Normal Equations Research Articles

Novel kinematic and geometric views for improving tooth contact analysis of spatial gears

AbstractTooth contact analysis (TCA) has been widely applied to evaluate the working performance of gear pairs. TCA is often formulated with five unknowns and five independent scalar equations. The solution process involves a global optimization problem with strong nonlinearity and numerical instability, especially for spatial gears with complicated tooth geometries. This study proposes novel kinematic and geometric views of gearing that reveal insights into the meshing process of spatial gears. One unknown can be removed from the position and normal equations of the TCA formulation. To solve the remaining four unknowns, a simplified optimization model with two unknowns is proposed, and the other two unknowns are obtained by using geometric iterative methods or directly from explicit expressions in some cases. A general algorithm was developed to solve the simplified TCA. The test results of both the spiral bevel and face gear drives validate the effectiveness of the proposed method.

A stable block adjustment method without ground control points using bound constrained optimization

ABSTRACT Block adjustment is a key technology for achieving large-scale accurate mapping of space via remote sensing. When block adjustment without ground control points (GCPs) is conducted, the adjustment model has large condition number of design matrix, and an ill-conditioned normal equation, which eventually leads to abnormal convergence of direct least squares solution. In this study, we investigated block adjustment without GCPs based on Rational Function Model (RFM), and applied image space affine model of RFM as the systematic error compensation model of single scene imagery. The value range of each parameter of the affine model was determined by calculating the inverse projection errors of tie points and the adjustment model parameters were solved by the optimization method with attached inequalities constraint. Finally, the abnormal convergence problem was solved by reducing the degree of freedom of convergence. Comparative tests of block adjustment without GCPs using various methods were conducted using multiple regional imagery from Ziyuan-3 (ZY-3), Tianhui-1 (TH-1), Luojia-1 (LJ-1) and Gaofen-3 (GF-3). The results show that the proposed method can achieve rapid and correct convergence, and has strong practicability.

Reference Values of Noninvasive Myocardial Work Indices Measured by Echocardiography in Healthy Children.

BackgroudNoninvasive myocardial work, estimated by left ventricular (LV) pressure-strain loop (PSL), has been introduced for assessing LV myocardial performance. Based on both blood pressure and speckle-tracking derived strain data, noninvasive myocardial work is considered to be less load-dependent than global longitudinal strain (GLS). In some conditions, such as hypertension or aortic coarctation, the increased afterload will affect strain measurements, and myocardial work can serve as a more robust metric.ObjectiveWe prospectively recruited healthy children to explore the relationship between myocardial work indices and body size parameters, and to determine the reference values of noninvasive myocardial work indices in healthy children.Methods183 healthy children (aged 1–18 years, males: 52.5%) were enrolled in the study. Global work index (GWI), global constructive work (GCW), global wasted work (GWW), global work efficiency (GWE), were assessed by LVPSL and compared according to age and sex.ResultsThe mean for GWI was 1,448.7 ± 265.0 mm Hg%, 1,859.8 ± 290.7 mm Hg% for GCW, and the median (interquartile range) for GWW was 54.0 (33.0–82.0) mm Hg% and 97.0 (95.0–99.0) % for GWE. male had greater GWI and GCW) than female (1,572.5 ± 250.2 mm Hg% vs. 1,312.2 ± 208.7 mm Hg% and 1,944.3 ± 299.2 mm Hg% vs. 1,766.6 ± 251.5 mm Hg%, respectively, all P < 0.001). GWI and GCW were significantly correlated with baseline parameters, including age, height, weight, BSA, body mass index, heart rate, and blood pressure. After indexed to BSA, GWI (BSA), GCW (BSA) remained significantly negatively correlated with age (P < 0.001).Conclusionswe proposed the normal reference values and regression equations for GWI and GCW based on age and BSA in healthy children. This might provide a basis of reference for the evaluation of cardiac function in children with cardiopulmonary disease.

2-cyclic splitting for mixed-valued least squares in engineering

We consider the least squares minimization problem in which both complex- and real-valued parameters are simultaneously present. A prominent example of this is the estimation of the frequency response function (FRF) in the presence of missing output data. In this case, the FRF parameters are complex-valued, while the missing output samples are real-valued. In the extended local polynomial method (ELPM), the missing samples are treated as global variables and are estimated simultaneously with the FRF parameters under the least squares criterion. This returns a mixed-valued least squares problem. We provide a formal setting, in which we show mixed-valued least squares problems are well-defined and have a unique solution. We introduce an iterative method for solving the resulting system of equations, which splits the complex-valued and real-valued least squares designs. We show that the resulting method is algebraically equivalent to applying a 2-cyclic matrix splitting to the mixed-valued normal equations, which ensures its convergence and provides an additional adaptive scheme to improve the convergence speed. Finally, we conduct a detailed case-study on the ELPM and compare our method to the original ad-hoc method. We present impressive improvements to the computation time by exploiting the separate physical structures within both regression matrices.

