Accurate Error Estimates Research Articles

Evaluation of dual-weighted residual and machine learning error estimation for projection-based reduced-order models of steady partial differential equations

Projection-based reduced-order models (pROMs) show great promise as a means to accelerate many-query applications such as forward error propagation, solving inverse problems, and design optimization. In order to deploy pROMs in the context of high-consequence decision making, accurate error estimates are required to determine the region(s) of applicability in the parameter space. The following paper considers the dual-weighted residual (DWR) error estimate for pROMs and compares it to another promising pROM error estimate, machine learned error models (MLEM). In this paper, we show how DWR can be applied to ROMs and then evaluate DWR on two partial differential equations (PDEs): a two-dimensional linear convection–reaction–diffusion equation, and a three-dimensional static hyper-elastic beam. It is shown that DWR is able to estimate errors for pROMs extrapolating outside of their training set while MLEM is best suited for pROMs used to interpolate within the pROM training set.

State of energy estimation of the echelon-use lithium-ion battery based on Takagi–Sugeno fuzzy optimization

Owing to the degradation of an echelon-use lithium-ion battery (EULIB), the Ohmic internal resistance (OIR) and actual capacity (AE) have both changed greatly, and the state of energy (SOE) can more accurately represent the state of a EULIB than the state of charge (SOC) because of the working voltage. To improve the accuracy and adaptability of SOE estimation, in the paper, we study the energy state estimation of a EULIB. First, the four-order resistor–capacitance equivalent model of a EULIB is established, and an unscented transformation is introduced to further improve the estimation accuracy of the SOE. Second, a EULIB’s SOE is estimated based on adaptive unscented Kalman filter (AUKF), and the OIR and AE of a EULIB are estimated based on the AUKF. Third, a Takagi–Sugeno fuzzy model is introduced to optimize the OIR and AE of the EULIB, and the SOE estimation method is established based on an adaptive dual unscented Kalman filter (ADUKF). Through simulation experiments, verification, and comparison, energy decayed to 80%, 60%, and 40% of the rated energy, respectively, even with a large initial error; with the initial value of the SOE starting at 100%, 60%, or 20%, the estimated SOE can track the actual value. It can be seen that the method has a strong adaptive ability, and the estimation accuracy error is less than 1.0%, indicating that the algorithm has high accuracy. The method presented in this paper provides a new perspective for SOE estimation of EULIBs.

Machine Learning-Based Shape Error Estimation Using the Servomotor Current Generated During Micro-Milling of a Micro-Lens Mold

The demand for the mass production of micro-lens arrays (MLAs) is increasing. An MLA is fabricated through an injection molding process, and its mold is manufactured by a five-axis high-precision machine tool using a small diameter endmill. A visual examination is not available to judge the quality of the mold while machining. Therefore, an effective process monitoring technology must be developed. A promising approach is to apply a servomotor current to in-process monitoring because as long as the servomotor works well, no external sensors, capital investment, or maintenance processes are required. From this perspective, a machine learning-based shape error estimation method using only the servomotor current is proposed. To explore the relationship between the motor current generated during micro-milling and the shape error of the mold, the servomotor current in X-, Y-, and Z-axes was recorded, and the corresponding shape error of the MLA mold was measured after machining. Input data were prepared by converting time-domain servomotor current data to frequency-domain data using short-time Fourier transform and reducing the dimensions of the data via principal component analysis. In terms of a meaningful label for the output data, the average shape error in the machined area corresponding to each window was provided. The input/output relationships were used to train five different machine learning models, and the accuracy of shape error estimation using each model was evaluated. In addition, the estimation accuracies using the X-, Y-, and Z-axes were compared to find the axis that senses the shape error with the highest accuracy. The results show that the non-linear method using the X-axis servomotor current information closest to the machining point achieved the highest shape error estimation accuracy.

