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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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All sourcesNASA NTRSDOE OSTI
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 OSTI2020

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

Here, we study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai’s class, regardless of the probability weights assigned to the self-similar measures.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Polynomial Scaling Localized Active Space Unitary Selective Coupled Cluster Singles and Doubles

We present a polynomial-scaling algorithm for the localized active space unitary selective coupled cluster singles and doubles (LAS-USCCSD) method. In this approach, cluster excitations are selected based on a threshold ϵ determined by the absolute gradients of the LAS-UCCSD energy with respect to cluster amplitudes. Using the generalized Wick’s theorem for multireference wave functions, we derive the gradient expression as a polynomial function of one-, two-, and three-body reduced density matrices and 1- and 2-electron integrals, valid for any multireference wave function. The resulting gradient implementation exhibits a memory scaling of 𝒪(N 6 ), with N spin orbitals in the combined active space of all fragments. The variational quantum eigensolver is used to optimize the selected cluster excitations on a quantum simulator. Furthermore, by plotting the energy error, defined as the difference between the LAS-USCCSD and corresponding CASCI energies, against the inverse cluster amplitude selection threshold (ϵ –1 ) for polyene chains containing 2 to 5 π-bond units, we establish a relationship between the energy error and the threshold. To further validate the accuracy of LAS-USCCSD, we computed the cis–trans isomerization energy of stilbene (a 20-qubit system) and the magnetic coupling constant of the tris-hydroxo-bridged chromium dimer [Cr 2 (OH) 3 (NH 3 ) 6 ] 3+ (evaluated as both 12- and 20-qubit systems) using the Qiskit-Qulacs simulator. Assessing such examples is important to determine the practical feasibility of quantum simulations for chemically realistic systems. Toward this goal, with the LAS-USCCSD algorithm we estimated the quantum resources required for simulating an active space of (30e,22o) in [Cr 2 (OH) 3 (NH 3 ) 6 ] 3+ , a size that remains beyond the reach of current quantum simulators for accurate treatment.

Algorithms↗
DOE OSTI2025

Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials

Abstract The remarkable complexity of a topologically ordered many-body quantum system is encoded in the characteristics of its anyons. Quintessential predictions emanating from this complexity employ the Fibonacci string net condensate (Fib SNC) and its anyons: sampling Fib-SNC would estimate chromatic polynomials while exchanging its anyons would implement universal quantum computation. However, physical realizations remained elusive. We introduce a scalable dynamical string net preparation (DSNP) that constructs Fib SNC and its anyons on reconfigurable graphs suitable for near-term superconducting processors. Coupling the DSNP approach with composite error-mitigation on deep circuits, we create, measure, and braids Fibonacci anyons; charge measurements show 94% accuracy, and exchanging the anyons yields the expected golden ratioϕwith 98% average accuracy. We then sample the Fib SNC to estimate chromatic polynomial atϕ + 2 for several graphs. Our results establish the proof of principle for using Fib-SNC and its anyons for fault-tolerant universal quantum computation and aim at a classically hard problem.

Science & Technology - Other Topics↗
DOE OSTI2026

Scalable quantum computational science: A perspective from block-encodings and polynomial transformations

Significant developments made in quantum hardware and error correction recently have been driving quantum computing toward practical utility. However, gaps remain between abstract quantum algorithmic development and practical applications in computational sciences. In this perspective article, we propose several properties that scalable quantum computational science methods should possess. We further discuss how block-encodings and polynomial transformations can potentially serve as a unified framework with the desired properties. Recent advancements on these topics are presented, including the construction and assembly of block-encodings, and various generalizations of quantum signal processing (QSP) algorithms to perform polynomial transformations. The scalability of QSP methods on parallel and distributed quantum architectures is also highlighted. Promising applications in simulation and observable estimation in chemistry, physics, and optimization problems are presented. We hope this perspective serves as a gentle introduction to state-of-the-art quantum algorithms for the computational science community and inspires future development of scalable quantum computational science methodologies that bridge theory and practice.

Bayesian inference↗
DOE OSTI2025

Chiral-odd generalized parton distributions in the large-𝑁 𝑐 limit of QCD: Spin-flavor structure, polynomiality, and sum rules

We study the nonperturbative properties of the nucleon’s chiral-odd generalized parton distributions (transversity GPDs) in the large-𝑁 𝑐 limit of QCD. This includes the parametric ordering of the spin-flavor components, the polynomiality property of the moments, and the sum rules connecting the GPDs with the tensor form factors. A multipole expansion in the transverse momentum transfer is used to enumerate and interpret the structures in the nucleon matrix element of the chiral-odd partonic operator, including monopole, dipole and quadrupole terms. The 1/𝑁 𝑐 expansion of the GPDs is performed using the abstract mean-field picture of baryons in the large-𝑁 𝑐 limit and its symmetries. We derive a large-𝑁 𝑐 relation between the flavor-nonsinglet GPDs 𝐸$^{𝑢−𝑑}_𝑇$ and $\tilde{𝐻}^{𝑢−𝑑}_𝑇$ and test it with recent lattice QCD results. We show that the polynomiality property and sum rules of the GPDs are fulfilled with the restricted realization of translational and rotational invariance in the mean-field picture. The results provide a basis for the phenomenological analysis of chiral-odd GPDs and hard exclusive processes in the large-𝑁 𝑐 limit, and for calculations in specific dynamical models.