Comparison of non-tidal loading data for application in a secular terrestrial reference frame

The Deutsches Geodätisches Forschungsinstitut der Technischen Universität München (DGFI-TUM) is one of the three Combination Centres of the International Earth Rotation and Reference Systems Service for the International Terrestrial Reference System (ITRS). In its upcoming realization of the ITRS, the DTRF2020, DGFI-TUM will again correct for non-tidal loading (NTL) effects at the normal equation level. Next to the dedicated NTL data set for the ITRS 2020 realization provided by the Global Geophysical Fluid Center (GGFC), we also considered the data provided by the Earth System Modelling group of the Deutsches GeoForschungsZentrum (ESMGFZ). Besides also comprising all NTL components (atmospheric, oceanic, hydrological) and being mass conserving, the ESMGFZ data has the advantage of daily availability and is already in use at DGFI-TUM. The decision for one or the other data set depends on their suitability for a secular terrestrial reference frame like the DTRF2020, which will be assessed in this work. Although we also compare the site displacements induced by NTL to the residuals of station positions of the Global Navigation Satellite Systems, we will not evaluate the quality of the underlying geophysical models per se. The two data sets differ w.r.t. the underlying hydrological models and the treatment of non-tidal oceanic loading, but the most relevant difference is given in terms of trends in the displacement time-series. After a close investigation of the latter, we finally decided to apply the GGFC contribution to the ITRS 2020 realization in the DTRF2020.Graphical

A new parallel algorithm for improving the computational efficiency of multi-GNSS precise orbit determination

The computational efficiency is critical with the increasing number of GNSS satellites and ground stations since many unknown parameters must be estimated. Although only active parameters are kept in the normal equation in sequential least square estimation, the computational cost for parameter elimination is still a heavy burden. Therefore, it is necessary to optimize the procedure of parameter elimination to enhance the computational efficiency of GNSS network solutions. An efficient parallel algorithm is developed for accelerating parameter estimation based on modern multi-core processors. In the parallel algorithm, a multi-thread guided scheduling scheme, and cache memory traffic optimizations are implemented in parallelized sub-blocks for normal-equation-level operations. Compared with the traditional serial scheme, the computational time of parameter estimations can be reduced by a factor of three due to the new parallel algorithm using a six-core processor. Our results also confirm that the architecture of computers entirely limits the performance of the parallel algorithm. All the parallel optimizations are also investigated in detail according to the characteristics of CPU architecture. This gives a good reference to architecture-oriented parallel programming in the future development of GNSS software. The performance of the multi-thread parallel algorithm is expected to improve further with the upgrade of new multi-core coprocessors.s

Алгоритм усреднения приращений координат и их ковариационных матриц при повторных ГНСС-измерениях

An algorithm for averaging part of GNSS measurements information obtained at different times and belonging to the same baseline is given. The averaging extends to the increments of such lines’ coordinates and the accompanying covariance matrix of the increments. It is implemented using the algorithm of the classic parametric version of data LS-optimization. The design matrix of the measured increasing is built with positive single blocks size 3×3; their total number is equal to that of the baseline observation sessions. Solving the system of relevant normal equations through the method of conversion, size 3×3, gives the vector of average increments of the base line coordinates. The reverse matrix of its normal equation ratios is an averaged covariance one of average increments (system roots). Processing observational materials is carried out in the Excel electronic computational table using a synthesized version of the parametric one of the data LS-optimization, taking into account the errors of the reference points’ coordinates. The two-way processing-determined coordinates and their RMSE (root mean square error) were identical both before and after averaging.

A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations

Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to achieve good convergence. In recent years, a number of preconditioners have been proposed. These are largely serial, and reported results demonstrate that none of the commonly used preconditioners for the normal equations matrix is capable of solving all sparse least-squares problems. Our interest is thus in designing new preconditioners for the normal equations that are efficient and robust and can be implemented in parallel. Our proposed preconditioners can be constructed efficiently and algebraically without any knowledge of the problem and without any assumption on the least-squares matrix except that it is sparse. We exploit the structure of the symmetric positive definite normal equations matrix and use the concept of algebraic local symmetric positive semidefinite splittings to introduce two-level Schwarz preconditioners for least-squares problems. The condition number of the preconditioned normal equations matrix is shown to be theoretically bounded independently of the number of subdomains in the splitting. This upper bound can be adjusted using a single parameter $\tau$ that the user can specify. We discuss how the new preconditioners can be implemented on top of the PETSc library using only 150 lines of Fortran, C, or Python code. Problems arising from practical applications are used to compare the performance of the proposed new preconditioner with that of other preconditioners.

