Normal Equations Research Articles

Multiple-Point Statistical Simulation of rock fracture network as a key control on the hydrogeology and salinity: a case study from the Qarabagh area, West Azarbayjan Province, Iran

Modeling and characterization of the geometry and distribution of Rock Facture Networks (RFNs) are essential in applications such as hydrogeological or environmental evaluations. It is widely accepted that RFNs are potentially associated with the hydrogeological (thus salinity) characteristics of the surrounding environments. Despite the complexity and inaccessibility of RFNs, stochastic methods provide a functional framework to predict their characteristics in the subsurface. An efficient tool for modeling RFNs is the Discrete Fracture Network (DFN) which also includes a number of geostatistical techniques that consider spatial variability structure. The advantages of these techniques are: realistic results, ease of application, and uncertainty assessments. Multiple-point geostatistics/statistics (MPS) is a modern and effective geostatistical tool for realistically simulating RFNs. In the present study, we modeled the RFNs in a location near the Qarabagh area, in the western Urmia Lake; in this regard, we used the Single Normal Equation Simulation (SNESIM) algorithm of the MPS geostatistical method using Training Images (TIs) instead of variograms. The required datasets and information for this modeling was provided using the field measurements of the fracture orientations and dips, as well as the outcrop photographs. The outcomes of these models can be used in predicting the salinity distribution in the surrounding area . Therefore, through the SNESIM algorithm, TIs obtained from the outcrop photographs, and direct measurements, 100 RFN realizations were generated at each station. These realizations were then averaged to predict the locations with higher and lower fracture probabilities and to assess the general trend of the fracture distributions.

High-precision downward continuation of potential fields algorithm utilizing adaptive damping coefficient of generalized minimal residuals

The downward continuation of potential fields is a process of calculating their values in a lower plane based on those of a certain plane. This technology is not only a data processing method for resource exploration but also plays an extremely important role in military applications. However, the downward continuation of potential fields is a typical linear inverse problem that is ill-posed. Generalized minimal residuals (GMRES) is an effective solution to ill-posed inverse problems, but it is unstable under the condition wherein the GMRES is directly applied in the calculation process. Moreover, the long-term behavior of its iterative computation is a disordered, divergent result. Therefore, to obtain stable solutions, GMRES is applied to solve the normal equations of the downward continuation of potential fields; it is also used to prequalify for occasional interruptions in the operation process by adding the damping coefficient, thus strengthening the stability conditions of the equations of residual minimization. Finally, the stable downward continuation of the potential fields method is proposed. As indicated by the theoretical data and the measured testing data, the method proposed in this paper has the advantages of high-precision and excellent stability. Furthermore, compared with the Tikhonov iteration method, the proposed method avoids the need to choose regularization parameters.

Identification of dynamic errors-in-variables systems with quasi-stationary input and colored noise

This paper studies the linear dynamic errors-in-variables (EIV) problem in a fairly general condition where the input–output disturbing noises are colored and the input is quasi-stationary. A novel formulation of the extended frequency domain maximum likelihood (ML) estimator is developed which reduces the number of nonlinear normal equations to be solved. Sufficient conditions are provided to achieve local identifiability of the EIV model for specified noise cases of interest. The parameter estimates are calculated via a numerically stable Gauss–Newton minimization scheme started by an initial value generation strategy. Also, both the consistency and accuracy of the extended ML estimate are analyzed in detail. The performance of the proposed method is finally demonstrated on simulated dynamic systems.

A new interior-point approach for large separable convex quadratic two-stage stochastic problems

ABSTRACT Two-stage stochastic models give rise to very large optimization problems. Several approaches have been devised for efficiently solving them, including interior-point methods (IPMs). However, using IPMs, the linking columns associated with first-stage decisions cause excessive fill-in for the solution of the normal equations. This downside is usually alleviated if variable splitting is applied to first-stage variables. This work presents a specialized IPM that applies variable splitting and exploits the structure of the deterministic equivalent of the stochastic problem. The specialized IPM combines Cholesky factorizations and preconditioned conjugate gradients for solving the normal equations. This specialized IPM outperforms other approaches when the number of first-stage variables is large enough. This paper provides computational results for two stochastic problems: (1) a supply chain system and (2) capacity expansion in an electric system. Both linear and convex quadratic formulations were used, obtaining instances of up to 38 million variables and 6 million constraints. The computational results show that our procedure is more efficient than alternative state-of-the-art IPM implementations (e.g. CPLEX) and other specialized solvers for stochastic optimization.