generalized parton distributions↗
DOE OSTI2023

Additive Polynomial Time Integrators, Part I: Framework and Fully Implicit-Explicit Collocation Methods

In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with arbitrary order-of-accuracy and varying degree of implicitness. In this first work, we focus on a new class of implicit-explicit polynomial block methods that are based on fully implicit Runge–Kutta methods with Radau nodes and that possess high stage order. Here, we show that the new fully implicit-explicit (FIMEX) integrators have improved stability compared to existing IMEX Runge–Kutta methods, while also being more computationally efficient due to recent developments in preconditioning techniques for solving the associated systems of nonlinear equations. For PDEs on periodic domains where the implicit component is trivial to invert, we will show how parallelization of the right-hand side evaluations can be exploited to obtain significant speedup compared to existing serial IMEX Runge–Kutta methods. For parallel (in space) finite element discretizations, the new methods can achieve orders of magnitude better accuracy than existing IMEX Runge–Kutta methods and/or achieve a given accuracy several times times faster in terms of computational runtime.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A Unified Approach to Unimodality of Gaussian Polynomials

In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ℓ, m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

Koutschan, Christoph↗
DOE OSTI2023

Use of Sobol’ Variance-Based Global Sensitivity Analysis and Multidimensional Legendre Polynomial Fitting for Reduced Order Modeling

Sandia National Laboratories (SNL) has developed a novel reduced order modeling approach. Prioritization of inputs is accomplished using Sobo' indices obtained through a more efficient variance-based global sensitivity analysis. To determine the Sobo' functions, simulated input values are aligned to collocation points to permit the use of Gauss-Lobatto integration, thereby reducing the number of simulation trials needed by more than an order of magnitude compared to standard Monte Carlo approaches. Furthermore, by leveraging the orthogonality of Legendre polynomials in conjunction with those same simulations at the collocation nodes, an efficient fitting method is developed to represent the Sobo' functions from which a reduced order model (ROM) is constructed. The developed method is both more efficient computationally, and the resulting ROM is more accurate. The efficacy of this technique is demonstrated on a nonlinear polynomial test function as well as the nonlinear Ishigami and Sobo' g functions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Data-Centric Approach to Capture Non-Polynomial Nonlinear Dynamics

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for applying EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or rely on machine learning algorithms to “learn” observable functions. Conversely, in this paper, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact and provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. We demonstrate the proposed approach with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

extended dynamic mode decomposition↗
DOE OSTI2026

Spectra-to-exposure conversion using polynomial response models for gamma-ray field characterization

Accurate measurement of exposure rate from gamma-ray spectral data remains a critical challenge during radiological emergency response operations. Conventional methods rely on pre-defined static conversion factors derived from fixed geometries and isotopic compositions, which often fail to capture real-world environmental variability. This study presents a generalized approach as a "next-step" for converting gamma-ray spectral data into exposure rate using polynomial response models. The method introduces a flexible weighting scheme based on the in-situ detector response to distributed sources, enabling a pathway towards improved correspondence between measured spectra and "ground-truth" exposure rates. Experimental data from sodium iodide NaI(Tl) detectors were used to validate the approach as, at least equivalent to the current count-to-exposure method employed in emergency response CONOPS. Results show that the polynomial weighting model is sufficiently equal to the count-to-exposure method and may help improve accuracy given its adaptability to real-world conditions.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2026

Bayesian Adaptive Polynomial Chaos Expansions

Polynomial chaos expansions (PCEs) are widely used for uncertainty quantification (UQ) tasks, particularly in the applied mathematics community. However, PCE has received comparatively less attention in the statistics literature, and fully Bayesian formulations remain rare—especially with implementations in R. Motivated by the success of adaptive Bayesian machine learning models such as BART, BASS and BPPR, we develop a new fully Bayesian adaptive PCE method with an efficient and accessible R implementation: khaos. Our approach includes a novel proposal distribution that enables data-driven interaction selection and supports a modified g-prior tailored to PCE structure. Through simulation studies and real-world UQ applications, we demonstrate that the Bayesian adaptive PCE provides competitive performance for surrogate modeling, global sensitivity analysis and ordinal regression tasks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Projection pursuit adaptation on polynomial chaos expansions

