Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “numerical errors”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 739 records · Page 41
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Numerical and boundary condition effects on the prediction of detonation engine behavior using detailed numerical simulations

High-fidelity numerical simulations of an experimental rotating detonation engine with discrete fuel/air injection were conducted. A series of configurations with different feed-plenum pressures but with constant equivalence ratio were studied. Detailed chemical kinetics for the hydrogen/air system is used. A resolution study for the full rotating detonation engine (RDE) system simulation is also conducted. Two kinds of boundary conditions, a total pressure boundary and a constant mass flow rate boundary, are used to assess the effects of the inlet boundary. As mass flow rate is increased, the total pressure boundary causes more error in the axial pressure distribution while the constant mass flow rate gives a better solution for all cases ran. The simulations confirm experimental findings, and reproduce qualitative as well as some of the quantitative trends. These results demonstrate that a) fuel-air mixing is highly non-uniform within the detonation chamber, leading to variations in local equivalence ratio, b) the fuel and oxidizer injectors experience significant backflow as the detonation wave passes over, but recover at different rates which further augments the inefficiencies in mixing, and c) parasitic combustion in the mixing region makes the detonation wave weak by extending the reaction zone across the wave.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2023

Inversion-based correction of Double-Torsion (DT) subcritical crack growth tests for crack profile geometry

Because of its simplicity and the ability to produce a stable, slow-propagating crack, the Double-Torsion (DT) method has been used widely for investigating the critical and subcritical propagation of a slow-propagating tensile (mode-I) crack. However, to determine the complex relationship between the crack velocity $\mathcal{v_c}$ vs. the strain energy release rate $\mathscr{G}$ (or the stress intensity factor K) from laboratory measurements, several corrections must be made to account for the impact of sample and crack geometry. Particularly, DT test typically produces a crack with a curved edge profile instead of a straight line, causing the local $\mathcal{v_c}$ and $\mathscr{G}$ vary along the crack front. The experimentally measured $\mathcal{v_c}$ and $\mathscr{G}$ data merely reflect collective, averaged behavior of the crack. This makes inversion for the intrinsic, “true” crack growth kinetics necessary, based upon the knowledge of the crack geometry. Simple and effective correction methods have been proposed and validated for the slow, chemical-reaction-controlled part (Region I) of the $\mathcal{v_c}$-$\mathscr{G}$ curve. However, reliable methods for the highly nonlinear, transport-dominated part (Region II) and its sudden transition to the dynamic propagation part (Region III) are still lacking. Here we propose a method for determining the intrinsic $\mathcal{v_c}$-$\mathscr{G}$ relationship across all three Regions based upon DT test data, using a simple model function and its numerical inversion. The performance of this approach is examined and demonstrated using both synthetic and laboratory data for subcritical crack growth in soda lime glass.

42 ENGINEERING↗
DOE OSTI2022

Parametrized process characterization with reduced resource requirements

Quantum process tomography (QPT) is a powerful tool to characterize quantum operations, but it requires considerable resources, making it impractical for more than two-qubit systems. This study proposes an alternative approach that requires significantly fewer resources for unitary process characterization with a built-in method for state preparation and measurement error mitigation. By measuring the quantum process as rotated through the X and Y axes on the Bloch sphere, we can acquire enough information to reconstruct the quantum process matrix χ and measure its fidelity. We test the algorithm's performance against standard QPT using simulated and physical experiments on several IBM quantum processors and compare the resulting process matrices. We demonstrate with numerical experiments that the method can improve gate fidelity via a noise reduction in the imaginary part of the process matrix, along with a stark decrease in the number of experiments needed to perform the characterization.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Supervisory Control and Data Acquisition for Electrochemical Separation Experimentation