Time Parameter Mapping and Contour Error Precompensation for Multiaxis Input Shaping

When the trajectory with high speed and high acceleration contains frequency contents matching with the structure mode, the inertial vibration of the structure is easily excited, reducing machining accuracy and surface finish. Input shaping can avoid vibration, but it introduces time-delay that causes multiaxis trajectory distortion and affects the accuracy of contour error estimation. This article presents time parameter mapping and contour error precompensation for multiaxis input shaping to suppress vibration. The mapping of time parameter domain of trajectories is established by deducing the relation between the local curvature extremums of the trajectories before and after shaping, which provides reference points for contour error estimation. The actual trajectory is predicted by the closed-loop servo dynamics model and reference trajectory, and then the Newton iterative algorithm based on time parameter mapping is used to seek the contour error point. Finally, the calculated contour error is low-pass filtered and projected to each axis, and then the shaped trajectory is precompensated to simultaneously reduce the contour error and avoid structural vibration. Comparative experiments are conducted to validate the effectiveness of the proposed method. Experimental results illustrate that the proposed method not only suppresses structural vibration, but also significantly improves the estimation accuracy and contouring performance by time parameter mapping.

Error estimation for the time to a threshold value in evolutionary partial differential equations

We develop an a posteriori error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals of the solution. Taylor’s theorem and an adjoint-based a posteriori analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Coherent plug-and-play artifact removal: Physics-based deep learning for imaging through aberrations

Current image-correction frameworks for sensors that employ optically coherent detection attempt to estimate phase errors in the data, like those caused by aberrations, and simultaneously reconstruct digitally enhanced images. In practice, these frameworks struggle to account for the effects of speckle. To address this gap, we develop a novel image-correction framework referred to as Coherent Plug-and-Play Artifact Removal (CPnP-AR), which couples a neural-network despeckler with a physics-based measurement model. We also develop the experimental protocol needed to quantitatively evaluate performance relative to multiple state-of-the-art frameworks. The results show that CPnP-AR produces higher-quality images and more accurate estimates of the phase errors for a wide range of objects, specifically without the need for object-dependent parameter tuning. This improvement in overall robustness is a key step towards employing this novel image-correction framework for numerous applications of interest.

Confirmatory Factor Analyses in Psychological Test Adaptation and Development

Confirmatory Factor Analyses in Psychological Test Adaptation and Development

Five-Axis Contour Error Control Based on Spatial Iterative Learning

In this paper, a contour error control strategy based on spatial iterative learning control (sILC) is developed for repetitive processing of five-axis computer numerical control (CNC) machine tools. A curve approximation method with an adaptive moving window is developed to achieve accurate tool position and orientation contour error estimation, and a five-axis contour error control strategy based on sILC is proposed. The compensation method is derived using an sILC algorithm to modify the geometric reference path instead of modifying the controller. The experimental results show that the proposed control strategy reduces contour errors of five-axis CNC machine tools, and it outperforms the traditional tracking error control. Note to Practitioners—This paper aims to propose an effective five-axis contour error control scheme. At present, most of the five-axis contour error control methods rely on the modification of the controller, which is not allowed in commercial CNC systems. Therefore, we propose a five-axis contour error compensation algorithm based on sILC, which modifies the system’s reference path. Experimental results show that the proposed method can reduce the five-axis contour errors, while maintaining the machining efficiency.

Direction‐of‐arrival estimation based on difference‐sum co‐array of a special coprime array

Coprime array can provide high degrees of freedom by using difference–sum co-array under the same physical sensors, and can estimate the source under uncertain conditions. A special coprime array structure is proposed in this paper. Its difference–sum co-array can improve the degrees of freedom. Finally, the proposed array structure estimation is used for direction-of-arrival. Simulation results and analysis show that the proposed array structure can effectively improve the accuracy of direction-of-arrival estimation. Under the same signal-to-noise ratio and snapshot number, it has lower root mean square error and higher estimation accuracy than other methods.

A Deep Fourier Residual method for solving PDEs using Neural Networks

When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides with – or is equivalent to – the H−1-norm of the residual; however, it is often difficult to accurately compute it. This work assumes rectangular domains and proposes the use of a Discrete Sine/Cosine Transform to accurately and efficiently compute the H−1 norm. The resulting Deep Fourier-based Residual (DFR) method efficiently and accurately approximate solutions to PDEs. This is particularly useful when solutions lack H2 regularity and methods involving strong formulations of the PDE fail. We observe that the H1-error is highly correlated with the discretised loss during training, which permits accurate error estimation via the loss.

