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At least 19 records

Efficient CP Rounding Using Alternating Least Squares with QR Decomposition

The CANDECOMP/PARAFAC (CP) decomposition is widely used for analyzing multidimensional data, and the alternating least squares (CP-ALS) algorithm is a common method for its computation. CP rounding is the problem of computing a lower-rank CP decomposition of an input already in a higher-rank CP format. While the normal equations (NE) approach in CP-ALS is efficient for the CP rounding problem and frequently used, it becomes unstable in the presence of ill-conditioned subproblems. This paper presents a new QR-based CP-ALS method for CP rounding that preserves both numerical stability and computational efficiency. Here, our experiments show that the proposed method offers significant speedup over a previous QR-based approach and the Tensor Toolbox's NE-based implementation, particularly for higher-order tensors. Furthermore, our approach demonstrates a marked reduction in error for ill-conditioned problems, with error reductions several orders of magnitude smaller compared to the NE-based method, while achieving faster convergence and more accurate solutions. By using a more numerically stable approach, we can solve more problems in reduced working precision, which enables further reduction in time to solution.

CANDECOMP/PARAFAC↗

Use of Fisher's Ratio assisted multivariate curve resolution- alternating least squares for discovery-based analysis using ultrahigh pressure liquid chromatography-high resolution mass spectrometry

Non-targeted analysis of complex chemical mixtures can be difficult considering the convoluted nature of the matrix and the potential unknown chemical differences between samples or classes of samples. Ultrahigh pressure liquid chromatography coupled to quadrupole time-of-flight mass spectrometry (UHPLC-QTOF) is an ideal technique to probe chemical differences for a wide variety of samples. While UHPLC-QTOF can discover minute chemical differences down to low part per billion (ppb) concentrations with a high degree of confidence, the application of high-resolution mass spectrometry can yield massive amounts of information (∼ 10 gb per sample) that cannot be analyzed manually. Therefore, the application of chemometric techniques is mandatory for the interrogation of complex samples. Fisher's ratio (FR) assisted multivariate curve resolution-alternating least squares (MCR-ALS) was used to the discover and identify the chemical differences between two classes of materials: 1) a pond water matrix and 2) the matrix spiked with a pharmaceutical standard mix containing 17 compounds. Thirteen of the seventeen spiked compounds were discovered using FR analysis, and then five were successfully deconvoluted using MCR-ALS wherein the number of curves chosen were automatically determined using singular value decomposition (SVD). In conclusion, the use of an automated FR assisted MCR-ALS will aid in discovering trace levels of chemical components without the need for the researcher to provide potentially biased input which will aid in non-targeted workflow.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Quantifying the Impact of Metal Population Distribution in MFI-Supported Mo Catalysts for Methane Dehydroaromatization

Precise evaluation of intrinsic kinetic behavior in Mo/MFI catalysts for methane dehydroaromatization (MDA) is confounded by variations in Mo dispersion and speciation, which are influenced by metal loading and zeolite acidity. This work advances the utility of H 2 -temperature programmed reduction (H 2 -TPR) for characterizing Mo/MFI catalysts by enabling quantitative comparison of MoO x populations distinguished by reduction behavior and linked to initial catalytic performance. A comprehensive H 2 reduction pathway is established through systematic H 2 - TPR studies varying catalyst composition (1–10 wt % Mo, Si/Al = 15, 40, ∞), supplemented by UV-Raman spectroscopy, X-ray powder diffraction, N 2 physisorption, NH 3 -TPD, and advanced spectroscopic analysis (in situ XAS with principal component analysis/multivariate curve resolution—alternating least squares). Two low-temperature H 2 -TPR regions capture distinct Mo populations undergoing initial Mo(VI)→Mo(IV) reduction: a lower-temperature population (Mo-RI) associated primarily with highly dispersed, anchored MoO x species expected to predominantly reside within MFI channels, and a higher-temperature population (Mo-RII) corresponding to a broader set of MoO x species that becomes increasingly bulk-like/extrazeolitic at higher Mo loading. Quantification of these populations provides practical, kinetically relevant descriptors for comparing initial MDA rates across catalysts with varying Mo loading, Si/Al ratio, and MoO x heterogeneity. Application of lower-temperature Mo-RI estimates to kinetic measurements reveals a minimum threshold of ∼0.12 × 10 –3 mol Mo-RI/g cat , above which initial forward benzene rates normalized to this population converge despite differences in metal loading and zeolite Brønsted acidity. This threshold coincides with a transition toward a common C 2 -mediated benzene-forming regime, as indicated by approach-to-equilibrium analysis of methane-to-ethane, ethane-to-ethylene, and ethylene-to-benzene reaction steps. Above this threshold, normalized initial benzene rates are nearly invariant with increasing Mo-RII/Mo-RI population ratio, indicating that excess Mo-RII populations, including bulk-like/extrazeolitic MoO x domains present at higher loading, do not measurably suppress benzene formation associated with anchored and mostly channel-confined Mo population under the initial-rate conditions examined.