The Python-based program is a laboratory automation tool designed to control and monitor electrochemical systems. The tool was developed for capacitive deionization (CDI) experiments, but it can be used for any system that requires controlled voltage or current segments and multi-parameter monitoring. The program integrates hardware components to run user-defined experimental parameters, providing operational control of a programmable power supply, peristaltic pump, and data acquisition devices. Currently, the program is structured with a workflow that includes an initialization (or pre-run) phase, a main loop, and a post-experiment stabilization (or post-run) phase. The initialization phase prepares and stabilizes the cell, ensuring that the electrodes and solution reach a baseline state before the experiment begins. The main loop consists of multiple voltage segments that repeat, controlling the experiment while recording key parameters such as time, voltage, current, pH, and conductivity. Finally, the post-experiment stabilization phase allows the system to stabilize after the experiment, returning the cell and solution to equilibrium conditions before ending the sequence. The program is designed with four variations, each tailored to different experimental needs. All variations include both the initialization and post-experiment stabilization stages, which run for a set amount of time, voltage, current, and flow rate before and after the main experiment block. The main loop runs for a set number of cycles, as defined by the user input, and each cycle is composed of 2 or 4 segments. The 4 program variations are described as follows: Program 1: The main program includes 2 segments. Each segment is defined to have a set duration, flow rate, voltage, and current. This program measures conductivity, flow rate, voltage, and current. Program 2: The main program expands Program 1 to include 4 segments. Each segment has a specified duration, flow rate, voltage, and current. Like Program 1, it measures conductivity, flow rate, voltage, and current. Program 3: The main program consists of 2 segments, each defined by time, flow rate, voltage, and current. In addition to conductivity, flow rate, voltage, and current, Program 3 collects pH and temperature data through a 4-channel data acquisition device. Program 4: This program independently controls two channels of a multi-channel power supply simultaneously. While conductivity can only be measured for one cell at a time, the dual-channel control makes it possible to operate two cells simultaneously under different voltage/current conditions. The main program includes 2 segments.For each program, all measurements are automatically logged and integrated into a single Excel output file. Data are displayed in numerical format and plotted, both in real time, to track system performance. A key feature of the program is its ability to synchronize all outputs so that every measurement shares a single timestamp, ensuring accurate alignment of voltage, current, pH, conductivity, and pH data.By combining hardware control, real-time monitoring, and unified data collection, this program significantly reduces manual workload and minimizes errors, making it a reliable platform for researchers, engineers, and laboratory technicians conducting CDI experiments, among other electrochemical tests.

Valentino, Lauren [Argonne National Laboratory (AN↗
DOE OSTI2024

Globally optimal interferometry with lossy twin Fock probes

Parity or quadratic spin (e.g., J z 2 ) readouts of a Mach–Zehnder (MZ) interferometer probed with a twin Fock (TF) input state allow saturating the optimal sensitivity attainable among all mode-separable states with a fixed total number of particles but only when the interferometer phase θ is near zero. When more general Dicke state probes are used, the parity readout saturates the quantum Fisher information (QFI) at θ = 0, whereas better-than-standard quantum limit performance of the J z 2 readout is restricted to an o ( N ) occupation imbalance. We show that a method of moments readout of two quadratic spin observables J z 2 and J + 2 + J − 2 is globally optimal for Dicke state probes; i.e., the error saturates the QFI for all θ . In the lossy setting, we derive the time-inhomogeneous Markov process describing the effect of particle loss on TF states, showing that the method of moments readout of four at-most-quadratic spin observables is sufficient for globally optimal estimation of θ when two or more particles are lost. The analysis culminates in a numerical calculation of the QFI matrix for distributed MZ interferometry on the four-mode state | N 4 , N 4 , N 4 , N 4 〉 and its lossy counterparts, showing that an advantage for the estimation of any linear function of the local MZ phases θ 1 and θ 2 (compared to independent probing of the MZ phases by two copies of | N 4 , N 4 〉 ) appears when more than one particle is lost.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Longitudinal Phase Space Tomography for the Booster Synchrotron

Efforts in the study of the longitudinal behavior of charged particles in the Fermilab Booster can be catalyzed with an image of the two-dimensional phase space distribution. In the past, tomography had been employed in the reconstruction of the phase space in accelerators such as the Recycler at Fermilab and the Proton Synchrotron Booster at CERN. However, such a capability had yet to realize for the Fermilab Booster synchrotron. In this work, the first successful tomographic phase space reconstruction of a low-energy Booster bunch is presented along with validation metrics. A numerical turn-by-turn model of the longitudinal particle dynamics in the Booster has been implemented, which utilizes a fast, map-based particle transport algorithm. Using a sinogram generated from the Wall Current Monitor signal, the iterative reconstruction algorithm recovers a discretized image of the original phase space distribution at variable resolution. The reconstruction result shows low root-mean-square error and a rapid convergence toward the solution, providing strong evidence of accuracy. Future and ongoing work includes modeling high-energy bunches above transition and using tomography to infer certain machine parameters such as synchronous phase, peak gap voltage, and synchronous energy in addition to the phase space distribution.

Ebeid, Safi [Unlisted; Fermilab]↗
DOE OSTI2024

Online randomized interpolative decomposition with a posteriori error estimator for temporal PDE data reduction

Traditional low-rank approximation is a powerful tool for compressing large data matrices that arise in simulations of partial differential equations (PDEs), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. Here, to address these issues, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non-adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. A strategy to improve the accuracy of the reconstructed data gradient, when desired, within the ID framework is also presented. We provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.