METHOD OF PROCESSING EMPIRICAL DATA IN THE STUDY OF THE INFLUENCE OF TECHNOLOGICAL PARAMETERS ON THE SUPPLY OF TOOLS FOR THE FINAL PROCESSING OF STEEL AND CAST IRON WORKPIECES

When implementing a multi-variant design of the machining process, one of the important tasks is to select rational technological parameters at the final transitions of the shaping process that ensure the specified surface quality on the finished part made of the appropriate material. When modeling most of the design procedures at all stages of machining, there are problems associated with algorithmization of the solution method, adequate choice of the regression equation and the range of acceptable solutions at the specified processing stages. In this paper, as an algorithm for solving the problem, we propose the use of a wellknown method of rational planning of the experiment, which allows us to obtain a mathematical model of the multi-factor process under study. In the process of implementing this algorithm, based on a priori information, the object of research was selected, its factors and the main levels of change of these factors were established. Next, a two-factor experiment plan was implemented using a numeric array from the wellknown CoroKey catalog by SANDVIK Coromant. At the next step, the experimental data was processed using the existing method and on the basis of this information, private graphical dependencies of the studied parameter on the levels of factors were constructed. At the final stage, using the types of graphical dependencies, the regression equation of the investigated value of the cutting tool feed from two independent factors at three levels of each of them was formulated. To implement this algorithm, the least squares method was used, and the system of normal equations obtained on its basis. The proposed algorithm for obtaining the regression equation for solving problems in the field of machining production can be effectively used in computer-aided design systems for selecting technological parameters in the modes of processing parts with a blade tool, using numerical data from technological directories and catalogs of cutting tool manufacturers.

Determination of precise Galileo orbits using combined GNSS and SLR observations

Galileo satellites are equipped with laser retroreflector arrays for satellite laser ranging (SLR). In this study, we develop a methodology for the GNSS-SLR combination at the normal equation level with three different weighting strategies and evaluate the impact of laser observations on the determined Galileo orbits. We provide the optimum weighting scheme for precise orbit determination employing the co-location onboard Galileo. The combined GNSS-SLR solution diminishes the semimajor axis formal error by up to 62%, as well as reduces the dependency between values of formal errors and the elevation of the Sun above the orbital plane—the β angle. In the combined solution, the standard deviation of the SLR residuals decreases from 36.1 to 29.6 mm for Galileo-IOV satellites and |β|> 60°, when compared to GNSS-only solutions. Moreover, the bias of the Length-of-Day parameter is 20% lower for the combined solution when compared to the microwave one. As a result, the combination of GNSS and SLR observations provides promising results for future co-locations onboard the Galileo satellites for the orbit determination, realization of the terrestrial reference frames, and deriving geodetic parameters.

Temporal Gravity Signals in Reprocessed GOCE Gravitational Gradients

The reprocessing of the satellite gravitational gradiometry (SGG) data from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission in 2018/2019 considerably reduced the low-frequency noise in the data, leading to reduced noise amplitudes in derived gravity field models at large spatial scales, at which temporal variations of the Earth’s gravity field have their highest amplitudes. This is the motivation to test the reprocessed GOCE SGG data for their ability to resolve time-variable gravity signals. For the gravity field processing, we apply and compare a spherical harmonics (SH) approach and a mass concentration (mascon) approach. Although their global signal-to-noise ratio is <1, SH GOCE SGG-only models resolve the strong regional signals of glacier melting in Greenland and Antarctica, and the 2011 moment magnitude 9.0 earthquake in Japan, providing an estimation of gravity variations independent of Gravity Recovery and Climate Experiment (GRACE) data. The benefit of combined GRACE/GOCE SGG models is evaluated based on the ice mass trend signals in Greenland and Antarctica. While no signal contribution from GOCE SGG data additional to the GRACE models could be observed, we show that the incorporation of GOCE SGG data numerically stabilizes the related normal equation systems.