Mo/MFI↗

Low Speed, 2-D Rotor/Stator Active Noise Control at the Source Demonstration

Wake/blade-row interaction noise produced by the Annular Cascade Facility at Purdue University has been modeled using the LINFLO analysis. Actuator displacements needed for complete cancellation of the propagating acoustic response modes have been determined, along with the associated actuator power requirements. As an alternative, weighted least squares minimization of the total far-field sound power using individual actuators has also been examined. Attempts were made to translate the two-dimensional aerodynamic results into three-dimensional actuator requirements. The results lie near the limit of present actuator technology. In order to investigate the concept of noise control at the source for active rotor/stator noise control at the source, various techniques for embedding miniature actuators into vanes were examined. Numerous miniature speaker arrangements were tested and analyzed to determine their suitability as actuators for a demonstration test in the Annular Cascade Facility at Purdue. The best candidates demonstrated marginal performance. An alternative concept to using vane mounted speakers as control actuators was developed and tested. The concept uses compression drivers which are mounted externally to the stator vanes. Each compression driver is connected via a tube to an air cavity in the stator vane, from which the driver signal radiates into the working section of the experimental rig. The actual locations and dimensions of the actuators were used as input parameters for a LINFLO computational analysis of the actuator displacements required for complete cancellation of tones in the Purdue experimental rig. The actuators were designed and an arrangement determined which is compatible with the Purdue experimental rig and instrumentation. Experimental tests indicate that the actuators are capable of producing equivalent displacements greater than the requirements predicted by the LINFLO analysis. The acoustic output of the actuators was also found to be unaffected by the presence of air flow representative of the Purdue experimental rig. A test of the active noise control at the source concept for rotor/stator active noise control was demonstrated. This 2-D test demonstrated conclusively the simultaneous reduction of two acoustic modes. Reductions of over 10 dB were obtained over a wide operating range.

Simonich, John C.↗

Crop area estimation based on remotely-sensed data with an accurate but costly subsample

Research activities conducted under the auspices of National Aeronautics and Space Administration Cooperative Agreement NCC 9-9 are discussed. During this contract period research efforts are concentrated in two primary areas. The first are is an investigation of the use of measurement error models as alternatives to least squares regression estimators of crop production or timber biomass. The secondary primary area of investigation is on the estimation of the mixing proportion of two-component mixture models. This report lists publications, technical reports, submitted manuscripts, and oral presentation generated by these research efforts. Possible areas of future research are mentioned.

Gunst, R. F.↗

Active Thermography Based on Tensor Rank Decomposition

Principal Component Thermography applies Singular Value Decomposition (SVD) to post-process data that are derived from active thermographic inspections. SVD provides useful compression of the data and allows for better understanding of substructure and indications of potential damage. In the standard approach, SVD is applied to a certain reshaping of a three-dimensional data stack into a two-dimensional array. This work applies the CANDECOMP-PARAFAC (CP) tensor rank decomposition directly to the three-dimensional data to avoid the initial reshaping step in order to begin to develop an inspection method that can more accurately detect defects in non-homogeneous and anisotropic materials. Tests against simulated data that compare the CP decomposition method with traditional Principal Component Thermography based on SVD are described. Finally, the method of Proper Generalized Decomposition (PGD) is used to derive the CP decomposition, and its performance against other algorithms is also discussed.