Column subset selection↗
DOE OSTI2020

Real-time monitoring and prediction of water quality parameters and algae concentrations using microbial potentiometric sensor signals and machine learning tools

We report the overarching hypothesis of this study was that temporal microbial potentiometric sensor (MPS) signal patterns could be used to predict changes in commonly monitored water quality parameters by using artificial intelligence/machine learning tools. To test this hypothesis, the study first examines a proof of concept by correlating between MPS's signals and high algae concentrations in an algal cultivation pond. Then, the study expanded upon these findings and examined if multiple water quality parameters could be predicted in real surface waters, like irrigation canals. Signals generated between the MPS sensors and other water quality sensors maintained by an Arizona utility company, including algae and chlorophyll, were collected in real time at time intervals of 30 min over a period of 9 months. Data from the MPS system and data collected by the utility company were used to train the ML/AI algorithms and compare the predicted with actual water quality parameters and algae concentrations. Based on the composite signal obtained from the MPS, the ML/AI was used to predict the canal surface water's turbidity, conductivity, chlorophyll, and blue-green algae (BGA), dissolved oxygen (DO), and pH, and predicted values were compared to the measured values. Initial testing in the algal cultivation pond revealed a strong linear correlation (R 2 = 0.87) between mixed liquor suspended solids (MLSS) and the MPSs' composite signals. The Normalized Root Mean Square Error (NRMSE) between the predicted values and measured values were <6.5%, except for the DO, which was 10.45%. The results demonstrate the usefulness of MPSs to predict key surface water quality parameters through a single composite signal, when the ML/AI tools are used conjunctively to disaggregate these signal components. The maintenance-free MPS offers a novel and cost-effective approach to monitor numerous water quality parameters at once with relatively high accuracy.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

3D nanolithography with metalens arrays and spatially adaptive illumination

The growing demand for advanced materials, miniaturized devices and integrated microsystems calls for the reliable fabrication of complex, multiscale, three-dimensional (3D) architectures, a need increasingly addressed through light-based and laser-based processes. However, owing to the field-of-view (FOV) limitations of conventional imaging optics, existing 3D laser nanofabrication techniques face fundamental challenges in throughput, proximity error and stitching defects on the path to scaling. Here, in this study, we present a scalable 3D nanofabrication platform that uses a metalens-generated focal spot array to parallelize two-photon lithography (TPL) beyond centimetre-scale write field areas. Metalenses are ideally suited for producing submicron-scale focal spots for high-throughput nanolithography, as they uniquely feature large numerical apertures (NAs), immersion media compatibility and large-scale manufacturability. We experimentally demonstrate a printing system that uses a 12-cm 2 metalens array to produce more than 120,000 cooperative focal spots, corresponding to a throughput exceeding 10 8 voxels s −1 . By programmatically patterning the focal spot array using a spatial light modulator (SLM), an adaptive parallel printing strategy is developed for precise greyscale linewidth modulation and choreographed printing of semiperiodic and fully aperiodic 3D geometries. We demonstrate parallel printing of replicated microstructures (>50 M microparticles per day), centimetre-scale 3D architectures with feature sizes down to 113 nm, and photonic and mechanical metamaterials. This work demonstrates the potential of 3D nanolithography towards wafer-scale production, showing how TPL could be used at scale for applications in microelectronics, biomedicine, quantum technology and high-energy laser targets.

Materials science↗
DOE OSTI2023

Second release of the CoRe database of binary neutron star merger waveforms

Abstract We present the second data release of gravitational waveforms from binary neutron star (BNS) merger simulations performed by the Computational Relativity ( CoRe ) collaboration. The current database consists of 254 different BNS configurations and a total of 590 individual numerical-relativity simulations using various grid resolutions. The released waveform data contain the strain and the Weyl curvature multipoles up to ℓ = m = 4 . They span a significant portion of the mass, mass-ratio, spin and eccentricity parameter space and include targeted configurations to the events GW170817 and GW190425. CoRe simulations are performed with 18 different equations of state, seven of which are finite temperature models, and three of which account for non-hadronic degrees of freedom. About half of the released data are computed with high-order hydrodynamics schemes for tens of orbits to merger; the other half is computed with advanced microphysics. We showcase a standard waveform error analysis and discuss the accuracy of the database in terms of faithfulness. We present ready-to-use fitting formulas for equation of state-insensitive relations at merger (e.g. merger frequency), luminosity peak, and post-merger spectrum.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