Factors influencing the temporal variability of atmospheric methane emissions from Upper Silesia coal mines: a case study from the CoMet mission

Abstract. Methane is a powerful greenhouse gas responsible for around 20 % of radiative forcing (relative to the pre-industrial era) caused by all long-lived greenhouse gases (WMO, 2021). About 60 % of the global emissions are from anthropogenic sources, and coal mining is one of the largest contributors. Emissions are either estimated by bottom-up approaches (based on inventories) or top-down approaches (based on atmospheric measurements). Combining those with an accurate error estimation allows us to better characterise model errors e.g. caused by transport mechanisms. Here we provide a detailed description of factors influencing the coal mine methane emission variability. We use high-frequency (up to hourly) temporal data from seven coal mines in the Upper Silesian Coal Basin during the Carbon dioxide and Methane (CoMet 1.0) mission from 14 May to 13 June 2018. Knowledge of these factors for the individual ventilation shaft is essential for linking the observations achieved during the CoMet 1.0 mission with models, as most publicly available data in the bottom-up worldwide inventories provide annual emissions only. The methane concentrations in examined shafts ranged from 0.10 % to 0.55 %±0.1 % during the study period. Due to the changing scope of mining works performed underground, they were subjected to a significant variation on a day-to-day basis. The yearly methane average emission rate calculated based on 1 month's set of temporal data of the analysed subset of mines was of the order of 142.68 kt yr−1 (σ=18.63 kt yr−1), an estimate 27 % lower than the officially published State Mining Authority (WUG) data and 36 % lower than reported to the European Pollutant Release and Transfer Register (E-PRTR). We also found that emissions from individual coal mine facilities were over- and underestimated by between 4 % to 60 %, compared to the E-PRTR, when short-term records were analysed. We show that the observed discrepancies between annual emissions based on temporal data and public inventories result from (1) the incorrect assumption that the methane emissions are time-invariant, (2) the methodology of measurements, and lastly, (3) the frequency and timing of measurements. From the emission monitoring perspective, we recommend using a standardised emission measurement system for all coal mines, similar to the Methane Fire Teletransmission Monitoring System (SMP-NT/A). Legal safety requirements require all coal mines to implement this system. After an adaptation, the system could allow for gas flow quantification, necessary for accurate and precise estimations of methane emissions at a high temporal resolution. Using this system will also reduce the emission uncertainty due to factors like frequency and timing of measurements. In addition, it would be beneficial to separately identify the emissions from individual ventilation shafts and methane drainage stations. That would bridge the gap between bottom-up and top-down approaches for coal mine emissions. The intermittent releases of unutilised methane from the drainage stations are currently not considered when constructing regional methane budgets.

Laser-Based Optical Wireless Communications for Internet of Things (IoT) Application

Internet of Things (IoT) is enabled by the integration of communication and sensor systems that are used to collect important information from objects around the world. In this article, we proposed the integration of laser-based optical wireless communication (OWC) with a fiber sensor system for IoT applications. OWC can be utilized in a harsh environment to send and receive communication and sensor signals in a wireless means. Moreover, fiber sensors are important to achieve real-time, accurate, and smart monitoring in IoT applications. As compared to mechanical and electrical sensors, optical fiber sensors have numerous benefits, including the tolerance to electromagnetic interference (EMI), small size, lightweight, high bandwidth, low cost, provide distributed sensing, high sensitivity, and electronic devices are not required at the sensor point. Moreover, we proposed a deep variational autoencoder (DVAE) model to estimate the strain changes and peak wavelength of multiple fiber Bragg grating (FBG) sensors using only the spectrum of FBGs obtained from the real experiment. The result showed that our proposed DVAE model can estimate the strain changes and peak wavelength of FBGs with low root mean squared error (RMSE) and high estimation accuracy. Furthermore, the performance of the proposed integration of OWC with the fiber sensor system achieves a clear eye diagram and excellent bit error rate (BER) curve.

