2, NH3, and H2O. By carrying out rigorous calculations, our calculated relaxation matrices are in good agreement with both experimental data and results derived from the ECS model.

Formalism↗

DOE OSTI2020

Pore-scale view of microbial turnover: combining 14 C imaging, μCT, and zymography after adding soluble carbon to soil pores of specific sizes

The location of microorganisms and substrates within soil pore networks plays a crucial role for organic carbon (C) processing, its microbial utilization and turnover, and has direct consequences for C and nutrient cycling. An optimal approach to quantify responses to new C inputs from microorganisms residing in specific pores is the addition of new C to pores of target sizes in undisturbed soil cores. We used the matric potential approach to add 14 C labeled glucose to small (< 40 μm, root free) or large (60-180 μm, potentially inhabited by roots) pores of undisturbed soil cores. Localization of glucose-derived C via 14 C imaging was related to pore size distributions and connectivity, characterized via X-ray computed micro-tomography (μCT), and to β-glucosidase activity, characterized via zymography. After two-week incubations, 1.3 times more glucose was mineralized ( 14 CO 2 ) when it was added to the large pores, however more 14 C remained in microbial biomass when glucose was added to the small pores. Consequently, while utilizing the same amounts of easily available C, the microorganisms localized in the large pores had faster turnover compared to microorganisms in small pores. Stronger associations between β-glucosidase activity and glucose-derived C were observed when glucose was added to the large pores. We conclude that 1) the matric potential approach allows placing, albeit not exactly, soluble substrates into pores of target diameter range, and 2) microorganisms localized in large pores respond to new C inputs with faster turnover, greater growth, and more intensive enzyme production compared to those inhabiting the small pores.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1992

Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n(sub 2)) arithmetic operations, as compared to O(n(sub 3)) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. An extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular, as well as ill-conditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials.

Freund, Roland W.↗

NASA NTRS2007

Quaternion Averaging

Many applications require an algorithm that averages quaternions in an optimal manner. For example, when combining the quaternion outputs of multiple star trackers having this output capability, it is desirable to properly average the quaternions without recomputing the attitude from the the raw star tracker data. Other applications requiring some sort of optimal quaternion averaging include particle filtering and multiple-model adaptive estimation, where weighted quaternions are used to determine the quaternion estimate. For spacecraft attitude estimation applications, derives an optimal averaging scheme to compute the average of a set of weighted attitude matrices using the singular value decomposition method. Focusing on a 4-dimensional quaternion Gaussian distribution on the unit hypersphere, provides an approach to computing the average quaternion by minimizing a quaternion cost function that is equivalent to the attitude matrix cost function Motivated by and extending its results, this Note derives an algorithm that deterniines an optimal average quaternion from a set of scalar- or matrix-weighted quaternions. Rirthermore, a sufficient condition for the uniqueness of the average quaternion, and the equivalence of the mininiization problem, stated herein, to maximum likelihood estimation, are shown.

Markley, F. Landis↗

DOE OSTI2021

Recycled Cardboard Containers as a Low Energy Source for Cellulose Nanofibrils and Their Use in Poly(l-lactide) Nanocomposites

Cellulose nanomaterials have attracted a great deal of interest as sustainable alternatives to nonrenewable or fossil fuel-derived materials, particularly in composite applications. Cellulose nanofibrils (CNF) are most often derived from wood sources through energy-intensive and costly mechanical fibrillation processes. Here, recycled cardboard, referred to as old corrugated cardboard (OCC), was investigated as an alternate CNF source. OCC is a recycled material that can be obtained at one-tenth the cost of the commonly used bleached softwood Kraft pulp and can be refined to similar levels with nearly one-half of the energy consumption. Additionally, the distinct composition and morphology of OCC-derived CNF provides a more favorable fiber–matrix interface and better reinforcement capabilities of the cellulose nanofibrils in polymer matrices, displaying an over 70% increase in tensile modulus at a loading of 40 wt % with no decrease in tensile strength. When paired with biobased poly(l-lactide) (PLLA), OCC-derived CNF is proven to be an attractive low-cost and low-energy reinforcing agent for sustainable, high performance nanocomposites.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1980

Improved orthogonality check for measured modes

A method is proposed for performing an orthogonality check for normal modes derived from ground vibration testing. The method utilizes partitioned mass and stiffness matrices for a linear undamped representation of a structure. The normalization of the modes by the proposed method inherently includes the effects of significant displacements which were not measured; and the method may allow the use of fewer measurement points than would be necessary with the conventional method.

Berman, A.↗