Error Estimation Process Research Articles

Error transport equations implementation for discontinuous Galerkin methods

The procedure for performing higher-order discretization error estimation for Discontinuous Galerkin (DG) methods using Error Transport Equations (ETE) is presented in this work. The results obtained using the linearized ETE and the nonlinear ETE are compared for several steady state inviscid test cases containing completely smooth cases and cases with singularities or discontinuities. Prior work with the linearized version of ETE achieved improved results by using a relinearization process. This process is generalized to the nonlinear version of ETE and is more appropriately termed iterative correction when applied to both the linear and nonlinear versions. For smooth test cases, the discretization error estimates show expected order of accuracy which is higher than the order of accuracy of the primal solution. This holds for both the linearized and the nonlinear ETE. Discontinuities and singularities in the solution cause trouble in the discretization error estimation process. Discussion is made on the behavior of the ETE error estimates for non-smooth test cases to figure out potential solutions to the existing problems.

An improved coastline inflection method for geolocation correction of Microwave Radiation Imager data

ABSTRACT Geolocation accuracy is an important factor affecting the application of microwave remote sensing data. For effective application of microwave radiation data in quantitative remote sensing, geolocation errors should be corrected. The coastline inflection method (CIM) is widely used for geolocation correction of microwave radiometer data, which consists of the inflection point identification process and the error estimation process. The detected coastline that is composed of the brightness temperature inflection points can be identified in the first process, and the error between the detected coastline and the true coastline can be estimated in the second process. However, both processes will be affected when the coastline is complex or disturbed by some clouds, rain, etc. Therefore, we improve the CIM in its two processes. Firstly, a method based on the dynamic threshold is used to identify inflection points. Secondly, a new geolocation error estimation method combining the Iterative Closest Point (ICP) algorithm and the Kernel Correlation (KC) algorithm is proposed to estimate geolocation errors. By applying the proposed method, the FengYun-3C (FY-3C) Microwave Radiation Imager (MWRI) dataset selected in the specified regions from 1 April 2019 to 1 October 2019 is processed. For the data with complex coastline structure or some interference factors, compared with the CIM based on a surface fitting interpolation model and ICP algorithm, the number of inflection points is increased by about 70%, and the mean geolocation errors in cross- and along-track directions are reduced by about 24.12% and 21.61%, respectively. For the data with simple coastline structure and less interference, the two methods have similar effects. For all selected data, the mean geolocation errors in cross- and along-track directions is reduced from [0.5376, 0.5528] pixels to [0.1581, 0.1110] pixels. Mean geolocation accuracy is improved by approximately 75%.

An Efficient Multichannel SAR Channel Phase Error Calibration Method Based on Fine-Focused HRWS SAR Image Entropy

Azimuth multichannel synthetic aperture radar (SAR) is a useful approach to solving minimum antenna area constraint and realizing high resolution and wide swath (HRWS) imaging by multichannel signal reconstruction. However, channel phase error will significantly degrade the reconstruction performance and leads to azimuth ghost. This article presents an efficient minimum entropy channel error estimation method based on fine-focused SAR image. The multichannel SAR images are obtained by using Range-Doppler imaging algorithm applied for preprocessed data of each channel. Then, the reconstruction and the residual range cell migration and residual second range compression compensation are performed to achieve multichannel fine-focused SAR images. After overlapping these multichannel fine-focused SAR images and iteratively calibrating the channel phase error, fine-focused HRWS SAR image can be achieved. Therefore, the SAR image used in the channel error estimation process can directly obtain the fine focused HRWS SAR image after minimum entropy iteration without additional imaging operations. Compared with the minimum entropy method by using the coarse-focused SAR image based on azimuth deramp, redundant reconstruction, and imaging operations are avoided. Thus, error estimation and imaging processing procedures are optimized. Processing efficiency improvement are analyzed, and simulation and acquired data processing verify the effectiveness of the proposed method.

