Normal Equations Research Articles

Flatness constraints in the estimation of GNSS satellite antenna phase center offsets and variations

Accurate information on satellite antenna phase center offsets (PCOs) and phase variations (PVs) is indispensable for high-precision geodetic applications. In the absence of consistent pre-flight calibrations, satellite antenna PCOs and PVs of global navigation satellite systems are commonly estimated based on observations from a global network, constraining the scale to a given reference frame. As part of this estimation, flatness and zero-mean conditions need to be applied to unambiguously separate PCOs, PVs, and constant phase ambiguities. Within this study, we analytically investigate the impact of different boresight-angle-dependent weighting functions for PV minimization, and we compare antenna models generated with different observation-based weighting schemes with those based on uniform weighting. For the case of the GPS IIR/-M and III satellites, systematic differences of 10 mm in the PVs and 65 cm in the corresponding PCOs are identified. In addition, new antenna models for the different blocks of BeiDou-3 satellites in medium Earth orbit are derived using different processing schemes. As a drawback of traditional approaches estimating PCOs and PVs consecutively in distinct steps, it is shown that different, albeit self-consistent, PCO/PV pairs may result depending on whether PCOs or PVs are estimated first. This apparent discrepancy can be attributed to potentially inconsistent weighting functions in the individual processing steps. Use of a single-step process is therefore proposed, in which a dedicated constraint for PCO-PV separation is applied in the solution of the normal equations. Finally, the impact of neglecting phase patterns in precise point positioning applications is investigated. In addition to an overall increase of the position scatter, the occurrence of systematic height biases is illustrated. While observation-based weighting in the pattern estimation can help to avoid such biases, the possible benefit depends critically on the specific elevation-dependent weighting applied in the user’s positioning model. As such, the practical advantage of such antenna models would remain limited, and uniform weighting is recommended as a lean and transparent approach for the pattern estimation of satellite antenna models from observations.

Method for verifying solutions of sparse linear systems with general coefficients

This paper proposes a verification method for sparse linear systems Ax=b with general and nonsingular coefficient matrices. A verification method produces the error bound for a given approximate solution. Practical methods use one of two approaches. One approach is to verify the computed solution of the normal equation ATAx=ATb by exploiting symmetric and positive definiteness; however, the condition number of ATA is the square of that for A. The other approach applies an approximate inverse of A; however, the approximate inverse of A may be dense even if A is sparse. Additionally, several other methods have been proposed; however, they are considered impractical due to various issues. Here, this paper provides a computing method for verified error bounds using the previous verification method and the latest equilibration. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, we will show the efficiency of an iterative refinement method to obtain accurate solutions.

Statistical characteristics of realistic fiber misalignments of unidirectional composites: Fitting distributions and scanning length effects

This paper presents a comprehensive study on the statistical characteristics of angle, tortuosity, curvature and wave magnitude to deepen the understanding of realistic fiber misalignments within unidirectional composites. A feasible fiber path reconstruction procedure has been optimized, which can be applicable to other types of composites. The high-resolution micrographs are acquired through X-ray computed tomography. The individual fiber segmentation is implemented using a U-Net deep learning method, and the fiber trajectories are reconstructed with the aid of a tracing algorithm. The stepped fiber trajectories are slightly smoothed and a polynomial fitting formula is adopted to quantitatively describe the fiber paths. The statistical characteristics corresponding to the differential tortuosity, misalignment angle, spatial curvature and wave magnitudes are comprehensively analyzed, with emphasis on their fitting distributions and scanning length effects. The collected data indicate that the statistical distributions of differential tortuosity and angle, curvature, and wave magnitude can be well fitted by normal, lognormal and Weibull equations, respectively. Particularly, the differential tortuosity and wave magnitude are overall features of individual fiber trajectory, which are highly correlated with the scanning length. In contrast, the angle and curvature are local features, so a smaller scanning length could obtain convergent results.