Thermography↗

Sequential filter design for precision orbit determination and physical constant refinement

Earth-based spacecraft tracking data have historically been processed with classical least squares filtering techniques both for navigation purposes and for physical constant determination. The small, stochastic nongravitational forces acting on the spacecraft are described to motivate the use of sequential estimation as an alternative to the least squares fitting procedures. The stochastic forces are investigated both in terms of their effect on the tracking data and their influence on estimation accuracy. A flexible sequential filter design which leaves the existing trajectory, variational equations, data observable and partial computations undisturbed is described. A detailed filter design is presented that meets the precision demands and flexibility requirements of deep space navigation and of scientific problems.

Curkendall, D. W.↗

Exact spacecraft detumbling and reorientation maneuvers with gimbaled thrusters and reaction wheels

The equations of rotational motion for a spacecraft equipped with external jets and internal reaction wheels are shown to be feedback-equivalent to those of a linear system in attitude parameter space. Reorientation maneuvers are thereby formulated as linear optimal control problems with least mean square acceleration in attitude parameter space, solved in closed form and implementable either with internal or external torque commands, the choice depending on power and throttling requirements. For prior detumbling, an alternative solution with least mean square torque by angular momentum feedback is also given, that is implementable with gimbaled pairs of thrusters at constant throttle. Such a detumbling maneuver may then be followed by an acceleration-commanded rest-to-rest maneuver by means of the reaction wheels.

Dwyer, T. A. W., III↗

First-Order System Least-Squares for Second-Order Elliptic Problems with Discontinuous Coefficients

The first-order system least-squares methodology represents an alternative to standard mixed finite element methods. Among its advantages is the fact that the finite element spaces approximating the pressure and flux variables are not restricted by the inf-sup condition and that the least-squares functional itself serves as an appropriate error measure. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. Ellipticity of an appropriately scaled least-squares bilinear form of the size of the jumps in the coefficients leading to adequate finite element approximation results. The occurrence of singularities at interface corners and cross-points is discussed. and a weighted least-squares functional is introduced to handle such cases. Numerical experiments are presented for two test problems to illustrate the performance of this approach.

Manteuffel, Thomas A.↗

Towards Robust and Accurate Implicit Gradient Methods for Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Discretizations on Tetrahedral Grids

In this paper, we introduce implicit gradient methods as alternatives to conventional least-squares gradient methods for second- and third-order nodal-gradient cell-centered finite-volume discretizations, where solutions are stored at cells but gradients are stored at nodes. Because of the unique configuration of solutions and gradients, implicit gradient systems developed for the node-centered edge-based discretization method can be directly applied once the numerical solutions are interpolated from cells to nodes with sufficient accuracy. The resulting defect-correction solver can be loosely coupled with a flow-equation solver, and at convergence, solutions and gradients that satisfy the corresponding residual equations are obtained. Each iteration is relatively cheap compared with least-squares methods involving hundreds of neighbors. Numerical results are presented for accuracy verification studies and some simple but realistic flow problems.

Computational Fluid Dynamics↗

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Jiang, Bo-Nan↗

Standardising the “Gregory method” for calculating equilibrium climate sensitivity