Turboelectric Uncertainty Quantification and Error Estimation in Numerical Modelling

Turboelectric systems can be considered complex systems that may comprise errors and uncertainty. Uncertainty quantification and error estimation processes can, therefore, be useful in achieving accurate system parameters. Uncertainty quantification and error estimation processes, however, entail some stages that provide results that are more positive. Since accurate approximation and power optimisation are crucial processes, it is essential to focus on higher accuracy levels. Integrating computational models with reliable algorithms into the computation processes leads to a higher accuracy level. Some of the current models, like Monte Carlo and Latin hypercube sampling, are reliable. This paper focuses on uncertainty quantification and error estimation processes in turboelectric numerical modelling. The current study integrates the current evidence with scholarly sources to ensure the incorporation of the most reliable evidence into the conclusions. It is evident that studies on the current subject began a long time ago, and there is sufficient scholarly evidence for analysis. The case study used to obtain this evidence is NASA N3-X, with three aircraft conditions: rolling to take off, cruising and taking off. The results show that the electrical elements in turboelectric systems can have decent outcomes in statistical analysis. Moreover, the risk of having overload branches is up to 2% of the total aircraft operation lifecycle, and the enhancement of the turboelectric system through electrical power optimisation management could lead to higher performance.

Wiener-based smoother and predictor for massive-MIMO downlink system under pilot contamination

One of the challenges in massive-MIMO system is pilot contamination during the channel estimation process. Pilot contamination can cause error or inaccurate channel estimation process for future fifth generation (5G) downlink transmissions. This paper considers using a Wiener-based filter to smooth and predict the channel estimation to reduce the pilot contamination for more accurate CSI during channel estimation. The simulation results show that the Wiener-based smoothing and predicting technique reduces the effect of pilot contamination and increases the accuracy of CSI during channel estimation process. Wiener smoother (WS) is implemented based on Wiener-based filtering technique. The previous estimated CSI and weight coefficient vector are used to smooth the current estimated CSI by using block data formulation to reduce the effect of pilot contamination. However, WS technique suffers from pilot contamination due to pilot training. This motivates the development of two Wiener predictors (WP), known as WP1 and WP2. The WP1 and WP2 run a prediction technique for CSI and number of pilot training during the prediction period, which is missing from the original WS. Comparison results show that the proposed WS and WP outperforms the conventional minimum mean square error and least square, in terms of channel estimation error and per-cell rate. WP2 perform better than WS and WP1 because of the algorithm complexity that required more information to be updated, stored and processed for prediction. Thus, WP2 requires large computation and matrix operation compared to WS and WP1. The results indicate that the channel estimation error due to pilot contamination can be reduced by using the Wiener-based approaches.

ERRORS IN QUANTITIES OF INTEREST IN THE LAMINATED PLATE BENDING PROBLEM USING HIERARCHIC SETS OF BASIS FUNCTIONS IN GFEM

Abstract. A formulation for error estimation is developed for the bending problem of composite laminated plates based on the Mindlin-Reissner kinematic model discritized by the Generalized Finite Element Method (GFEM). The error estimation process starts with an upper bound in energy norm, which is obtained following the basic CRE (Constitutive Relation Error) framework of the Ladev`eze formulation, that is, the estimate is obtained from a statically admissible stress field computed at element level in a Neumann problem where the element boundary forces are equilibrated. The authors have previously shown that an accurate description of the in plane stresses in a laminate is essential to obtain an accurate approximation to the transverse shear stresses at the layers interfaces. Since important failure modes in laminated composite plates, like the delamination, are linked to the transverse stresses, it is essential to develop both, accurate post-processing procedures to compute improved transverse stresses, and also estimate techniques for the discretization errors. The first condition is adequately satisfied by GFEM. Therefore, the aim of the present work is to extend the general CRE technology to develop formulations to estimation of errors in Quantity of Interest (QI) identified preferably with the stress field in the laminated plate problem. One of the steps necessary in the CRE procedure is the computation of and admissible stress field in each element, in a Neumann problem where the boundary forces have been previously equilibrated. For a GFEM basis with high order enrichment, adequate procedures have to be sought. Here we use one single higher order finite element, based on displacement FEM, to obtain an approximation to the equilibrated field. The formulation is implemented for arbitrary degree of the basis, which allows an arbitrarily close approximation to the equilibrium condition. The sharpness of the QI’s error bounds is increased with the accuracy of the primal and dual global energy norm of errors. In the present work we investigate the effectiveness of a local GFEM p-enrichment as a tool to improve the approximability of the model in capturing the local gradients which characterizes response of the dual loading. The GFEM p-enrichment is implemented in a simple and straightforward way, as opposed to some other possible forms of enrichment, e.g. local h-refinement or a sub-domain approach. Numerical tests are performed to asses the effect of the different parameters in the modeling over the errors in the quantities of interest. Keywords: Strict error bounds estimation, Generalized Finite Element Method, Admissible stress field, Element Equilibrated Technique, Laminated plate Mindlin model.