Depth-based sparse bundle adjustment.

It is demonstrated in this paper that due to error model inconsistency, a certain degree of accuracy loss would be incurred to the estimated parameters when the traditional bundle adjustment method is directly applied to the scenario where a fraction of observations is implicitly error free (e.g., the reference image points in commonly used least squares matching refinement). To this end, a depth-based object point model and corresponding depth-based sparse bundle adjustment method are proposed in this paper, in which the position of an object point is represented by its 1D depth relative to its reference image. A corresponding projection model is derived, the sparse block structures of normal equations are studied depending on whether there are shared image parameters to be optimized or not, and corresponding sparse solutions of the normal equations and parameter covariance matrices are derived. Compared with the traditional sparse bundle adjustment method, simulated experiments demonstrate that our method matches the error model of the target scenario, and thus can avoid further accuracy loss. Moreover, both simulated and real data experiments demonstrate that our method can effectively improve computational efficiency.

GMRES methods for tomographic reconstruction with an unmatched back projector

Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they are caused by the need for fast algorithms that best utilize the computer hardware, and it is an interesting and challenging task to develop fast and easy-to-use algorithms for these cases. Our approach is to use preconditioned GMRES, in the form of the AB- and BA-GMRES algorithms, to handle the unmatched normal equations associated with an unmatched pair. These algorithms are simple to implement, they rely only on computations with the available forward and back projectors, and they do not require the tuning of any algorithm parameters. We show that these algorithms are equivalent to well-known LSQR and LSMR algorithms in the case of a matched projector. Our numerical experiments demonstrate that AB- and BA-GMRES exhibit a desired semi-convergence behavior that is comparable with LSQR/LSMR and that standard stopping rules work well. Hence, AB- and BA-GMRES are suited for large-scale CT reconstruction problems with noisy data and unmatched projector pairs.

The method of spectral analysis of the determination of random digital signals

The methods of spectral analysis based on the use of any model to describe the signal are considered in the article, while using them some assumptions about the behavior of the signal outside the observation interval are made. The task of spectral analysis or evaluation, in this case, is to find the parameters of the model used, which is selected based on the available a priori information about the studied process.A new spectral analysis method is proposed, which uses the partially classical Prony method, and this method has been improved by replacing the damping sine wave with the use of damped sine wave.Replacing damped sine wave with the use of non-damped sine wave, allows you to very accurately isolate the signal and determine its characteristics against the background of very rich in interference with air space, against the background of radio devices that work legally. For the first time, a fast conversion algorithm was applied to solve the normal equations for finding variables to sequentially determine the parameters of random short-term signals such as amplitude, frequency, and phase. It is suggested to determine not only static parameters but also the rate of change of these parameters. Speed of change of parameters allows determining more carefully the signal of the means of silent receiving of information.Modeling of processes of determination of random short-term pulses simulating digital signals of the means of silent receiving of information is carried out, on the basis of the proposed method of spectral analysis, the simulation results are presented in the form of three-dimensional graphs.The main difference is the use not only of the analysis of the amplitude, frequency, phase, and spectrum of the signal but the main analysis of the spectral density of the signal.Analyzes of the simulation result fully confirm the advantages of the proposed method for the determination of random short-term pulses.

Tangential elastic contact model of the cylindrical contact with different axis crossing angles.

In this paper, a cylinder contact model with different axis crossing angles is proposed, and the axis crossing angle is introduced into the normal and tangential constitutive equations. This paper uses the method of formula derivation to establish the analytical solution of the tangential elastic contact of the cylinder with different axis cross angles, and gives the relational expression of the tangential load and the tangential displacement. The finite element method (FEM) is used to verify the tangential displacement load curve and the tangential stiffness curve of the two cylinders. The influence of various factors on mechanical properties such as tangential stiffness, tangential displacement and contact area is explored, and the distribution of stress and strain on the contact surface is analyzed. The research results show that the tangential displacement and tangential stiffness of the cylindrical contact model correspond well to the simulation results.