Parallel resolution of large GNSS networks using carrier ranges

Owing to the increasing numbers of navigation satellites, available frequencies, and tracking stations, the computational efficiency of large GNSS networks has become a critical issue, especially when undifferenced (UD) pseudorange and carrier phase observations are used. By converting UD carrier phases into carrier ranges before implementing the integrated processing of a whole network, the UD calculation time can be greatly decreased. However, because a common multicore hardware platform is not fully utilized to optimize the algorithm, the processing of a whole network is still time consuming. In this paper, this problem is overcome by applying the multicore parallel computing technology, namely the Task Parallel Library (TPL). To improve the processing efficiency of large GNSS networks using UD carrier ranges and based on the task parallelism and data parallelism offered through TPL, the wide-lane (WL) and narrow-lane (NL) fractional cycle biases (FCBs), and carrier ranges are computed in parallel, and the integrated processing of a whole network is performed by establishing the normal equation in parallel and adopting parallel computing for Cholesky decomposition. The proposed strategy is validated using GPS, BDS and Galileo observation data recorded from January 6, 2019, to January 26, 2019 by the IGS network with a global distribution of approximately 500 stations. The average positioning accuracies in the east, north and vertical components are 2.54, 2.34 and 4.41 mm, respectively, and the average SDBS ambiguity fixing rate is 93.18%. The average time consumptions for step 1, 2 and 3 are reduced from 151.45, 248.80 and 44.35 min to 32.20, 52.71 and 14.25 min, respectively, with acceleration ratios of 4.70, 4.72 and 3.11 times, respectively, on a six-core platform.

Accuracy and Conditioning of Surface-Source Based Near-Field to Far-Field Transformations

The conditioning and accuracy of various inverse surface-source formulations are investigated. First, the normal systems of equations are discussed. Second, different implementations of the zero-field condition are analyzed regarding their effect on solution accuracy, conditioning, and source ambiguity. The weighting of the Love-current side constraint is investigated in order to provide an accurate problem-independent methodology. The transformation results for simulated and measured near-field data show a comparable behavior regarding accuracy and conditioning for most of the formulations. Advantages of the Love-current solutions are found only in diagnostic capabilities. Regardless of this, the Love side constraint is a computationally costly way to influence the iterative solver threshold, which is more conveniently controlled with the appropriate type of normal equation. The solution behavior of the inverse surface-source formulations is mostly influenced by the choice of the reconstruction surface. A spherical Huygens surface leads to the best conditioning, whereas the most accurate solutions are found with a tight, possibly convex hull around the antenna under test.

Solving interval linear least squares problems by PPS-methods

In our work, we consider the linear least squares problem for m × n-systems of linear equations Ax = b, m ≥ n, such that the matrix A and right-hand side vector b can vary within an interval m × n-matrix A and an interval m-vector b, respectively. We have to compute, with a prescribed accuracy, outer coordinate-wise estimates of the set of all least squares solutions to Ax = b for A ∈A and b ∈b. Our article is devoted to the development of the so-called PPS-methods (based on partitioning of the parameter set) to solve the above problem. We reduce the normal equation system, associated with the linear lest squares problem, to a special extended matrix form and produce a symmetric interval system of linear equations that is equivalent to the interval least squares problem under solution. To solve such symmetric system, we propose a new construction of PPS-methods, called ILSQ-PPS, which estimates the enclosure of the solution set with practical efficiency. To demonstrate the capabilities of the ILSQ-PPS-method, we present a number of numerical tests and compare their results with those obtained by other methods.

Analytic integrability for holographic duals with Joverline{T} deformations

We probe warped BTZ ×S3 geometry with various string solitons and explore the classical integrability criteria of the associated phase space configurations using Kovacic’s algorithm. We consider consistent truncation of the parent sigma model into one dimension and obtain the corresponding normal variational equations (NVE). Two specific examples have been considered where the sigma model is reduced over the subspace of the full target space geometry. In both examples, NVEs are found to possess Liouvillian form of solutions which ensures the classical integrability of the associated phase space dynamics. We address similar issues for the finite temperature counterpart of the duality, where we analyse the classical phase space of the string soliton probing warped BTZ black string geometry. Our analysis reveals a clear compatibility between normal variational equations and the rules set by the Kovacic’s criteria. This ensures the classical integrability of the parent sigma model for the finite temperature extension of the duality conjecture.

Evaluating Processing Choices for the Geodetic Estimation of Earth Orientation Parameters With Numerical Models of Global Geophysical Fluids

AbstractDifferent Earth orientation parameter (EOP) time series are publicly available that typically arise from the combination of individual space geodetic technique solutions. The applied processing strategies and choices lead to systematically differing signal and noise characteristics particularly at the shortest periods between 2 and 8 days. We investigate the consequences of typical choices by introducing new experimental EOP solutions obtained from combinations at either normal equation level processed by Deutsches Geodätisches Forschungsinstitut at the Technical University of Munich (DGFI‐TUM) and Federal Agency for Cartography and Geodesy (BKG), or observation level processed by European Space Agency (ESA). All those experiments contribute to an effort initiated by ESA to develop an independent capacity for routine EOP processing and prediction in Europe. Results are benchmarked against geophysical model‐based effective angular momentum functions processed by Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences (ESMGFZ). We find, that a multitechnique combination at normal equation level that explicitly aligns a priori station coordinates to the ITRF2014 frequently outperforms the current International Earth Rotation and Reference Systems Service (IERS) standard solution 14C04. A multi‐Global Navigation Satellite System (GNSS)‐only solution already provides very competitive accuracies for the equatorial components. Quite similar results are also obtained from a short combination at observation level experiment using multi‐GNSS solutions and SLR from Sentinel‐3A and Sentinel‐3B to realize space links. For ΔUT1, however, very long baseline interferometry (VLBI) information is known to be critically important so that experiments combining only GNSS and possibly SLR at observation level perform worse than combinations of all techniques at normal equation level. The low noise floor and smooth spectra obtained from the multi‐GNSS solution nevertheless illustrates the potential of this most rigorous combination approach so that further efforts to include in particular VLBI are strongly recommended.