Adaptive polynomial rendering

In this paper, we propose a new adaptive rendering method to improve the performance of Monte Carlo ray tracing, by reducing noise contained in rendered images while preserving high-frequency edges. Our method locally approximates an image with polynomial functions and the optimal order of each polynomial function is estimated so that our reconstruction error can be minimized. To robustly estimate the optimal order, we propose a multi-stage error estimation process that iteratively estimates our reconstruction error. In addition, we present an energy-preserving outlier removal technique to remove spike noise without causing noticeable energy loss in our reconstruction result. Also, we adaptively allocate additional ray samples to high error regions guided by our error estimation. We demonstrate that our approach outperforms state-of-the-art methods by controlling the tradeoff between reconstruction bias and variance through locally defining our polynomial order, even without need for filtering bandwidth optimization, the common approach of other recent methods.

On the generalization of the error and error estimation process of Ortiz’s recursive formulation of the tau method

In this paper, the generalization of the Lanczos–Ortiz’s Recursive formulation of the tau method for general non-overdetermined ordinary differential equations is presented. The generalization of the canonical polynomials and their derivatives for both overdetermined and non-overdetermined cases were reported in the earlier works of these authors, thus the emphasis here is on the error and the error estimate procedures. The accuracy of the results were established using some numerical examples and the induction principle.

An improved motion compensation method for high resolution UAV SAR imaging

The polynomial and sinusoidal motion errors always exist in the unmanned aero vehicle (UAV) SAR due to the small size and low velocity of the platform, causing serious spectrum compressing/stretching and significant spectral replicas of the azimuth signal. The motion errors induce serious blurring of the SAR image and “ghost targets”, and can hardly be precisely estimated by the conventional motion compensation (MOCO) method. In this paper, an improved MOCO method is proposed to estimate and eliminate the motion errors in the high resolution UAV SAR without high precision inertial navigation system (INS) data. The time domain range walk correction (RWC) operation in the coarse phase error estimation process of the proposed MOCO method is the key operation that ensures the estimation accuracy of the whole MOCO method. Finally, the validity of the improved MOCO method is verified by computer simulations and real UAV SAR data processing.

A-posteriori error estimation for second order mechanical systems

One important issue for the simulation of flexible multibody systems is the reduction of the flexible bodies degrees of freedom. As far as safety questions are concerned knowledge about the error introduced by the reduction of the flexible degrees of freedom is helpful and very important. In this work, an a-posteriori error estimator for linear first order systems is extended for error estimation of mechanical second order systems. Due to the special second order structure of mechanical systems, an improvement of the a-posteriori error estimator is achieved. A major advantage of the a-posteriori error estimator is that the estimator is independent of the used reduction technique. Therefore, it can be used for moment-matching based, Gramian matrices based or modal based model reduction techniques. The capability of the proposed technique is demonstrated by the a-posteriori error estimation of a mechanical system, and a sensitivity analysis of the parameters involved in the error estimation process is conducted.