The equilibrium climate sensitivity (ECS) – the equilibrium global mean temperature response to a doubling of atmospheric CO 2 – is a high-profile metric for quantifying the Earth system's response to human-induced climate change. A widely applied approach to estimating the ECS is the “Gregory method” (Gregory et al., 2004), which uses an ordinary least squares (OLS) regression between the net radiative flux, N, and surface air temperature anomalies, ΔT, from a 150 year experiment in which atmospheric CO 2 concentrations are quadrupled. The ECS is determined by extrapolating the linear fit to N=0, i.e. the ΔT-intercept, indicating the point at which the system is back in equilibrium. This method has been used to compare ECS estimates across the CMIP5 and CMIP6 ensembles and will likely be a key diagnostic for CMIP7. Despite its widespread application, there is little consistency or transparency between studies in how the climate model data is processed prior to the regression, leading to potential discrepancies in ECS estimates. We identify 32 alternative data processing pathways, varying by differences in global mean weighting, net radiative flux variable, anomaly calculation method, and linear regression fit. Using 44 CMIP6 models, we systematically assess the impact of these choices on ECS estimates and calculate uncertainty ranges using two bootstrap approaches. While the inter-model ECS range is insensitive to the data processing pathway, individual outlier models exhibit notable differences. Approximating a model's native grid cell area (if irregular) with cosine of the latitude can decrease the ECS by 11 %, the choice of N-variable can change the ECS by 6 %, and some anomaly calculation methods can introduce spurious temporal correlations in the processed data. Beyond data processing choices, we also evaluate an alternative linear regression method – total least squares (TLS) – which has a more statistically robust basis than OLS. However, for consistency with previous literature, and given TLS may reduce the ECS compared to OLS (by up to 24 %), thereby making a known bias in the Gregory method worse, we do not feel there is sufficient clarity to recommend a transition to TLS in all cases. To improve reproducibility and comparability in future studies, we recommend a standardised Gregory method: weighting the global mean by cell area, using the top of the atmosphere (as opposed to the top of model) N-variable, and calculating anomalies by first applying a rolling average to the preindustrial control timeseries then subtracting from the raw CO 2 quadrupling experiment. This approach accounts for model drift while reducing noise in the data to best meet the pre-conditions of the linear regression. While CMIP6 results of the multi-model mean ECS appear insensitive to these processing choices, similar assumptions may not hold for CMIP7, underscoring the need for standardised data preparation in future climate sensitivity assessments.

Geosciences↗

Binning of satellite magnetic anomalies

Crustal magnetic anomaly signals over satellite orbits were simulated to investigate numerical averaging as an anomaly estimator. Averaging as an anomaly estimator involves significant problems concerning spatial and amplitude smoothing of the satellite magnetic observations. The results of simulations suggest that the error of numerical averaging constitutes a small and relatively minor component of the total error-budget of higher orbital anomaly estimates, whereas for lower orbital estimates numerical averaging error increases substantially. As an alternative to numerical averaging, least-squares collocation was investigated and observed to produce substantially more accurate anomaly estimates, particularly as the orbital elevation of prediction was decreased towards the crustal sources. In contrast to averaging, collocation is a significantly more resource-intensive procedure to apply because of the practical, but surmountable problems related to establishing and inverting the covariance matrix for accurate anomaly prediction. However, collocation may be much more effectively used to exploit the anomaly details contained in the lower orbital satellite magnetic data for geologic analysis.

Goyal, H. K.↗

Vector potential methods

Vector potential and related methods, for the simulation of both inviscid and viscous flows over aerodynamic configurations, are briefly reviewed. The advantages and disadvantages of several formulations are discussed and alternate strategies are recommended. Scalar potential, modified potential, alternate formulations of Euler equations, least-squares formulation, variational principles, iterative techniques and related methods, and viscous flow simulation are discussed.

Hafez, M.↗

Statistical magnetic anomalies from satellite measurements for geologic analysis

The errors of numerically averaging satellite magnetic anomaly data for geologic analysis are investigated using orbital anomaly simulations of crustal magnetic sources by Gauss-Legendre quadrature integration. These simulations suggest that numerical averaging errors constitute small and relatively minor contributions to the total error-budget of higher orbital estimates (approx. 400 km), whereas for lower orbital estimates the error of averaging may increase substantially. Least-squares collocation is also investigated as an alternative to numerical averaging and found to produce substantially more accurate anomaly estimates as the elevation of prediction is decreased towards the crustal sources.

Goyal, H. K.↗