An Algebraic Quantum Circuit Compression Algorithm for Hamiltonian Simulation

Quantum computing is a promising technology that harnesses the peculiarities of quantum mechanics to deliver computational speedups for some problems that are intractable to solve on a classical computer. Current generation noisy intermediate-scale quantum (NISQ) computers are severely limited in terms of chip size and error rates. Shallow quantum circuits with uncomplicated topologies are essential for successful applications in the NISQ era. In this work, based on matrix analysis, we derive localized circuit transformations to efficiently compress quantum circuits for simulation of certain spin Hamiltonians known as free fermions. The depth of the compressed circuits is independent of simulation time and grows linearly with the number of spins. The proposed numerical circuit compression algorithm behaves backward stable and scales cubically in the number of spins enabling circuit synthesis beyond O(10 3 ) spins. The resulting quantum circuits have a simple nearest-neighbor topology, which makes them ideally suited for NISQ devices.

Hamiltonian simulation↗
DOE OSTI2020

Improvements to CTF for Modeling of Boiling Water Reactor Geometry and Operating Conditions

This report documents work done to improve CTF modeling of boiling water reactor (BWR) designs during FY 2020. Several improvements were made to the geometry modeling capabilities, including adding support for modeling of inlet flow orifices, axially varying large water rods, channel boxes with large corner round radii, and quarter symmetry models. Work was also done to improve on the numerical stability and run-time performance of CTF in Virtual Environment for Reactor Applications (VERA) for core-scale BWR problems. An outer iteration loop was implemented that ensures mass and energy are conserved before moving on to the next time step in the solution. The ability to use a direct solver for the pressure matrix was also added, which reduces the matrix solution error and improves code stability and run-time performance. Improvements to the pressure balance iteration loop were needed to allow CTF to be used in a coupled VERA solution of a model of Hatch that was developed. Finally, various improvements have led to a 2×reduction in the total VERA walltime for the Hatch model.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2026

Development of local hybrid density functionals to treat self-interaction error and many-electron effects

The goal of this project was to improve density functional theory (DFT) for systems where conventional semilocal approximations are limited by self-interaction error (SIE) and by near-degeneracy or strong many-electron effects. While DFT remains the only broadly practical first-principles framework for large-scale materials simulations, its predictive accuracy is often challenged in situations involving stretched bonds, charge transfer, transition-metal chemistry, magnetic couplings, band gaps, and correlated electronic states. This project addressed these limitations by developing physically grounded and numerically robust exchange–correlation functionals that retain the efficiency of modern DFT while extending its predictive scope.

36 MATERIALS SCIENCE↗
DOE OSTI2022

A unified thermionic and thermionic-field emission (TE–TFE) model for ideal Schottky reverse-bias leakage current

We present a unified thermionic emission (TE) and thermionic-field emission (TFE) model for the ideal reverse-bias leakage current in Schottky junctions. The unified TE–TFE analytical model advances upon previous analytical TFE models by Murphy–Good and Padovani–Stratton, which are the two most widely adopted models by the community, in two major aspects: (i) the applicability of the TFE expression therein is extended to near-zero surface electric fields by an error-function correction, allowing for the calculation of the total current by a nontrivial sum of TE (over-the-barrier current) and TFE (below-the-barrier current) contributions; therefore, an accurate description of the TE-to-TFE transition region is captured analytically for the first time; (ii) image-force lowering is considered with much-simpler correction terms. Comparisons with the reference numerical model show that the unified TE–TFE model has excellent accuracy, as well as a 10 000× reduction in computation time. The unified model is further tested against experimental data from Schottky barrier diodes based on Si, 4H-SiC, GaN, and β-Ga 2 O 3 , revealing accurate extractions of barrier heights and correct descriptions of the ideal reverse leakage characteristics. With the extended applicable range, improved accuracy, and computational efficiency, the unified TE–TFE model is highly valuable for the design and analysis of devices based on Schottky junctions, as well as for potential integration in technology computer-aided design (TCAD) tools.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Toward ultra-efficient high-fidelity predictions of wind turbine wakes: Augmenting the accuracy of engineering models with machine learning

This study proposes a novel machine learning (ML) methodology for the efficient and cost-effective prediction of high-fidelity three-dimensional velocity fields in the wake of utility-scale turbines. The model consists of an autoencoder convolutional neural network with U-Net skipped connections, fine-tuned using high-fidelity data from large-eddy simulations (LES). The trained model takes the low-fidelity velocity field cost-effectively generated from the analytical engineering wake model as input and produces the high-fidelity velocity fields. The accuracy of the proposed ML model is demonstrated in a utility-scale wind farm for which datasets of wake flow fields were previously generated using LES under various wind speeds, wind directions, and yaw angles. Comparing the ML model results with those of LES, the ML model was shown to reduce the error in the prediction from 20% obtained from the Gauss Curl hybrid (GCH) model to less than 5%. In addition, the ML model captured the non-symmetric wake deflection observed for opposing yaw angles for wake steering cases, demonstrating a greater accuracy than the GCH model. The computational cost of the ML model is on par with that of the analytical wake model while generating numerical outcomes nearly as accurate as those of the high-fidelity LES.