Experimental study of bound and autoionizing Rydberg states of the europium atom

An isolated-core-excitation (ICE) scheme and stepwise excitation are employed to study the highly excited states of the europium atom. The bound europium spectrum with odd parity in a region of 42400–43500 cm−1 is measured, from which spectral information on 38 transitions, such as level position and relative intensity, can be deduced. Combined with information about excitation calibration and the error estimation process, the selection rules enable us to determine the possible values of total angular momentum J for the observed states. The autoionization spectra of atomic europium, belonging to the 4f76pnl (l = 0, 2) configurations, are systematically investigated by using the three-step laser resonance ionization spectroscopy (RIS) approach. With the ICE scheme, all the experimental spectra of the autoionizing states have nearly symmetric profiles whose peak positions and widths can be easily obtained. A comparison between our results and those from the relevant literature shows that our work not only confirms many reported states, but also discovers 14 bound states and 16 autoionizing states.

Accuracy Enhancement for Precision Angle Measuring Structures

Producing high-quality optical limbs, circular scales, rasters, multibit code disks (key metrological elements hereinafter called “angle measuring structures” (AMS) is an important engineering problem for any manufacturer, making angle measuring gauges. The analysis of the precision parameters of the AMS that were made by the method of circular scanning has revealed that the distorted circular trajectory of the recording laser beam moving across the surface of a semimanufactured article to be the main source of errors in the AMS produced. The main reason causing the distortion has been the rotor imbalance in the spindle unit of the laser generator of images. The article offers solutions for accuracy enhancement in manufacturing precision angle measuring structures. It also describes error estimation process in the raster, manufactured upon implementing the solutions offered.

An algorithm for estimation and separation of ephemeris and clock errors in SBAS

The estimation and separation of ephemeris and clock errors is an integral part of a SBAS (Space Based Augmentation System). Generally, the global solution is based on the full state approach for satellite errors (ephemeris and clock) and station errors, using a large least square estimator; or the other way is to sequentially estimate the ephemeris and clock through a Kalman filter, using a complex model of the satellite dynamics. In this paper, the estimation and separation of ephemeris and clock errors is addressed through a unique approach of combining both the methods. The algorithm employs measurements, which are pre-processed for various errors and known biases. A single difference technique is used to separately estimate the ephemeris and clock components. The ephemeris Kalman filter uses a priori information of ephemeris errors along with measurements through a minimum variance estimator to provide ephemeris error estimate. A similar approach is adopted in the clock error estimation process, to provide clock and clock rate estimates. The algorithm results are presented using simulated data for known errors in ephemeris/clock and subsequent retrieval. This algorithm estimates these errors as corrections to the broadcast Global Positioning System (GPS) navigation data, required by a SBAS user for accuracy improvement.

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM.

A fully digital fast convergence algorithm for nonlinearity correction in multistage ADC

This paper describes a fast digital background calibration algorithm for pipeline or cyclic analog-to-digital converters. The proposed method corrects for the finite gain/bandwidth of the interstage operational amplifier and for capacitor mismatch in multiplying digital-to-analog converters stages. The new algorithm, fast gain error correction, converges more than 100 times faster than other correlation-based correction techniques presented in literature. The high speed of convergence of the error estimation is due to the use of a polynomial interpolation that cancel the interference in the error estimation process caused by the input signal. This new method does not need any added analog circuitry, does not perturb the output samples, and requires only a digital finite-impulse response filter to implement the polynomial interpolation as opposed to existing fast converging correlation techniques.

A Representation Theorem for the Error of Recursive Estimators

The ultimate objective of this paper is to develop new techniques that can be used for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems. For this we need new technical tools similar to those used in [L. Gerencser, Statist. Plann. Inference, 41 (1994), pp. 303-325]. The immediate technical objective is to extend the previous technical results to the Djereveckii--Fradkov--Ljung scheme with enforced boundedness. Our starting point is a standard approximation of the estimation error used in the asymptotic theory of recursive estimation. Tight control of the difference between the estimation error and its standard approximation, referred to as residuals, is a crucial point in our applications. The main technical advance of the present paper is a set of strong approximation theorems for three closely related recursive estimation algorithms in which, for any $q \ge 1$, the $L_q$-norms of the residual terms are shown to tend to zero with rate $N^{-1/2-\varepsilon}$ with some $\varepsilon > 0$. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencser, Systems Control Lett., 21 (1993), pp. 347-351]. Two useful corollaries will be derived. In the first a standard transform of the estimation-error process for the basic recursive estimation method, Algorithm CR\@, will be shown to be $L$-mixing, while in the second the asymptotic covariance matrix of the estimator for the same method will be given. Applications to multivariable adaptive prediction and the minimum-variance self-tuning regulator for ARMAX systems will be described.