Benefits of non-tidal loading applied at distinct levels in VLBI analysis

In the analysis of very long baseline interferometry (VLBI) observations, many geophysical models are used for correcting the theoretical signal delay. In addition to the conventional models described by Petit and Luzum (eds) (IERS Conventions, 2010), we are applying different parts of non-tidal site loading, namely the atmospheric, oceanic, and hydrological ones. To investigate their individual contributions, these parts are considered both separately and combined to a total loading. The application of the corresponding site displacements is performed at two distinct levels of the geodetic parameter estimation process (observation and normal equation level), which turn out to give very similar results in many cases. To validate our findings internally, the site displacements are provided by two different data centres: the Earth-System-Modelling group at the Deutsches GeoForschungsZentrum in Potsdam (ESMGFZ, see Dill and Dobslaw, J Geophys Res Solid Earth, 2013. https://doi.org/10.1002/jgrb.50353)] and the International Mass Loading Service [IMLS, see Petrov (The international mass loading service, 2015)]. We show that considering non-tidal loading is actually useful for mitigating systematic effects in the VLBI results, like annual signals in the station height time series. If the sum of all non-tidal loading parts is considered, the WRMS of the station heights and baseline lengths is reduced in 80–90% of all cases, and the relative improvement is about -,3.5% on average. The main differences between our chosen providers originate from hydrological loading.

A Simplified Model, Dynamic Analysis and Force Estimation for a Large-scale Orinthopter in Forward Flight Based on Flight Data

Similarities and differences of a large-scale flapping-wing robot with fixed-wing UAVs in equations of motion, trim curves, and aerodynamic forces in forward flight are discussed in this paper and a simplified model for flapping flight is presented. Due to the high Wing to Total Weight (WTW) ratio of large-scale ornithopters, simple rigid body dynamics is not accurate enough for flight dynamics modeling. On the other hand, the multi-body dynamics associated with flapping gives little insight into the behavior of the resulting model due to complexity of equations. It is also difficult to design proper controllers for such complicated models. In this paper, the effects of different terms of multi-body equations of ornithopter on the estimated aerodynamic forces are studied via experimental flight data. A simpler but yet accurate set of equations is obtained by removing less effective terms from original relations. The presented model is in the form of normal aircraft equations plus some additional terms which can be used in different control and estimation processes. In addition, trim conditions of forward flight are extracted using several flight tests, and corresponding periodic behavior of states and forces are studied. These studies are applicable for identifying time-periodic models.

EFFICIENT COMPUTATION OF POSTERIOR COVARIANCE IN BUNDLE ADJUSTMENT IN DBAT FOR PROJECTS WITH LARGE NUMBER OF OBJECT POINTS

Abstract. One of the major quality control parameters in bundle adjustment are the posterior estimates of the covariance of the estimated parameters. Posterior covariance computations have been part of the open source Damped Bundle Adjustment Toolbox in Matlab (DBAT) since its first public release. However, for large projects, the computation of especially the posterior covariances of object points have been time consuming.The non-zero structure of the normal matrix depends on the ordering of the parameters to be estimated. For some algorithms, the ordering of the parameters highly affect the computational effort needed to compute the results. If the parameters are ordered to have the object points first, the non-zero structure of the normal matrix forms an arrowhead.In this paper, the legacy DBAT posterior computation algorithm was compared to three other algorithms: The Classic algorithm based on the reduced normal equation, the Sparse Inverse algorithm by Takahashi, and the novel Inverse Cholesky algorithm. The Inverse Cholesky algorithm computes the explicit inverse of the Cholesky factor of the normal matrix in arrowhead ordering.The algorithms were applied to normal matrices of ten data sets of different types and sizes. The project sizes ranged from 21 images and 100 object points to over 900 images and 400,000 object points. Both self-calibration and non-self-calibration cases were investigated. The results suggest that the Inverse Cholesky algorithm is the fastest for projects up to about 300 images. For larger projects, the Classic algorithm is faster. Compared to the legacy DBAT implementation, the Inverse Cholesky algorithm provides a performance increase by one to two orders of magnitude. The largest data set was processed in about three minutes on a five year old workstation.The legacy and Inverse Cholesky algorithms were implemented in Matlab. The Classic and Sparse Inverse algorithms included code written in C. For a general toolbox as DBAT, a pure Matlab implementation is advantageous, as it removes any dependencies on, e.g., compilers. However, for a specific lab with mostly large projects, compiling and using the classic algorithm will most likely give the best performance. Nevertheless, the Inverse Cholesky algorithm is a significant addition to DBAT as it enables a relatively rapid computation of more statistical metrics, further reinforcing its application for reprocessing bundle adjustment results of black-box solutions.