Constraints on Testing Post-Newtonian Gravity with Scanning Space Astrometry

High-precision astrometry has the capability of testing General Relativity in the framework of the parametrized post-Newtonian formalism (PPN). Accurate measurements of gravitational light deflection over the entire celestial sphere open up a potential opportunity for determining the post-Newtonian parameter γ, which governs the deflection strength. We examine both analytically and numerically the coupling between a global parallax shift and the PPN parameter γ in Gaia-like astrometry. We also study effect of a parallax zero point on PPN γ estimation is scanning space astrometry. Least squares technique is applied to a direct astrometric solution including PPN γ and astrometric parameters (positions, parallax and proper motions). Analytical solution of the normal equations provides covariance information for the parameter γ and parallaxes. We present a new treatment of the correlation between parallax zero point and PPN γ, based on a statistical approach. We also derive a formula describing the bias in PPN γ estimation resulting from a global parallax offset. These theoretical results are verified by Monte Carlo experiments using astrometric solutions with simulated Gaia observations. Parallax zero point at a sub-μas level is needed to obtain a PPN γ estimate comparable with today’s best determination. We generalize the problem to two-dimensional observations and derive formula for the correlation between parallax zero point and PPN γ in the case when both along- and across-scan measurements are used for PPN γ determination.

Improvements to Quantum Interior Point Method for Linear Optimization

Quantum linear system algorithms (QLSA) have the potential to speed up Interior Point Methods (IPM). However, a major bottleneck is the inexactness of quantum Tomography to extract classical solutions from quantum states. In addition, QLSAs are sensitive to the condition number, and this sensitivity is exacerbated when the Newton systems arising in IPMs converge to a singular matrix. Recently, an Inexact Feasible Quantum IPM (IF-QIPM) has been developed that addresses the inexactness of QLSAs. However, this method requires a large number of gates and qubits to be implemented. Here, we propose a new IF-QIPM using the normal equation system, which requires less number of gates and qubits. To mitigate the sensitivity to the condition number and other input data-related parameters, we use preconditioning coupled with iterative refinement to obtain better complexity. Finally, we demonstrate the effectiveness of our approach on IBM Qiskit simulators.

Combination of altimetry crossovers and Doppler observables for orbit determination and geodetic parameter recovery: Application to Callisto

An accurate knowledge of the orientation, the tidal deformability, and the gravity field of a celestial body is fundamental to provide constraints on its internal structure. These quantities may be retrieved by processing radiometric tracking and altimetry data from a probe in orbit around such body. This paper presents a method to combine altimetry crossovers with two-way Doppler tracking observations at normal equation level, using the Bernese GNSS Software and the pyXover software library. This method was applied to a proposed 200km altitude orbiter around Callisto, a privileged destination for the upcoming phase of Solar System exploration. Enhancing “standard” Doppler tracking with altimetry generally benefited both orbit determination and a joint estimation of the orientation of the north pole and of planetary librations. The retrieval of low-degree gravity field parameters was also improved by the addition of altimetry data. However, the improvements on the estimated parameters were highly dependent on the characteristics of the simulation, e.g., the underlying topography roughness. Overall, combining radioscience with altimetry data accounted for a visible reduction of correlations among estimated parameters, while also allowing for a consistent estimation of the “vertical” Love number h2 along with gravity.

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

A new extension of the core inverse

There are a number of extensions of the core inverse represented by the products of known generalized inverses, like the core-EP inverse, DMP inverse, GC inverse and so on. However, none of them is an inner inverse of the matrix, and especially for an arbitrary nilpotent matrix, such inverses are always null. Our goal is to use a new technic to introduce an extension of the core inverse that preserves the interesting property of being an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. We define the extended core inverse for square complex matrices combining the sum and the difference of three known generalized inverses. Various properties and representations of the extended core inverse are developed. The extended dual core inverse is investigated too. We apply the extended core inverse to solve some systems of linear equations and one minimization problem. A significant normal equation related to least-squares solutions can be solved using the extended core inverse.

