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Results for “least-squares method”

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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NASA NTRS1972

On the mass-luminosity relation.

The results of a least-squares study of the mass-luminosity relation for eclipsing and visual binary stars consisting of main sequence components are presented. Two methods are discussed. First, the values of the coefficients A and B in the relation log M = A + BM sub BOL are determined. Then a technique which permits the determination of alpha and beta in the relation M = alpha L beta, when only the sum of the masses, and not the individual masses of each component, is known. The results and a comparison of the two methods are discussed. It is found that the following mass-luminosity relation represents the observational data satisfactorily: log M = 0.504 - 0.103 M sub BOL, -8 less than or equal to M sub BOL less than or equal to +10.5. A discussion of the data and of the possibility that separate mass-luminosity relations may exist for visual and eclipsing binaries is given. The possibility that more than one mass-luminosity relation is required in the range -8 less than or equal to M sub BOL less than or equal to +13 is also discussed.

Mccluskey, G. E., Jr.↗
DOE OSTI2026

Two-Level Sketching Alternating Anderson Acceleration for Complex Physics Applications

We present a novel two-level sketching extension of the Alternating Anderson–Picard (AAP) method for accelerating fixed-point iterations in challenging single- and multiphysics simulations governed by discretized PDEs. Our approach combines a static, physics-based projection that reduces the least-squares (LS) problem to the most informative field (e.g., via Schur-complement insight) with a dynamic, algebraic sketching stage driven by a backward stability analysis under Lipschitz continuity. We introduce inexpensive estimators for stability thresholds and cache-aware randomized selection strategies to balance computational cost against memory access overhead. The resulting algorithm solves reduced LS systems in place, minimizes memory footprints, and seamlessly alternates between low-cost Picard updates and Anderson mixing. Implemented in Julia, our two-level sketching AAP achieves up to 50% time-to-solution reductions compared to standard Anderson acceleration—without degrading convergence rates—on benchmark problems including Stokes, 𝑝-Laplacian, bidomain, and Navier–Stokes formulations at varying problem sizes. These results demonstrate the method’s robustness, scalability, and potential for integration into high-performance scientific computing frameworks. Our implementation is available open source in the AAP.jl library.

Barnafi, Nicolas [University of Chile, Santiago]↗
DOE OSTI2026

4D beam matrix reconstruction in particle accelerators

Transverse beam parameters in particle accelerators are commonly described using the Twiss parameters, which are experimentally accessible yet inherently limited because they neglect correlations between different transverse coordinates. Such correlations frequently arise from uncompensated cathode magnetic fields or misaligned focusing quadrupoles, affecting beam quality and accelerator performance. To address this limitation, we propose and validate a novel diagnostic method for the complete four-dimensional (4D) transverse beam matrix. Our method involves passing the beam through a beamline comprising both conventional and skew quadrupole magnets, followed by downstream measurements of the resulting two-dimensional (2D) beam profiles. These measurements represent distinct 2D projections of the underlying 4D transverse phase–space distribution. By systematically varying quadrupole strengths, multiple independent projections of the beam phase space are obtained. We reconstruct the original 4D beam matrix from these measured projections using an optimization-based least-square fit, providing fast and robust reconstruction regardless of the specific beamline configuration. Through extensive numerical simulations and realistic particle-tracking studies, we demonstrate the diagnostic’s accuracy, robustness, and capability to achieve reconstruction uncertainties smaller than measurement errors, particularly when employing sufficient numbers of quadrupole scans. This method presents a powerful and flexible approach for comprehensive beam characterization and accelerator tuning.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

Optimized Auxiliary Functions for Robust Mitigation of Finite-Size Errors in Periodic Hybrid Density Functional Theory