An analysis of lipid membrane morphology in the presence of coordinate-dependent non-uniformity

A model for the mechanics of lipid membranes with non-uniform (coordinate-dependent) properties is discussed. The coordinate-dependent responses of the membranes are incorporated via the augmented non-uniform energy function and material parameters, which are dependent explicitly on the surface coordinates. We formulate the associated normal and tangential Euler equilibrium equations through which the coordinate-dependent responses of membranes are characterized. The admissible boundary conditions are taken from the existing non-linear model but reformulated and adopted to the present framework. Within the prescription of superposed incremental deformations, a compatible linear model is also formulated, from which a complete analytical solution describing the non-uniform responses of the membrane subjected to substrate–membrane interactions is obtained.

An extension of the landweber regularization for a backward time fractional wave problem

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Theoretical analysis of thermal characteristics of casson nano fluid flow past an exponential stretching sheet in Darcy porous media

The existent problem involves, the steady two-dimensional flow adjacent to boundary, mass as well as heat transfer of non-Newtonian Casson nano fluid in presence of an exponential stretching sheet. Then at this time work relating the BVP which forms a set of nonlinear normal differential equations, borrowed from a set of P.D.E's using similarity transformation, are cracked numerically with Runge-Kutta method of 4th order procedure with most efficient shooting technique. The brunts of an assortment of other dimensionless variables on temperature, velocity along with nano particle concentration are worked out, as well as the consequential values are depicted in graphical forms. The dimensionless variables under examination are Suction/Injection parameter F*w, Casson fluid Parameter β, Lewis number Le, Brownian Parameter Nb, Prandtl number Pr, Thermopheresis Parameter Nt, Porous parameter Kk*, whose brunt on Temperature profile G*(η), velocity profile F*(η), nanoparticle concentration profile H*(η) are exhibited in graphs in the course of Figs. 1 to 9 with detailed explanation.

A fast and accurate bundle adjustment method for very large-scale data

Bundle adjustment with very large scale datasets has drew much concern recently in both photogrammetry and computer vision communities. Different from the existing out-of-core and distributed methods for large scale datasets, we propose a fast and accurate bundle adjustment method which still uses the framework of the traditional Levenberg Marquardt (LM) algorithm while adopting preconditioned conjugate gradient (PCG) to iteratively solve normal equation, and using point resampling scheme and normal matrix compression to decrease the memory requirement and computational complexity. Preliminary results show that our method running on a single laptop computer with i7 2.6 GHz CPU and 8 GB RAM is even faster than the state-of-the-art distributed method deployed on a large distributed computer system with multiple computers each of which is equipped with CPU i7-4770K 3.5 GHz with 8 threads and 32 GB RAM and connected with each other at the speed of 10 MB/s. The proposed method is also more accurate.

Impulsive Synchronization of Derivative Coupled Neural Networks With Cluster-Tree Topology

This article is devoted to discussing the exponential synchronization for a kind of delay derivative coupled neural networks with stochastic disturbance and multiple time-varying delays. To simulate more practical situations and widen the synchronization application fields in network science, the coupled neural networks with cluster-tree topology structure is studied by applying a novel impulsive pinning control strategy, which skillfully considered the neural networks in current cluster that directly linked to the neural networks in other clusters. Since the existence of delayed impulses, the general comparison principle for normal impulsive differential equations is efficiently extended. In view of the concept of average impulsive interval, the parameters classification discussion method and the mathematical induction method, some judgement conditions for achievement of the cluster synchronization on derivative coupled neural networks are derived. Additionally, the exponential convergence velocity of the derivative coupled neural networks is accurately estimated. Finally, numerical examples are presented to demonstrate the effectiveness of the control strategy and the theoretical results.