Real-Time Aerodynamic Load Estimation for Hypersonics via Strain-Based Inverse Maps

This work develops an efficient real-time inverse formulation for inferring the aerodynamic surface pressures on a hypersonic vehicle from sparse measurements of the structural strain. The approach aims to provide real-time estimates of the aerodynamic loads acting on the vehicle for ground and flight testing, as well as guidance, navigation, and control applications. Specifically, the approach targets hypersonic flight conditions where direct measurement of the surface pressures is challenging due to the harsh aerothermal environment. For problems employing a linear elastic structural model, the inference problem can be posed as a least-squares problem with a linear constraint arising from a finite element discretization of the governing elasticity partial differential equation. Due to the linearity of the problem, an explicit solution is given by the normal equations. Precomputation of the resulting inverse map enables rapid evaluation of the surface pressure and corresponding integrated quantities, such as the force and moment coefficients. The inverse approach additionally allows for uncertainty quantification, providing insights for theoretical recoverability and robustness to sensor noise. Numerical studies demonstrate the estimator performance for reconstructing the surface pressure field, as well as the force and moment coefficients, for the Initial Concept 3.X (IC3X) conceptual hypersonic vehicle.

Escape from compact sets of normal curves in Carnot groups

In the setting of subFinsler Carnot groups, we consider curves that satisfy the normal equation coming from the Pontryagin Maximum Principle. We show that, unless it is constant, each such a curve leaves every compact set, quantitatively. Namely, the distance between the points at time 0 and time t grows at least of the order t^(1/s), where s denotes the step of the Carnot group. In particular, in subFinsler Carnot groups there are no periodic normal geodesics.

On the Integration of VLBI Observations to GENESIS into Global VGOS Operations

The upcoming European Space Agency (ESA) satellite mission GENESIS is an Earth-orbiting satellite carrying instruments of all four space geodetic techniques. The onboard transmitter for Very Long Baseline Interferometry (VLBI) will allow the observation of the satellite with VLBI radio telescopes. The objective of this study is to investigate the integration of VLBI observations of GENESIS into the operations of the VLBI Global Observing System (VGOS). Based on both current and foreseeable modern VGOS antenna networks, we consider the realistic observability of both geodetic radio sources and GENESIS. We conduct a comprehensive scheduling and perform extensive simulations of the VLBI observations. We assume that observations of GENESIS are scheduled within regular, geodetic experiments. The integration of GENESIS as an additional source in the scheduling results in a minimal degradation in the geodetic parameter estimation of station positions and dUT1 of less than 0.09 mm and 0.06 μs, respectively. The results suggest to schedule scans of GENESIS at intervals of about 5 min to limit the decrease in the number of observations of geodetic sources to less than 5% with respect to schedules containing only geodetic radio sources. The schedules for 24 h experiments comprise about 150 to 200 scans and 1000 to 5000 observations of GENESIS, depending on the size of the utilized network. The frame tie accuracy between the VLBI and GENESIS frames is assessed in the form of station positions, which are solely estimated from observations of GENESIS. Multiple 24 h experiments are simulated over 52 weeks with assumed session cadences of two to three experiments per week. By stacking the normal equations from three months of experiments, we obtain station position estimates with a precision of less than 10 mm. After 12 months, the repeatabilites are reduced to less than 5 mm.