When calculating properties of periodic systems at the thermodynamic limit (TDL), the dominant source of finite size error (FSE) arises from the long-range Coulomb interaction, and can manifest as a slowly converging quadrature error when approximating an integral in the reciprocal space by a finite sum. The singularity subtraction (SS) method offers a systematic approach for reducing this quadrature error and thus the FSE. Here, in this work, we first investigate the performance of the SS method in the simplest setting, aiming at reducing the FSE in exact exchange calculations by subtracting the Coulomb contribution with a single, adjustable Gaussian auxiliary function. We demonstrate that a simple fitting method can robustly estimate the optimal Gaussian width and leads to rapid convergence toward the TDL. Furthermore, we suggest new forms of the auxiliary function, whose optimal parameters could also be determined through least-squares fitting. For a range of semiconductors and insulators, the proposed auxiliary functions achieve robust, millihartree-level accuracy in hybrid density functional theory calculations, including cases with sparse k-meshes and large basis sets.

Quiton, Stephen Jon [University of California, Ber↗
DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗
NASA NTRS1972

A two-spacecraft test of a single spacecraft method of estimating shock normals

By assuming the validity of a subset of the Rankine-Hugoniot conservation relations for interplanetary (IP) shocks in an isotropic medium it has been demonstrated, in principle, that improved shock normals can be calculated by using a least-squares technique on combined magnetic field and plasma data from a single spacecraft. The scheme uses those six conservation relations not involving pressure and temperature. This paper deals with a test of the scheme by examining in detail a shock across which the magnetic field changed direction by a small amount (approximately 10 deg). On January 26, 1968 at about 1430 UT this shock was observed by the plasma and magnetic field instruments in Explorers 33 and 35. The spacecraft were 76.6 and 56.9 R sub E sunward of the earth, respectively (and 43.5 R sub E from each other), and therefore well outside the earth's bow shock region, a necessary condition for a valid test.

Lepping, R. P.↗
DOE OSTI2025

Deciphering the Scattering of Mechanically Driven Polymers Using Deep Learning

Here, we present a deep learning approach for analyzing two-dimensional scattering data of semiflexible polymers under external forces. In our framework, scattering functions are compressed into a three-dimensional latent space using a Variational Autoencoder (VAE), and two converter networks establish a bidirectional mapping between the polymer parameters (bending modulus, stretching force, and steady shear) and the scattering functions. The training data are generated using off-lattice Monte Carlo simulations to avoid the orientational bias inherent in lattice models, ensuring robust sampling of polymer conformations. The feasibility of this bidirectional mapping is demonstrated by the organized distribution of polymer parameters in the latent space. By integrating the converter networks with the VAE, we obtain a generator that produces scattering functions from given polymer parameters and an inferrer that directly extracts polymer parameters from scattering data. While the generator can be utilized in a traditional least-squares fitting procedure, the inferrer produces comparable results in a single pass and operates 3 orders of magnitude faster. This approach offers a scalable automated tool for polymer scattering analysis and provides a promising foundation for extending the method to other scattering models, experimental validation, and the study of time-dependent scattering data.

Ding, Lijie [Oak Ridge National Laboratory (ORNL),↗
DOE OSTI2025

A framework for discrete optimization of stellarator coils

Designing magnets for three-dimensional plasma confinement is a key task for advancing the stellarator as a fusion reactor concept. Stellarator magnets must produce an accurate field while leaving adequate room for other components and being reasonably simple to construct and assemble. In this paper, a framework for coil design and optimization is introduced that enables the attainment of sparse magnet solutions with arbitrary restrictions on where coils may be located. The solution space is formulated as a 'wireframe' consisting of a mesh of interconnected wire segments enclosing the plasma. Two methods are developed for optimizing the current distribution on a wireframe: Regularized Constrained Least Squares, which uses a linear least-squares approach to optimize the currents in each segment, and Greedy Stellarator Coil Optimization, a fully discrete procedure in which loops of current are added to the mesh one by one to achieve the desired magnetic field on the plasma boundary. Examples are presented of solutions obtainable with each method, some of which achieve high field accuracy while obeying spatial constraints that permit easy assembly.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

3D seismic imaging of a fracture damage zone controlling reservoir compartmentalization at the Raft River EGS using multi-azimuth walkaway VSP