A vision-based error identification method for separating geometric errors of rotating axes in five-axis platforms

In this paper, a programmable identification method for visually measuring and identifying position-independent and position-dependent geometric errors (PIGEs and PDGEs) of rotary axes in compact five-axis platforms is presented. Firstly, the kinematic model of the five-axis platform is established via screw algebraic theory. Then, considering the measurement data diversity, the orientation errors identification mechanism for PIGEs is proposed. The mechanism optimally integrates the coordinate search and the least-squares method. Additionally, a least-squares sample regression model is developed to identify the positional deviation of the rotation axis. After identifying PIGEs, the PDGE sources are identified based on the right multiplication principle. Furthermore, the error estimation accuracy of the proposed method is investigated. Finally, a five-axis dispensing platform is employed to validate the identification accuracy and evaluate its feasibility.

Robust error assessment for reduced order vibro-acoustic problems

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region.

An Estimation and Compensation Method for Motion Trajectory Error in Bistatic SAR

Bistatic synthetic aperture radar (BiSAR) has drawn increasing attention in recent studies benefiting from its ability for forward-looking imaging, its capability of receiver radio silence and its resistance to jamming. However, the motion trajectory error compensation of BiSAR is a challenging task due to multiple error sources and complex effects. In this paper, an estimation and compensation method for three-dimensional (3D) motion trajectory error of BiSAR is proposed. In this method, the Doppler error of multiple scattering points is estimated firstly by using the time–frequency analysis method. Next, a local autofocus process is introduced to improve the Doppler error estimation accuracy. Then, the 3D trajectory error of BiSAR is estimated by solving a series of linear equations of the trajectory error and the Doppler error with the least squares method, and a well-focused BiSAR image is produced by using the corrected 3D trajectories. Finally, simulation and experiment results are presented to demonstrate the effectiveness of the proposed method.

VECBEE: A Versatile Efficiency–Accuracy Configurable Batch Error Estimation Method for Greedy Approximate Logic Synthesis

Approximate computing is an emerging strategy to improve the energy efficiency of many error-tolerant applications. To design an approximate circuit automatically, many approximate logic synthesis (ALS) methods have been proposed, among which many are greedy. To improve the synthesis quality of these greedy methods, one key is to calculate the errors of all candidate approximate transformations accurately. However, the traditional simulation-based method is time consuming. Instead, many existing methods just perform quick but inaccurate error estimation. In this work, to improve both the accuracy and runtime of error estimation, we propose VECBEE, a versatile efficiency–accuracy configurable batch error estimation method for greedy ALS. It is based on Monte Carlo simulation and an efficient technique to capture whether a signal change due to an introduced approximation will be propagated to each primary output. VECBEE is generally applicable to any statistical error measurement, such as error rate and average error magnitude, and any graph-based circuit representation. It allows a flexible tradeoff between the error estimation accuracy and the runtime, while even the fully accurate version is much faster than the traditional simulation-based method. We apply VECBEE to two representative greedy ALS methods and demonstrate its effectiveness in generating better approximate circuits. The code of VECBEE is made open source.

An optimally accurate second-order time-stepping algorithm for the nonstationary magneto-hydrodynamics equations

In this paper, we study an optimally accurate second-order time-stepping algorithm for nonstationary magneto-hydrodynamics equations. The algorithm deals with a linear treatment for the nonlinear terms in both momentum equations and magnetic equations, and adds some stabilization terms of the discrete solutions for the velocity, the pressure and the magnetic. We derive that this algorithm satisfies unconditionally stable and optimally accurate error estimate. Numerical tests are given to validate the predicted convergence rate.

Exercising choice over feedback schedules during practice is not advantageous for motor learning.

The idea that there is a self-controlled learning advantage, where individuals demonstrate improved motor learning after exercising choice over an aspect of practice compared to no-choice groups, has different causal explanations according to the OPTIMAL theory or an information-processing perspective. Within OPTIMAL theory, giving learners choice is considered an autonomy-supportive manipulation that enhances expectations for success and intrinsic motivation. In the information-processing view, choice allows learners to engage in performance-dependent strategies that reduce uncertainty about task outcomes. To disentangle these potential explanations, we provided participants in choice and yoked groups with error or graded feedback (Experiment 1) and binary feedback (Experiment 2) while learning a novel motor task with spatial and timing goals. Across both experiments (N = 228 participants), we did not find any evidence to support a self-controlled learning advantage. Exercising choice during practice did not increase perceptions of autonomy, competence, or intrinsic motivation, nor did it lead to more accurate error estimation skills. Both error and graded feedback facilitated skill acquisition and learning, whereas no improvements from pre-test performance were found with binary feedback. Finally, the impact of graded and binary feedback on perceived competence highlights a potential dissociation of motivational and informational roles of feedback. Although our results regarding self-controlled practice conditions are difficult to reconcile with either the OPTIMAL theory or the information-processing perspective, they are consistent with a growing body of evidence that strongly suggests self-controlled conditions are not an effective approach to enhance motor performance and learning.