Modeling of Growth in Turkeys by Nonlinear Regression Models

Nonlinear regression models are commonly used in many areas, such as physics, chemistry, biology, and engineering. In these models, the solution of normal equations is more difficult than normal equations of linear regression models. Iterative algorithms are used to solve these equations. It is highly important to choose an appropriate initial value while using these algorithms. The values obtained during the study or from previous studies can be used as an initial value. With an inappropriately chosen initial value, the number of iterations will increase, and convergence may also not occur. In this study, the nonlinear Gompertz, Richards, and Weibull models of turkey growth were considered by taking the data on female turkey weight, and it was investigated which of these models was the most appropriate model for turkey growth using the R program

Utilizing Infrared Thermometry to Assess the Crop Water Stress Index of Wheat Genotypes in Arid Regions under Varying Irrigation Regimes

Researchers are depending more than ever on remote sensing techniques to monitor and assess the agricultural water status, as well as to estimate crop water usage or crop actual evapotranspiration. In the current work, normal and stressed baselines for irrigated wheat genotypes were developed in an arid part of the Sohag governorate, Egypt, using infrared thermometry in conjunction with weather parameters. The experiment was carried out in a randomized complete block design in the normal and drought stress conditions based on three replicates using ten bread wheat genotypes (G1–G10), including five accessions, under drought stress. A standard Class-A-Pan in the experimental field provided the daily evaporation measurements (mm/day), which was multiplied by a pan factor of 0.8 and 0.4 for normal and stressed conditions, respectively. The relationship between the vapor pressure deficit (VPD) and canopy-air temperature differences (Tc − Ta) was plotted under upper (fully stressed) and lower baseline (normal) equations. Accordingly, the crop water stress indexes (CWSIs) for the stressed and normal baselines for wheat genotypes were developed. Additionally, the intercept (b) and the slope (a) of the lower baseline equation were computed for different genotypes. The results indicate that, before applying irrigation water, the CWSI values were high in both growing seasons and under all irrigation regimes. After that, the CWSI values declined. G10 underwent stress treatment, which produced the greatest CWSI (0.975). Conversely, the G6 condition that received well-watered irrigation yielded the lowest result (−0.007). When compared to a well-watered one, the CWSI values indicated a trend toward rising stress. There existed an inverse link between the CWSI and grain yield (GY); that is, a lower CWSI resulted in better plant water conditions and a higher GY. Under standard conditions, the wheat’s highest GY was recorded in G2, 8.36 Ton/ha and a WCSI of 0.481. In contrast, the CWSI result for the stress treatment was 0.883, indicating a minimum GY of 5.25 Ton/ha. The Water Use Efficiency (WUE) results demonstrated that the stress irrigation regime produced a greater WUE value than the usual one. This study makes a significant contribution by investigating the techniques that would allow CWSI to be used to estimate irrigation requirements, in addition to determining the irrigation time.

An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method

Several methods have been used to estimate the Weibull parameters such as least square method (LSM), weighted least square method (WLSM), method of moments (MOM), and maximum likelihood (MLE). The maximum likelihood method is the most popular method (MLE). Newton-Raphson method has been applied to solve the normal equations of MLE’s in order to estimate the Weibull parameters. The method was used to find the optimal values of the Weibull distribution parameters for which the log-likelihood function is maximized. We tried to find the approximation solution to the normal equations of the MLE’s because there is no close form for get analytical solution. In this work, we tried to carry out a study that show the difference between two strategies to solve the MLE equations using Newton-Raphson algorithm. Both two strategies are provided an optimal solution to estimate the Weibull distribution parameters but which one more easer and which one converges faster. Therefore, we applied both strategies to estimate the Weibull’s shape and scale parameters using two different types of data (Real and simulation). We compared between the results that we got by applying the two strategies. Two studies have been done for comparing and selecting the optimal strategy to estimate Weibull distribution parameters using maximum likelihood method. We used some measurements to compare between the results such as number of steps for convergence (convergence condition), the estimated values for AIC, BIC and the RMSE value. The results show the numerical solution that we got by applying first strategy convergence faster than the solution that we got by applying second strategy. Moreover, the MRSE estimated by applying the first strategy is lower than the MRSE estimated by applying second strategy for the simulation study with different noise levels and different samples size.