Accurate imaging of steeply dipping fracture zones in crystalline enhanced geothermal systems (EGS) is critical for constraining permeability architecture and guiding stimulation design. However, such structures remain poorly resolved by conventional surface seismic methods. We present a fully three-dimensional (3D) elastic-waveform inversion-migration workflow applied to multi-azimuth walkaway vertical seismic profiling (VSP) data acquired in a deviated borehole at the Raft River EGS. The workflow integrates first-arrival traveltime tomography, multi-scale elastic waveform inversion (EWI), and elastic least-squares reverse-time migration (ELSRTM) to recover high-resolution compressional-and shear-wave velocity models and to image structural discontinuities in the crystalline basement. The results reveal a laterally continuous low-velocity anomaly, with shear-wave velocity reductions of 25-30%, consistent with fractureinduced mechanical weakening. Two steeply dipping discontinuities bound a 50-80 m wide fracture damage zone. Independent constraints from microseismic clustering and geochemical compartmentalization corroborate the geometry and structural significance of this feature. Synthetic modeling further confirms that structures of this scale are resolvable (~30 m). These findings indicate that the Narrows structure is a distributed fracture damage zone rather than a discrete fault plane. By resolving fracture-zone geometry at the tens-of-meters scale using a single borehole, this workflow provides a practical and transferable approach for improving structural characterization, reducing uncertainty in permeability architecture, and supporting reservoir modeling and stimulation design in fractured crystalline EGS reservoirs.

58 GEOSCIENCES↗
DOE OSTI2026

Gearbox bearing crack growth prognostics and uncertainty quantification with physics-informed machine learning

This paper introduces the extreme theory of functional connections (X-TFC), a physics-informed machine learning algorithm, and tailors it to estimate the remaining useful life (RUL) of wind turbine gearbox bearings experiencing fatigue crack growth. Unlike purely data-driven methods, X-TFC embeds a physics model, based on Head's theory in this work, into its training objective. The core of X-TFC is a random-projection single-layer neural network trained via an extreme learning machine, which requires only limited damage progression data and solves for output weights with a least-squares optimization algorithm. A composite loss function balances the network's fit to observed degradation data against the residuals of the governing crack growth differential equation, ensuring the learned damage trajectory remains physically plausible. When applied to a vibration-based health-index (HI) dataset measured during the growth of a crack on the inner ring of a high-speed bearing in a wind turbine gearbox (Bechhoefer and Dubé, 2020), X-TFC achieves near-zero prediction bias. Even when trained on only the first 10 %–20 % of the damage progression data, with sufficient physics weighting its predictions remain monotonic and smooth, delivering high prognosability and trendability. To quantify the epistemic uncertainty, we employ a Monte Carlo ensemble of independently initialized X-TFC models trained on noise-perturbed data, which yields confidence intervals around each RUL estimate and captures both model-parameter and epistemic uncertainty. In addition to a vibration-based HI, we demonstrate that the proposed framework can be directly applied to a supervisory control and data acquisition (SCADA) data-based HI (Eftekhari Milani et al., 2026) measured during similar wind turbine gearbox bearing crack faults, preserving its accuracy and interpretability. This extension shows the versatility of our approach, which is applicable to bearings of multiple gearbox manufacturers, models, and ratings using only SCADA data. By integrating domain knowledge with machine learning, X-TFC offers a rapid, reliable tool for crack prognostics. Its adaptability to other bearing failure modes, such as pitch bearing ring cracks, positions X-TFC as a powerful enabler of data-driven, physics-informed asset management in the wind energy sector and beyond.