"Split ADC" architecture for deterministic digital background calibration of a 16-bit 1-MS/s ADC

Self-calibration in approximately 10 000 conversions is demonstrated in a 16-bit, 1-MS/s algorithmic analog-to-digital converter (ADC). Continuous digital background calibration is enabled by introduction of a "split ADC" architecture, in which the die area of a single ADC design is split into two independent converters, each converting the same input signal. The two independent outputs are averaged to produce the ADC output code. The difference of the two outputs provides information for the background calibration process. Since both ADCs convert the same input, when correctly calibrated their outputs should be equal and the difference should be zero. Any nonzero difference provides information to an error estimation process which adjust calibration parameters in each ADC. For the specific realization of an algorithmic ADC described in this paper, a multiple residue mode amplifier is used to ensure different decision trajectories and provide valid calibration information. The analog sub-system of the ADC is implemented in 0.25-/spl mu/m CMOS, consumes 105 mW, and has a die size of 1.2 mm /spl times/ 1.4 mm.

On error estimation and adaptive refinement for element free Galerkin method: Part I: stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the element free Galerkin method (EFGM) for the solution of 2D elastostatic problem is suggested. This refinement procedure is based on the well-known Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation and a simple point refinement scheme for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the EFGM. A comprehensive study on the theories and natures of some commonly used recovery schemes for the EFGM is given. The essential features and relationships among the studied recovery schemes are highlighted. It will be demonstrated that all these recovery schemes are particular cases of a general recovery scheme called the weighed continuous moving-least-square over stationary least square (MLS-SLS) recovery scheme. A numerical experience has also been carried out to study the performances of a number of different MLS-SLS recovery procedures generated by applying different fitting parameters to the continuous MLS-SLS recovery scheme. In Part II of the study, attention will be focused on the mesh refinement and the development of the adaptive algorithms for the proposed automatic adaptive EFGM procedure.

Enhancement and Error Estimation of Neural Network Prediction of Niño-3.4 SST Anomalies

Abstract A procedure to enhance neural network (NN) predictions of tropical Pacific sea surface temperature anomalies and calculating their estimated errors is presented. A simple linear correction enables more accurate predictions of warm and cold events but can result in introduction of larger errors in other cases. The prediction error estimates aid recognizing erroneously magnified anomalies and are used to sort the predictions into El Nino, La Nina, and neutral states. The error estimation process is based on bootstrap resamplings of the data and construction of a large number of bootstrap prediction replicas. A statistic calculated on the set of bootstrap replicas that corresponds to each of the actual predictions is used to estimate the prediction’s errors. The method is demonstrated on NN prediction of the Nino-3.4 index.

Selective regional refinement procedure for adaptive finite element analysis

An alternative approach to carry out adaptive finite element analysis based on the concept of selective refinement is proposed to enhance the computational efficiency of the adaptive refinement procedure. In this approach only elements that are inside some selected refinement regions will be refined, and all remaining elements will be reused for the next cycle of analysis. The definition of the refinement region can simply be determined from the refinement ratios of the elements which are calculated in the conventional adaptive procedures, and hence the suggested technique can easily be incorporated into any conventional adaptive refinement scheme. Numerical experiments show that by refining elements only within the selected zones, significant saving in CPU time is possible in the mesh generation and the error estimation processes. Furthermore, in comparing with the traditional adaptive refinement procedures, the efficiency of the new scheme is enhanced such that equally high rate of convergence and solution accuracy is maintained using similar degrees of freedom, but at a reduced total CPU time.