Spatiotemporal estimation of model error to improve soil moisture analysis in ensemble Kalman filter data assimilation

As a key variable in the surface hydrological cycle and energy balance, soil moisture is often analyzed using data assimilation to improve precise and high spatiotemporal resolution from multisource data. However, accurate estimation of model error in soil moisture data assimilation is not sufficient in spatial and temporal dimensions. This paper proposes a spatiotemporal error estimation method that integrates the spatial and temporal dimensions of the study area into model error estimation. The in-situ data was regarded as the “true value” for the temporal dimensions model error estimation. The triple collocation (TC) technique was used to estimate the spatial error of the model. Based on the aforementioned estimation, a linear method to construct the relationship between spatial and temporal information was proposed to fuse the spatiotemporal model error. Result from the proposed method was compared against the overall model error estimation and spatial estimation methods in ensemble Kalman filter (EnKF) data assimilation. The ERA-Interim reanalysis data and in-situ soil moisture data from Washington State in 2016 were used to evaluate the performance of EnKF data assimilation. Using statistical metrics, including the root mean square error (RMSE), the mean bias error (MBE), and the Pearson correlation coefficient (i.e., R-value), the results show that our proposed approach, which uses spatiotemporal approach to model error estimation in EnKF data assimilation, yielded more accurate soil moisture estimates. When ERA-Interim reanalysis soil moisture data was used as the reference data, compared with the overall model error estimation EnKF Data Assimilation Experiment, RMSE and MBE of spatiotemporal model error estimation EnKF Data Assimilation Experiment were reduced by 0.0062 and 0.0061 m3 / m3, respectively, and R remained basically unchanged. When using in-situ measurements as reference data, RMSE and MBE decreased by 0.0034 and 0.0038 m3 / m3, respectively, and R increased by 0.0365. Compared with the other estimation approaches in the experiments, the spatiotemporal error estimation approach greatly improved the estimation accuracy of soil moisture analysis, yielding the closest values and variation trends with the in-situ measurements. Fully considering the temporal and spatial variation of model error and improving the estimation accuracy of model error can provide more accurate model information for data assimilation. The proposed method and framework in this paper can be used in model error estimation and data assimilation, particularly in the fields of numerical weather forecasting, natural environment monitoring, and geographical environment research.

Estimating a Causal Exposure Response Function with a Continuous Error-Prone Exposure: A Study of Fine Particulate Matter and All-Cause Mortality.

Numerous studies have examined the associations between long-term exposure to fine particulate matter (PM2.5) and adverse health outcomes. Recently, many of these studies have begun to employ high-resolution predicted PM2.5 concentrations, which are subject to measurement error. Previous approaches for exposure measurement error correction have either been applied in non-causal settings or have only considered a categorical exposure. Moreover, most procedures have failed to account for uncertainty induced by error correction when fitting an exposure-response function (ERF). To remedy these deficiencies, we develop a multiple imputation framework that combines regression calibration and Bayesian techniques to estimate a causal ERF. We demonstrate how the output of the measurement error correction steps can be seamlessly integrated into a Bayesian additive regression trees (BART) estimator of the causal ERF. We also demonstrate how locally-weighted smoothing of the posterior samples from BART can be used to create a more accurate ERF estimate. Our proposed approach also properly propagates the exposure measurement error uncertainty to yield accurate standard error estimates. We assess the robustness of our proposed approach in an extensive simulation study. We then apply our methodology to estimate the effects of PM2.5 on all-cause mortality among Medicare enrollees in New England from 2000-2012.