17 WIND ENERGY↗
DOE OSTI2025

Karhunen–Loève deep learning method for surrogate modeling and approximate Bayesian parameter estimation

We evaluate the performance of the Karhunen-Loève Deep Neural Network (KL-DNN) framework for surrogate modeling and approximate Bayesian parameter estimation in partial differential equation models. In the surrogate model, the Karhunen-Loève (KL) expansions are used for the dimensionality reduction of the number of unknown parameters and variables, and a deep neural network is employed to relate the reduced space of parameters to that of the state variables. The KL-DNN surrogate model is used to formulate a maximum-a-posteriori-like least-squares problem, which is randomized to draw samples of the posterior distribution of the parameters. We test the proposed framework for a hypothetical unconfined aquifer via comparison with the forward MODFLOW and inverse PEST++ iterative ensemble smoother (IES) solutions as well as the state-of-the-art Fourier neural operator (FNO) and deep operator networks (DeepONets) operator learning surrogate models. Our results show that the KL-DNN surrogate model outperforms FNO and DeepONet for forward predictions. For solving inverse problems, the randomized algorithm provides the same or more accurate Bayesian predictions of the parameters than IES as evidenced by the higher log-predictive probability of both the estimated parameter field and the forecast hydraulic head. The posterior mean obtained from the randomized algorithm is closer to the reference parameter field than that obtained with FNO as the maximum a posteriori estimate.

Approximate Bayesian inference↗
DOE OSTI2026

RegularizedOptimization.jl: A Julia framework for regularized and nonsmooth optimization

RegularizedOptimization.jl is a Julia package that implements families of quadratic regularization and trust-region methods for solving the nonsmooth optimization problem $^{\textrm{minimize}}_{𝑥∈ℝ^𝑛}$ 𝑓(𝑥) + ℎ(𝑥) subject to 𝑐(𝑥) = 0, (1) where 𝑓 ∶ ℝ 𝑛 → ℝ and 𝑐 ∶ ℝ 𝑛 → ℝ 𝑚 are continuously differentiable, and ℎ ∶ ℝ 𝑛 → ℝ∪{+∞} is lower semi-continuous. The nonsmooth objective ℎ can be a regularizer, such as a sparsity inducing penalty, model simple constraints, such as 𝑥 belonging to a simple convex set, or can be a combination of both. All 𝑓, ℎ, and 𝑐 can be nonconvex. RegularizedOptimization.jl provides a modular and extensible framework for solving (1), and developing novel solvers. Currently, the following solvers are implemented: • Trust-region solvers TR and TRDH (Aravkin et al., 2022; Leconte & Orban, 2025) • Quadratic regularization solvers R2, R2DH and R2N (Aravkin et al., 2022; Diouane, Habiboullah, et al., 2024) • Levenberg-Marquardt solvers LM and LMTR (Aravkin et al., 2024) used when 𝑓 is a least-squares residual. • Augmented Lagrangian solver AL (De Marchi et al., 2023). All solvers rely on first derivatives of 𝑓 and 𝑐, and optionally on their second derivatives in the form of Hessian-vector products. If second derivatives are not available, quasi-Newton approximations can be used. In addition, the proximal mapping of the nonsmooth part ℎ, or adequate models thereof, must be evaluated. At each iteration, a step is computed by solving a subproblem of the form (1) inexactly, in which 𝑓, ℎ, and 𝑐 are replaced with appropriate models around the current iterate. The solvers R2, R2DH, and TRDH are particularly well suited to solve the subproblems, though they are general enough to solve (1). All solvers are allocation-free, so re-solves incur no additional allocations. To illustrate our claim of extensibility, a first version of the AL solver was implemented by an external contributor. Furthermore, a nonsmooth penalty approach, described in Diouane, Gollier, et al. (2024), is currently being developed, that relies on the library to efficiently solve the subproblems.

