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Results for “least-squares optimization”

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 OSTI2026

Measurement-informed Dynamic Aggregation of Distribution Systems

This paper proposes a measurement-informed dynamic aggregation methodology in order to create equivalent representations of distribution systems that are compatible with large-scale transmission analysis. By optimizing an equivalent feeder parameters using time-series measurements of active power, reactive power, and voltage at the Point of Interconnection (POI), the approach yields simplified yet dynamically accurate equivalents. Implemented in PSCAD with models of photovoltaic–battery systems, three-phase motors, and static loads, the method employs hybrid differential evolution and bounded least-squares optimization laying the foundation for for real-time state estimation and optimized sensor placement in distribution networks.

Ahmed, Kazi Ishrak [University of Tennessee, Knoxv↗
DOE OSTI2025

A machine learning framework for accurate and robust analysis of radiation detector pulses

The microscopic properties of atomic nuclei are used to study various scientific questions. They are essential for understanding the fundamental forces of nature and the chemical evolution of the universe. Detecting decay radiation from radioactive nuclei makes it possible to probe these fundamental nuclear properties. Detector waveform traces may contain additional information about the radiation. Generally, advanced signal processing techniques are needed to extract this additional information, often involving fitting the waveform with model response functions using non-linear least-squares optimization with second-order gradient methods. While this is a powerful technique, it is also computationally expensive, leading to slow processing time, which scales with the volume of data. To address this problem, we have developed a machine learning (ML) approach that infers the characteristics of traces from a model detector response function. In particular, we are interested in classifying whether a single recorded trace consists of one or two pulse constituents and estimating the pulse parameters. Furthermore, our proposed ML method can precisely extract the pulses’ parameters, such as energy and timing information, and accurately classify the pulse multiplicity of a trace. Unlike non-learning-based approaches, our ML approach uses neural networks that are significantly faster at inference, as they do not require any optimization during this stage.

Curve fitting↗
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 OSTI2026

Framework for X-ray mirror surface shape fitting

For accurate characterization of grazing-incidence X-ray mirrors, we present a comprehensive framework to fit measured surface shapes (either slope or height) of X-ray mirrors used in synchrotron radiation and free-electron laser facilities. We summarize the closed-form expressions of some typical surface shapes of X-ray mirrors including elliptic cylinders, hyperbolic cylinders, ellipsoids, hyperboloids, and diaboloids. This framework is composed of four layers: definition of standard shapes with closed-form expressions, generation of theoretical surface with pose parameters (six degrees of freedom defining an object's position and orientation relative to a coordinate system), parameter optimization with the ability to select which parameters are fit and which are held constant, and the development of user-friendly fitting function wrappers for particular fitting tasks. A few practical fitting examples are demonstrated to verify the effectiveness of the proposed fitting framework. We discuss the physical meanings of the fitting parameters, and provide several examples using the elliptic cylinder and ellipsoid shapes to highlight some features of the framework. Moreover, we provide the presented framework as open-source codes (MATLAB and Python codes available at https://github.com/nsls2omf/xmf) to the community to encourage academic collaboration and further improvements.

36 MATERIALS SCIENCE↗
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

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 OSTI2024

How Much Reserve Fuel: Quantifying the Maximal Energy Cost of System Disturbances

Motivated by the design question of additional fuel needed to complete a task in an uncertain environment, this paper introduces metrics to quantify the maximal additional energy used by a control system in the presence of bounded disturbances, compared to a nominal, disturbance-free system. In particular, we consider the task of finite-time stabilization for a linear, time-invariant system. We compare the nominal energy required to achieve this task in the disturbance-free system to the worst-case energy over all feasible disturbances. Solving for the worst-case energy over all disturbances first leads to an optimal control problem with a least-squares solution, and then an infinite-dimensional optimization problem where we derive an upper bound on the solution. The comparison of energies is accomplished using additive and multiplicative metrics, for which we derive bounds. Simulation examples on an ADMIRE fighter jet model demonstrate the practicability of these metrics, and their variation with the distance of the initial condition from the origin and the task completion time.

koopman operator, resilience↗
NASA NTRS1976

Design analogy between optimal time-fuel and rate-ledge relay controllers

An empirically obtained reaction control jet relay control law with deadband is used as the basis for determining an equivalent weighted time-fuel optimal switching curve according to a least-squares criterion. The derived transformation from the empirical to the optimal law is found to be reversible and to yield a unique transformed control law. The proposed method provides a basis for determining the behavior of an easily implemented relay control law using well-known optimal control results, as well as determining the equivalent relay law corresponding to an analytically determined optimal control law. A numerical example illustrates the transformation technique and simulation results are presented to compare the two control laws.

White, L. R.↗
NASA NTRS1979

Ambiguity resolution for satellite Doppler positioning systems

The implementation of satellite-based Doppler positioning systems frequently requires the recovery of transmitter position from a single pass of Doppler data. The least-squares approach to the problem yields conjugate solutions on either side of the satellite subtrack. It is important to develop a procedure for choosing the proper solution which is correct in a high percentage of cases. A test for ambiguity resolution which is the most powerful in the sense that it maximizes the probability of a correct decision is derived. When systematic error sources are properly included in the least-squares reduction process to yield an optimal solution the test reduces to choosing the solution which provides the smaller valuation of the least-squares loss function. When systematic error sources are ignored in the least-squares reduction, the most powerful test is a quadratic form comparison with the weighting matrix of the quadratic form obtained by computing the pseudoinverse of a reduced-rank square matrix. A formula for computing the power of the most powerful test is provided. Numerical examples are included in which the power of the test is computed for situations that are relevant to the design of a satellite-aided search and rescue system.