Existence of optimal virtual element weights for mmWave FMCW MIMO radar based heart rate estimation

AbstractMeasuring heart rate is a critical component of assessing cardiac health and detecting potential heart diseases at an early stage. Various sensors, including electrocardiogram and photoplethysmograph, are commonly employed for this purpose. However, these conventional methods necessitate direct contact with the patient's skin which may be impractical or uncomfortable in situations involving patients with skin diseases or burn injuries. To address these limitations, a concerted effort has been to develop non‐contact methods leveraging frequency‐modulated continuous‐wave multiple‐input multiple‐output radar systems. Recent studies have illustrated the sensitivity of the phase component of received signals to micro‐motion, presenting a promising avenue for effective heart rate estimation. However, existing literature predominantly focuses on single‐antenna setups, overlooking the potential benefits offered by multiple‐input multiple‐output systems, which provide diverse channels with varying precision levels. This correspondence introduces a novel phase extraction model grounded in the argument of the analytic signal derived from both in‐phase and quadrature channels. Furthermore, leveraging the normal equation, we establish the feasibility of optimizing weights assigned to individual virtual antennas to achieve a robust approximation of ground truth data.

APPLICATION OF THE SINGULAR DECOMPOSITION OF THE MATRIX IN SOLVING INCORRECT GEODESIC PROBLEMS

The most reliable method for solving linear equations of the least squares principle, which can be used to solve incorrect geodetic problems, is based on matrix factorization, which is called a singular expansion. Some other methods require less machine time and memory. However, they are less effective in taking into account the errors of the source information, rounding errors, and linear dependence. The methodology of such research is that for any matrix A and any two orthogonal matrices U and V, there is a matrix , which is determined as T U AV  . The idea of a singular decomposition is that by choosing the right matrices U and V, you can convert most elements of the matrix  to zero and make it diagonal with non-negative elements. The novelty and relevance of scientific solutions lie in the feasibility of using a singular decomposition of the matrix to obtain linear equations of the least squares method, which can be used to solve incorrect geodetic problems. The purpose of scientific research is to obtain a stable solution of parametric equations of corrections to the results of mea- surements in incorrect geodetic problems. Based on the performed research on the application of the singular decomposition method for solving incorrect geodetic problems, we can summarize the following results. A singular expansion of a real matrix A is any factorization T A U W  of a matrix with orthogonal columns U, an orthogonal matrix W , and a diagonal matrix , the elements of which are called singular numbers of the matrix A, and the columns of matrices U and W — left and right singular vectors. If the matrix A has a full rank, then its solution will be unique and stable, which can be obtained by different methods. However, the method of singular decomposition, in contrast to other methods, makes it possible to solve problems with incomplete rank. Research shows that the method of solving normal equations by sequential exclusion of unknowns (Gaussian method), which is quite common in geod- esy, does not provide stable solutions for poorly conditioned or incorrect geodetic problems. Therefore, in the case of unstable systems of equations, it is proposed to use the method of singular matrix decomposition, which in computational mathematics is called SVD. The SVD singular decomposition method makes it possible to obtain stable solutions to both stable and by nature unstable problems. This possibility to solve incorrect geodetic problems is associated with the application of some limit , the choice of which can be made by the relative errors of the matrix of coefficients of parametric equations of corrections A and the vector of results of geodetic measurements L . Moreover, the solution of the system of normal equations obtained by the SVD method will have the shortest length. Thus, applying the apparatus of the singular decomposition of the matrix of coefficients of parametric equations of correc- tions to the results of geodetic measurements, we obtained new formulas for estimating the accuracy of the least squares method in solving incorrect geodetic problems. The derived formulas have a compact form and allow the easy calculation of elements  and XQ estimates of accuracy, almost ignoring the complex procedure of rotation of the matrix of coefficients of normal equations.

The relationship between children's emotional development tasks fulfilment and Strength and Difficulties Rutter Questionnaire assessment on elementary school-aged children's mental health.