Here, the present work addresses the issue of accurate stochastic approximations in high-dimensional parametric space using tools from uncertainty quantification (UQ). The basis adaptation method and its accelerated algorithm in polynomial chaos expansions (PCE) were recently proposed to construct low-dimensional approximations adapted to specific quantities of interest (QoI). The present paper addresses one difficulty with these adaptations, namely their reliance on quadrature point sampling, which limits the reusability of potentially expensive samples. Projection pursuit (PP) is a statistical tool to find the “interesting” projections in high-dimensional data and thus bypass the curse-of-dimensionality. In the present work, we combine the fundamental ideas of basis adaptation and projection pursuit regression (PPR) to propose a novel method to simultaneously learn the optimal low-dimensional spaces and PCE representation from given data. While this projection pursuit adaptation (PPA) can be entirely data-driven, the constructed approximation exhibits mean-square convergence to the solution of an underlying governing equation and thus captures the supports and probability distributions associated with the physics constraints. The proposed approach is demonstrated on a borehole problem and a structural dynamics problem, demonstrating the versatility of the method and its ability to discover low-dimensional manifolds with high accuracy with limited data. In addition, the method can learn surrogate models for different quantities of interest while reusing the same data set.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Polynomial chaos expansions on principal geodesic Grassmannian submanifolds for surrogate modeling and uncertainty quantification

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemannian K-means and the minimization of the sample Fréchet variance on the Grassmann manifold to identify “local” principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. Here, the method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Bénard convection problem.

42 ENGINEERING↗
DOE OSTI2023

Adaptive Online Multivariate Signal Extraction With Locally Weighted Robust Polynomial Regression

High-frequency, multivariate data collected in real-time and used to control or make decisions regarding a process’ operation often contain some noise and outliers. Thus, a method to extract the signal is needed in order to reduce the number and magnitude of control-based adjustments that are implemented. Such a method must be (i) online, depending only on past and current observations; (ii) fast, producing a smooth value more quickly than the measurement frequency; (iii) robust, ignoring brief bursts of erroneously measured values; (iv) multivariate, ignoring observations that are jointly unusual; (v) adaptive, adjusting to periods of rapid fluctuation in the signal versus periods of stability; and (vi) purely data-driven, not incorporating any information about the process from which the data are collected. Most existing methods are only able to address a subset of these six features. Furthermore, we also require the method to be nonlinear, providing a local nonlinear estimate of the signal. In this work, we propose a novel, real-time signal extraction method based on a local, robust polynomial fit. We demonstrate the performance of our method compared to a state-of-the-art competitor through simulation. For illustration, the methodology is applied to data collected from a reverse osmosis water treatment process.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A New Proposal Generalized Predictive Control Algorithm With Polynomial Reference Tracking Applied for Sodium Fast Reactors

This paper proposes a generalized predictive control (GPC) with constraints and orthonormal Laguerre functions using the simplified model of the primary system (reactor core and intermediate heat exchanger (IHX)) of a prototypical sodium fast reactor (SFR). This paper develops a multiple-input multiple-output (MIMO) GPC with input constraints able to track polynomial references of any degree applied in coolant temperature difference across the core and fractional power. The manipulated variables of the GPC-SFR are the reactivity and the sodium flow rate of the primary and secondary pipes. Moreover, orthonormal Laguerre functions and step down condition number techniques were also applied to avoid the numerical ill-conditioning issue in quadratic programming of large systems. Thus, a GPC type-2 was designed to control fractional power, coolant temperature difference across the core and sodium tank temperature of the SFR primary system when temperature references change according to a linear ramp after reaching their steady-state operation, sustaining 100% power operation on the reactor. In order to analyze the load tracking capability of the GPC-SFR type-2, the load following from 100% fractional power (FP) to 60% FP at 0.8% FP/min rate is simulated. Constraints on the rate of coolant temperature difference across the core and reactivity were applied for the design safety. For comparison criteria, this paper compares the GPC-SFR type-2 with the GPC-SFR type-1, i.e., standard model predictive control (MPC), to verify the viability and superior performance of the proposal regarding: (a) ramp-tracking capability of temperature and load; (b) the rejections of a reactivity disturbance of -1 cent and a secondary sodium inlet temperature disturbance of +10°F; and (c) a simulation with uncertainty in reactor design. The simulations show that the GPC-SFR type-2 overcome the GPC-SFR type-1 robustness and performance.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2021

A Generalized Copula-Polynomial Chaos Expansion for Probabilistic Power Flow Considering Nonlinear Correlations of PV Injections

This paper develops a generalized Copula-polynomial chaos expansion (PCE) framework for power system probabilistic power flow that can handle both linear and nonlinear correlations of uncertain power injections, such as wind and PVs. A data-driven Copula statistical model is used to capture the correlations of uncertain power injections. This allows us to resort to the Rosenblatt transformation to transform correlated variables into independent ones while preserving the dependence structure. This paves the way of leveraging the PCE for surrogate modeling and uncertainty quantification of power flow results, i.e., achieving the probabilistic distributions of power flows. Simulations carried out on the IEEE 57-bus system show that the proposed framework can get much more accurate results than other alternatives with different linear and nonlinear power injection correlations.

copula↗
DOE OSTI2025

Polynomial Chaos Surrogate Construction for Random Fields with Parametric Uncertainty

Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, which are evaluated via kernel density estimation of the associated conditional cumulative distributions. Furthermore, we extend the construction to random field data via the Karhunen–Loève expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model and a synthetic example controlled by a parameter that enables a switch from unimodal to bimodal response distributions.

97 MATHEMATICS AND COMPUTING↗