Gollier, Maxence [Polytechnique Montréal, QC (Cana↗
NASA NTRS1971

Development of techniques for measuring pilot workload

An objective method of assessing information workload based on physiological measurements was developed. Information workload, or reserve capacity, was measured using a visual discrimination secondary task and subjective rating of task difficulty. The primary task was two axis (pitch and roll) tracking, and the independent variables in this study were aircraft pitch dynamics and wind gust disturbances. The study was structured to provide: (1) a sensitive, nonloading measure of reserve capacity, and (2) an unencumbering reliable measurement of the psychophysiological state. From these, a measured workload index (MWI) and physiological workload index (PWI) were extracted. An important measure of the success of this study was the degree to which the MWI and PWI agreed across the 243 randomly-presented, four-minute trials (9 subjects X 9 tasks X 3 replications). The electrophysiological data collected included vectorcardiogaram, respiration, electromyogram, skin impedance, and electroencephalogram. Special computer programs were created for the analysis of each physiological variable. The digital data base then consisted of 82 physiological features for each of the 243 trials. A prediction of workload based on physiological observations was formulated as a simultaneous least-squares prediction problem. A best subset of 10 features was chosen to predict the three measures of reserve capacity. The cannonical correlation coefficient was .754 with a chi squared value of 91.3 which allows rejection of the null hypothesis with p of .995.

Spyker, D. A.↗
DOE OSTI2026

Improved Accuracy in Semi-Experimental Structure Determination by Resolving Problems Associated with Rotation of Principal Inertial Axes of Isotopologues: Structures of 1,3-Oxazole ( c -C 3 H 3 NO)

The rotational spectrum of the normal isotopologue of 1,3-oxazole (c-C 3 H 3 NO) was observed from 43 to 750 GHz. Over 3900 transitions for the ground vibrational state are measured, assigned, and least-squares fit to sextic centrifugally distorted-rotor Hamiltonians. The measured frequencies and resulting spectroscopic constants from this extended spectral range, combined with previous measurements of the nuclear quadrupole coupling constants, will facilitate astronomical searches for oxazole across the majority of the range of modern radiotelescopes. Spectra for a set of 30 oxazole isotopologues, which include multiple isotopic substitutions of each atom, are used to determine the first semi-experimental equilibrium ($r$$^{SE}_{e}$) structure and semi-experimental substitution structure ($r$$^{SE}_{e}$), each using CCSD(T) computed values for the vibration–rotation interaction and electron-mass corrections. The large number of isotopologues, including 21 isotopologues observed for the first time, and the redundant substitutions of each atom provide sufficient spectroscopic information to determine the $r$$^{SE}_{e}$ structure with the expected high level of accuracy and precision (0.0001 or 0.0002 Å in bond distances and 0.013 to 0.025° in bond angles). In the course of this study, we analyzed a known issue for some $r$$^{SE}_{e}$ structure determinations of near-oblate asymmetric tops in which inclusion of individual isotopologues degrades the structure determination. We demonstrate that this problem primarily arises from the difference in the values of the computed vibration–rotation interaction corrections as evaluated at the computed re geometry vs the $r$$^{SE}_{e}$ geometry of the “real” molecule. Our solution to this problem substantially improves the $r$$^{SE}_{e}$ structure of oxazole and likely can be generalized to many other molecules.

Chemical structure↗
DOE OSTI2025

Spatial Signatures of Electron Correlation in Least-Squares Tensor Hypercontraction

Least Squares Tensor Hypercontraction (LS-THC) has received some attention in recent years as an approach to reduce the significant computational costs of wavefunc- tion based methods in quantum chemistry. However, previous work has demonstrated that the LS-THC factorization performs disproportionately worse in the description of wavefunction components (e.g. cluster amplitudes T 2 ) than Hamiltonian compo- nents (e.g. electron repulsion integrals (pq|rs)). This work develops novel theoretical methods to study the source of these errors in the context of the real-space T 2 kernel, and reports, for the first time, the existence of a “correlation feature” in the errors of the LS-THC representation of the “exchange-like” correlation energy EX and T 2 that is remarkably consistent across ten molecular species, three correlated wavefunctions, and four basis sets. This correlation feature portends the existence of a “pair-point kernel” missing in the usual LS-THC representation of the wavefunction, which critically depends upon pairs of grid points situated close to atoms and with inter-pair distances between one and two Bohr radii. These findings point the way for future LS-THC developments to address these shortcomings.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