To assess the emotional development of school-aged children. The descriptive, analytical study was carried out on children of elementary schools in the Semarang City area in January 2022. Included were children aged 6-12 years with active elementary school status. Data was collected using the 25-item Strength and Difficulties Rutter Questionnaire. The questionnaire was filled manually by the subject along with the guardian or teacher. Data was analysed using SPSS 20. Of the 326 children, 174(53.21%) were girls and 153(46.79%) were boys. Overall, 171(52.3%) subjects were aged <10 years, while 156(47.7%) were aged 10-12 years. There were 295(90.21%) children with normal prosocial behaviour, 206(63%) with normal emotional status, 264(80.73%) with normal conduct, 133(40.67%) with normal hyperactivity level, and 91(27.83%) with normal equation with peers. Different domains of the Strength and Difficulties Questionnaire showed varying degrees of pro-social behaviour, emotional status, conduct, hyperactivity level and peer interaction among the subjects.

Studies on meshing clearance of thread pairs of planetary roller screw mechanism in different directions

The helix surface model of planetary roller screw mechanism (PRSM) is established by utilizing the parametric equations of space surface according to the structural features of screw, roller and nut. The helix surface model of PRSM is improved in consideration of the PRSM assembly relationship. The normal vector equations of helix surfaces are obtained based on the helix surface model. The meshing clearance models of PRSM in axial, radial and circumferential directions are built considering the movement direction of parts utilizing surface meshing conditions. The influence of structural parameters on the meshing clearance of PRSM is also analyzed. The results show that the meshing points between the roller and the screw in three directions are not in the middle diameter. The meshing points in the three directions of the roller and the nut are in the middle diameter of the roller. The pitch, tooth angle and tooth thickness of the thread all have an influence on the meshing clearance of PRSM. Only by increasing the tooth thickness of the thread can the meshing clearance in the three directions be reduced at the same time. The pitch and tooth angle should be selected properly to meet the practical requirements.

A three-dimensional inverse method based on moving no-slip boundary for profiled end wall design

How to utilize existing flow control mechanisms to make profiled end wall design more flexible, efficient, and physical is a meaningful challenge. This study presents a three-dimensional inverse method for profiled end wall design to achieve the application of flow control mechanisms. The predetermined pressure distribution on the end wall is reached by modifying the end wall geometry during flow field calculation. A motion velocity model is derived from the normal momentum equation of the moving no-slip boundary to modify the end wall geometry. A Reynolds-Averaged Navier-Stokes (RANS) solver based on the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is adopted to simulate the flow field. Based on the mechanism understanding obtained through numerical optimization results, this study adopts the inverse method to redesign an optimized end wall in a compressor cascade. The results indicate that the redesigned end wall exhibits better loss reduction, reducing the overall total pressure loss by 5.5%, whereas the optimized end wall reduces it by 3%. The inverse method allows the imposition of desired influences on the end wall flow without constructing a database, making it highly flexible, efficient, and physical.

Sparse Approximate Pseudoinverse Preconditioning for Sparse Supervised Learning Problems with More Features than Samples

In this paper, we propose a new technique for handling a class of sparse supervised learning problems, specifically focusing on regression, binary, and multi-class classification problems. Sparse regression problems are associated with datasets including both arithmetic and categorical features and by encoding the dataset leads to an underdetermined sparse linear system. Furthermore, logistic regression can be used for both sparse binary and multi-class classification problems, solving an underdetermined sparse linear system. A new technique called Sparse Approximate Pseudoinverse Preconditioning (SAPP), namely the Explicit Preconditioned Conjugate Gradient for Normal Equations (EPCGNE) method based on Generic Approximate Sparse Pseudoinverse matrices is introduced for solving underdetermined sparse least square problems. Numerical experiments were carried out demonstrating a significant improvement of the performance metrics for the proposed SAPP scheme compared to other learners.