Cuckoo Search Combined with PID Controller for Maximum Power Extraction of Partially Shaded Photovoltaic System

In the case of partial shading conditions (PSCs), normal equations cannot be completely implemented. The mathematical model of the Photovoltaic (PV) array needs to be modified and re-established with the existence of bypass diodes connected to the PV module, which can alleviate the negative effects of the PSCs and generate several peaks on the PV output characteristics curve. The first aim of this study is to modify and re-establish the mathematical model of the PV array under PSCs. Second, it aims to improve and validate the reliable Cuckoo Search Algorithm (CSA) by integrating it with PID (hybrid CSA-PID) to diminish the impact of PSCs problems. The hybrid CSA-PID was proposed to both track the global maximum power point (GMPP) of PV systems and reduce the tracking time to eliminate the fluctuations around the GMPP. Further, the PID controller was used to eliminate the error percentage obtained by CSA under PSCs to generate the required duty cycle, which provides the required and desired maximum voltage accordingly. The proposed CSA-PID technique has been implemented using both Matlab/Simulink and Hardware-In-Loop experiments on the MT real-time control platform NI PXIE-1071. For validation, the Hybrid CSA-PID method is evaluated and compared with CSA, modified particle swarm optimization (MPSO), PSO, and modified perturb and observe (MP&amp;O) methods under similar conditions. Finally, the obtained findings demonstrated the efficacy and superiority of the proposed hybrid CSA-PID technique, demonstrating its resilience, fast reaction, and good performance in terms of tracking time and GMPP tracking.

A novel model to optimize multiple imputation algorithm for missing data using evolution methods

The concept of missing data is considered significant when applying statistical methods to a dataset and the quality of the data analysis results is based on the correct data completeness. As a result, improving missing data filling processes is vital in order to give more reliable data throughout the phase of analysis. Here, we present a novel method for optimizing multiple regression imputation processes and obtaining the best fitness values for missing data from patients by combining multiple imputations with a genetic algorithm. To train and assess our proposed method, we employed 583 patient records from a publicly available database, divided into 416 records of liver patients and 167 records of the non-liver patients. The proposed approach offers the largest improvement for missing data findings, according to the results. Instead of employing the normal equation in multiple imputations, which yielded 92.72 as the utmost fitness value with Mean Absolute Error (MAE) 0.5877 from 1.1840 after our second optimization, we were able to achieve a fitness value of 233. The proposed approach might be tested using a large database and used in Hepatocellular carcinoma (HCC) labs to help clinicians make accurate diagnoses.

A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.

Computational Optimization of the 3D Least-Squares Matching Algorithm by Direct Calculation of Normal Equations

3D least-squares matching is an algorithm that allows to measure subvoxel-precise displacements between two data sets of computed tomography voxel data. The determination of precise displacement vector fields is an important tool for deformation analyses in in-situ X-ray micro-tomography time series. The goal of the work presented in this publication is the development and validation of an optimized algorithm for 3D least-squares matching saving computation time and memory. 3D least-squares matching is a gradient-based method to determine geometric (and optionally also radiometric) transformation parameters between consecutive cuboids in voxel data. These parameters are obtained by an iterative Gauss-Markov process. Herein, the most crucial point concerning computation time is the calculation of the normal equations using matrix multiplications. In the paper at hand, a direct normal equation computation approach is proposed, minimizing the number of computation steps. A theoretical comparison shows, that the number of multiplications is reduced by 28% and the number of additions by 17%. In a practical test, the computation time of the 3D least-squares matching algorithm was proven to be reduced by 27%.

A Statistical Interpolation Code for Ocean Analysis and Forecasting

Abstract We present a data assimilation package for use with ocean circulation models in analysis, forecasting, and system evaluation applications. The basic functionality of the package is centered on a multivariate linear statistical estimation for a given predicted/background ocean state, observations, and error statistics. Novel features of the package include support for multiple covariance models, and the solution of the least squares normal equations either using the covariance matrix or its inverse—the information matrix. The main focus of this paper, however, is on the solution of the analysis equations using the information matrix, which offers several advantages for solving large problems efficiently. Details of the parameterization of the inverse covariance using Markov random fields are provided and its relationship to finite-difference discretizations of diffusion equations are pointed out. The package can assimilate a variety of observation types from both remote sensing and in situ platforms. The performance of the data assimilation methodology implemented in the package is demonstrated with a yearlong global ocean hindcast with a 1/4° ocean model. The code is implemented in modern Fortran, supports distributed memory, shared memory, multicore architectures, and uses climate and forecasts compliant Network Common Data Form for input/output. The package is freely available with an open source license from www.tendral.com/tsis/.