Argentiero, P.↗
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

Multi-plane moment-of-fluid interface reconstruction in 3D

Moment-of-fluid (MOF) methods for interface reconstruction approximate the region occupied by material in each mesh element only through reference to its geometric moments. Here, we present a 3D MOF method that represents the material (POM) in each cell as the convex intersection of the cell and multiple half-spaces, each selected to minimize the least-squares error between computed moments of the approximated material and provided reference moments. This optimization problem is highly non-linear and non-convex, making the numerical result very sensitive to the initial guess. To create an effective initial guess in each cell, we construct an ellipsoid from 0th–2nd order reference moments such that its shape corresponds with that of the POM. Within this ellipsoid we inscribe a polyhedron, and initialize the minimization problem with the half-spaces defined by each of its faces. The inscribed polyhedron has minimally 4 faces, and using up to 3rd order moments permits optimization over up to 20 unknown values. We therefore define MOF methods that utilize 4, 5, or 6 half-spaces, correspondingly initialized with the faces of a single inscribed tetrahedron, triangular prism, or hexahedron. Stability of the non-linear optimization is further improved with a prepossessing step that normalizes the reference moments according to the axes of the reference ellipsoid. Using this approach, the non-linear least-squares solver reliably converges to a near-global minimum from a single initial guess. We demonstrate accuracy and robustness using single-cell and multi-cell examples over a wide spectrum of geometry. In particular, we demonstrate our ability to exactly reproduce several important and complex features defined by up to four half-spaces, such as corners, filaments, filament tips, and embedded material in the cell.

3D interface reconstruction↗
NASA NTRS1975

Development of a digital adaptive optimal linear regulator flight controller

Digital adaptive controllers have been proposed as a means for retaining uniform handling qualities over the flight envelope of a high-performance aircraft. Towards such an implementation, an explicit adaptive controller, which makes direct use of online parameter identification, has been developed and applied to the linearized lateral equations of motion for a typical fighter aircraft. The system is composed of an online weighted least-squares parameter identifier, a Kalman state filter, and a model following control law designed using optimal linear regulator theory. Simulation experiments with realistic measurement noise indicate that the proposed adaptive system has the potential for onboard implementation.

Berry, P.↗
NASA NTRS1979

State and model error estimation for distributed parameter systems

In-flight estimation of large structure model errors in order to detect inevitable deficiencies in large structure controller/estimator models is discussed. Such an estimation process is particularly applicable in the area of shape control system design required to maintain a prescribed static structural shape and, in addition, suppress dynamic disturbances due to the vehicle vibrational modes. The paper outlines a solution to the problem of static shape estimation where the vehicle shape must be reconstructed from a set of measurements discretely located throughout the structure. The estimation process is based on the principle of least-squares that inherently contains the definition and explicit computation of model error estimates that are optimal in some sense. Consequently, a solution is provided for the problem of estimation of static model errors (e.g., external loads). A generalized formulation applicable to distributed parameters systems is first worked out and then applied to a one-dimensional beam-like structural configuration.

Rodriguez, G.↗
DOE OSTI2026

Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces

This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven least-squares regression problem for low-dimensional matrix operators. Our approach instead leverages regularized kernel interpolation, which yields an optimal approximation of the ROM dynamics from a user-defined reproducing kernel Hilbert space. We show that our kernel-based approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel. The approach is flexible and allows a combination of informed structure through feature maps and closure terms via more general nonlinear terms in the kernel. We also derive a computable a posteriori error bound that combines standard error estimates for intrusive projection-based ROMs and kernel interpolants. In conclusion, the approach is demonstrated in several numerical experiments that include comparisons to operator inference using both proper orthogonal decomposition and quadratic manifold dimension reduction.

Data-driven model reduction↗
DOE OSTI2026

Quantifying Temperature Dependence of Pu(IV) Absorbance Spectra for Advanced Online Monitoring of Nuclear Processes

This article presents a systematic study of Pu(IV) absorbance spectral features as a function of temperature to develop an understanding of this parameter’s effect on chemometric models that can be used as online monitoring tools to support nuclear processing. The descriptive and predictive models that provide real-time feedback of these processes are usually constructed with data collected in conditions typical of a laboratory environment, which can differ drastically from a processing environment. To assess the impact of temperature on Pu(IV) absorbance spectra, 11 samples of Pu(IV) were synthesized with varying HNO 3 concentrations ranging from 0.6 to 9.5 M and heated between 15 and 45 °C. Ultraviolet (UV)–visible (vis)–near-infrared (NIR) absorption spectra collected at different HNO 3 concentrations and temperatures revealed that features associated with Pu(IV) are sensitive to temperature at all HNO 3 concentrations and that changes in features depend on HNO 3 concentration. The contributions of temperature and HNO 3 concentration to variation in Pu(IV) spectral features were evaluated using the principal component analysis of spectra that were baseline-corrected with an asymmetric least-squares method. Furthermore, predictive modeling for HNO 3 concentration with partial least-squares regression of UV–vis–NIR spectra highlighted the importance of accounting for temperature in the calibration set to optimize model performance. This methodology constitutes a new, systematic approach to account for the effect of temperature on the absorption spectra of metal ions and is useful for process monitoring applications in many industries.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1979

Optimal estimation of large structure model errors

In-flight estimation of large structure model errors is usually required as a means of detecting inevitable deficiencies in large structure controller/estimator models. The present paper deals with a least-squares formulation which seeks to minimize a quadratic functional of the model errors. The properties of these error estimates are analyzed. It is shown that an arbitrary model error can be decomposed as the sum of two components that are orthogonal in a suitably defined function space. Relations between true and estimated errors are defined. The estimates are found to be approximations that retain many of the significant dynamics of the true model errors. Current efforts are directed toward application of the analytical results to a reference large structure model.

Rodriguez, G.↗
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↗