Mechanics↗
DOE OSTI2021

Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a simple approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, nearly matching or even outperforming the best previous results. Our applications include: (i) a simulation of second-quantized plane-wave electronic structure, nearly matching the interaction-picture algorithm of Low and Wiebe; (ii) a simulation of $k$-local Hamiltonians almost with induced one-norm scaling, faster than the qubitization algorithm of Low and Chuang; (iii) a simulation of rapidly decaying power-law interactions, outperforming the Lieb-Robinson-based approach of Tran et al.; (iv) a hybrid simulation of clustered Hamiltonians, dramatically improving the result of Peng, Harrow, Ozols, and Wu; and (v) quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets, tightening previous analyses of Bravyi and Gosset. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound nearly matching a recent result of Tran et al. Our analysis reproduces known tight bounds for first- and second-order formulas. We further investigate the tightness of our bounds for higher-order formulas. For quantum simulation of a one-dimensional Heisenberg model with an even-odd ordering of terms, our result overestimates the complexity by only a factor of $5$. Our bound is also close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both the asymptotic scaling and the constant prefactor.

quantum computing, numerical analysis↗
DOE OSTI2022

Implementation of a Model Predictive Control Strategy to Regulate Temperature Inside Plug-Flow Solar Reactor With Countercurrent Flow

Abstract Solar-driven thermochemical energy storage systems are proven to be promising energy carriers (solar fuels) to utilize solar energy by using reactive solid-state pellets. However, the production of solar fuel requires a quasi-steady-state process temperature, which represents the main challenge due to the transient nature of solar power. In this work, an adaptive model predictive controller (MPC) is presented to regulate the temperature inside a tubular solar reactor to produce solid-state solar fuel for long-term thermal storage systems. The solar reactor system consists of a vertical tube heated circumferentially over a segment of its length by concentrated solar power, and the reactive pellets (MgMn2O4) are fed from the top end and flow downwards through the heated tube. A countercurrent flowing gas supplied from the lower end interacts with flowing pellets to reduce it thermochemically at a temperature range of 1000—1500 °C. A low-order physical model was developed to simulate the dynamics of the solar reactor including the reaction kinetics, and the proposed model was validated numerically by using a 7-kW electric furnace. The numerical model then was utilized to design the MPC controller, where the control system consists of an MPC code linked to an adaptive system identification code that updates system parameters online to ensure system robustness against external disturbances (sudden change in the flow inside the reactor), model mismatches, and uncertainty. The MPC controller parameters are tuned to enhance the system performance with minimum steady-state error and overshoot. The controller is tested to track different temperature ranges between 500 °C and 1400 °C with different particles/gas mass flowrates and ramping temperature profiles. Results show that the MPC controller successfully regulated the reactor temperature within ± 1 °C of its setpoint and maintained robust performance with minimum input effort when subjected to sudden changes in the amount of flowing media and the presence of chemical reaction.

Engineering↗
DOE OSTI2024

Learning Functions Varying along a Central Subspace

Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This paper studies regression of an s-Hölder function in $R^D$ which varies along a central subspace of dimension $d$ while $d \ll D$. A direct approximation of $f$ in $R^D$ with an accuracy $\varepsilon$ requires the number of samples in the order of $\varepsilon^{-(2s+D)/s}$. In this paper, we analyze the generalized contour regression (GCR) algorithm for the estimation of the central subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the central subspace, but its sample complexity is an open question. In this paper, we partially answer this questions by proving that if a variance quantity is exactly known, GCR leads to a mean squared estimation error of $O(n^{-1})$ for the central subspace. The estimation error of this variance quantity is also given in this paper. The mean squared regression error of $f$ is proved to be in the order of $(n/\log n)^{-\frac{2s}{2s+d}}$, where the exponent depends on the dimension of the central subspace instead of the ambient space . This result demonstrates that GCR is effective in learning the low-dimensional central subspace. We also propose a modified GCR with improved efficiency. Here, the convergence rate is validated through several numerical experiments.

97 MATHEMATICS AND COMPUTING↗