DTRF2014: DGFI-TUM’s ITRS realization 2014

The DTRF2014 solution is the latest ITRS realization of DGFI-TUM computed in the framework of ITRF2014. It is mainly characterized by two innovations: it is the first secular ITRS realization considering non-tidal loading corrections derived from geophysical models. Secondly, the DTRF2014 release includes all information necessary to approximate the observed instantaneous station positions at any epoch of the observation time span. Therefore, besides the classical SINEX and EOP files, the applied non-tidal loading corrections, the residual time series of station positions as well as the translation time series of the DTRF2014 origin are provided. The DTRF2014 is computed from the same input as ITRF2014 comprising the full history of observation data of the four space geodetic techniques Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), Global Navigation Satellite System (GNSS), and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS). The DTRF2014 solution is computed using the DGFI-TUM approach based on the combination of normal equation systems. With reference to the IERS Conventions, the origin of DTRF2014 is realized from SLR input data only. The scale of the DTRF2014 solution is assumed to be statistically equal for SLR and VLBI. This assumption is based on tests performed in the framework of the DTRF2014 computation showing a range for the scale offset at epoch 2000.0 of ±3.3 mm. For the GNSS and the DORIS subnetwork of the DTRF2014 solution, a weighted mean scale of SLR and VLBI is realized. Cross-validations performed between DTRF2014 and ITRF2014 show the high consistency of present ITRS realizations. For GNSS, VLBI and SLR the obtained transformation parameters range within ±1.7 mm for positions (except for the VLBI and SLR scale and the VLBI z-translation) and ±0.3 mm/yr for the velocities, the parameters for DORIS are all below 4.2 mm and 0.25 mm/yr (except for the scale rate). The RMS of the transformations, reflecting the agreement of network geometry, range between 0.55 mm for VLBI and 2.25 mm for DORIS. The RMS of station velocities is between 0.08 and 0.62 mm/yr. The DTRF2014 solution provides a high precision and reliability confirmed by various validations performed internally and by external groups. For instance, applications of DTRF2014 for a precise orbit determination (POD) of the altimetry satellite TOPEX and Jason-2 show small radial crossfit residuals between SLR and DORIS and indicate a good performance of DTRF2014.

Toward an Optimal Selection of Constraints for Terrestrial Reference Frame (TRF)

Given that the observations from current space geodetic techniques do not carry all the necessary datum information to realize a Terrestrial Reference System (TRS), and each of the four space geodetic techniques has limits, for instance: Very Long Baseline Interferometry (VLBI) ignores the center of mass and satellite techniques lack the TRS orientation, additional constraints have to be added to the observations. This paper reviews several commonly used constraints, including inner constraints, internal constraints, kinematic constraints, and minimum constraints. Moreover, according to their observation equations and normal equations, the similarities and differences between them are summarized. Finally, we discuss in detail the influence of internal constraints on the scale of VLBI long-term solutions. The results show that there is a strong correlation between the scale parameter and the translation parameter introduced by the combination model at the Institut National de l’Information Géographique et Forestière (IGN), and internal constraints force these two groups of parameters to meet certain conditions, which will lead to the coupling of scale and translation parameters and disturbing the scale information in VLBI observations. The minimum or kinematic constraints are therefore the optimum choices for TRF.

Elastic least-squares reverse time migration of steeply dipping structures using prismatic reflections

The migration of prismatic reflections can be used to delineate steeply dipping structures, which is crucial for oil and gas exploration and production. Elastic least-squares reverse time migration (ELSRTM), which considers the effects of elastic wave propagation, can be used to obtain reasonable subsurface reflectivity estimations and interpret multicomponent seismic data. In most cases, we can only obtain a smooth migration model. Thus, conventional ELSRTM, which is based on the first-order Born approximation, considers only primary reflections and cannot resolve steeply dipping structures. To address this issue, we develop an ELSRTM framework, called Pris-ELSRTM, which can jointly image primary and prismatic reflections in multicomponent seismic data. When Pris-ELSRTM is directly applied to multicomponent records, near-vertical structures can be resolved. However, the application of imaging conditions established for prismatic reflections to primary reflections destabilizes the process and leads to severe contamination of the results. Therefore, we further improve the Pris-ELSRTM framework by separating prismatic reflections from recorded multicomponent data. By removing artificial imaging conditions from the normal equation, primary and prismatic reflections can be imaged based on unique imaging conditions. The results of synthetic tests and field data applications demonstrate that the improved Pris-ELSRTM framework produces high-quality images of steeply dipping P- and S-wave velocity structures. However, it is difficult to delineate steep density structures because of the insensitivity of the density to prismatic reflections.