Non-iterative Solution Research Articles

Solving Unsteady Boundary Value Problems Using Discrete-Analytic Method for Non-Iterative Simulation of Temperature Processes in Time

Method for solving a boundary value problem of inhomogeneous unsteady-state heat conduction transfer is considered. This physical process can be described by a boundary value problem for a partial differential equation of the 2nd order. Discrete-analytical method, which turns out the mathematical formulation of the initial problem to be normal system of differential equations, was used. There is the non-iterative solution of such system, which is the set of analytic functions. The theory of matrix functions, particularly the properties of matrix exponential, was applied to get the solution. This approach allows us to model the unsteady-state heat conduction processes with unstationary boundary conditions of different types, defined as time-dependent functions. Such modeling describes the real physical processes in structural materials more accurately.

Stereo-Based Visual Odometry for Autonomous Robot Navigation

Mobile robots should possess accurate self-localization capabilities in order to be successfully deployed in their environment. A solution to this challenge may be derived from visual odometry (VO), which is responsible for estimating the robot's pose by analysing a sequence of images. The present paper proposes an accurate, computationally-efficient VO algorithm relying solely on stereo vision images as inputs. The contribution of this work is twofold. Firstly, it suggests a non-iterative outlier detection technique capable of efficiently discarding the outliers of matched features. Secondly, it introduces a hierarchical motion estimation approach that produces refinements to the global position and orientation for each successive step. Moreover, for each subordinate module of the proposed VO algorithm, custom non-iterative solutions have been adopted. The accuracy of the proposed system has been evaluated and compared with competent VO methods along DGPS-assessed benchmark routes. Experimental results of relevance to rough terrain routes, including both simulated and real outdoors data, exhibit remarkable accuracy, with positioning errors lower than 2%.

Systematic comparison of DFT and CCSD dipole moments, polarizabilities and hyperpolarizabilities

A comparative study of dipole moments, polarizabilities and hyperpolarizabilities of 16 different molecules is performed employing two completely different theoretical approaches namely, density functional theory (DFT) and coupled cluster singles and doubles (CCSD). Both methods include electron correlation. The CCSD method is more accurate but highly expensive. DFT with auxiliary density allows non-iterative solutions which is computational advantage and useful for large molecules. Dipole moments and polarizability calculations from DFT are in very good agreement with CCSD calculations. However, negative hyperpolarizability values from DFT differ significantly from their CCSD counterparts, whereas positive hyperpolarizabilities show reasonable agreement between these methodologies.

Comparison of Noniterative Algorithms Based on Different Forms of Richards’ Equation

Through linearizing an implicit differencing scheme, several noniterative numerical solutions of Richards’ equation in different forms are derived here. Paniconi et al. (Water Resources Research, 27(6), 1147–1163, 1991) have developed a first-order accurate linearization of the head-based Richards’ equation (RE) and a second-order accurate linearization of the implicit-factored head-based RE. Considering other forms of RE, we propose a second-order accurate linearization of the moisture-based RE and a second-order accurate linearization of the mixed form RE combined with the primary variable switching technique. Extensive comparisons between the noniterative solutions are conducted through three numerical experiments. Their accuracies, efficiencies, and mass balance behaviors are analyzed. The results indicate that the first-order accurate scheme is not efficient compared to iterative models. The noniterative schemes of head-based RE suffer from the mass imbalance problem without iteration. The linearized moisture-based RE can obtain mass conservative, accurate results effectively, while it may confront numerical problems when the soil approaches saturation. Among these noniterative schemes, the linearized mixed form RE combined with the primary variable switching technique is superior in terms of accuracy, mass balance, and efficiency compared to traditional iterative methods.

An Update of Non-iterative Solutions for Surface Fluxes Under Unstable Conditions

A new non-iterative approach has been developed to calculate surface fluxes. The range \(-5 \le Ri_\mathrm{B} < 0, 10 \le \,z/z_\mathrm{0} \,\le 10^{5}\) and \(-0.5 \le \ln (z_\mathrm{0}/z_\mathrm{0h}) \,\le 30\) is divided into eight regions, and in each of the regions, multiple linear regression is performed to obtain non-iterative solutions for surface fluxes. As compared to the other two most recent non-iterative schemes, we show that the suggested scheme has the smallest bias. The maximum relative errors of turbulent transfer coefficients for momentum \((C_\mathrm{M})\) and sensible heat \((C_\mathrm{H})\), as compared to those obtained from the classic iterative method, are always smaller than 2 % from our new non-iterative scheme.

An Adaptive PMU-Based Wide Area Backup Protection Scheme for Power Transmission Lines

This paper proposes a novel adaptive wide area backup protection scheme for transmission lines. In the proposed scheme, both the faulted line and fault location are determined by a limited number of synchronized phasor measurements. Based on phasor measurement unit placement and network topology, subsets of lines and buses called backup protection zones (BPZs) are formed. After a fault occurs in the transmission network, the sum of zero- and/or positive-sequence currents entering the faulted BPZ highly increases, and hence, the faulted BPZ can be determined. The linear least squares method is then used to determine the faulted line, as well as the fault location by voltage and current phasors of the faulted BPZ. Accordingly, the proposed scheme provides a closed-form and noniterative solution for the faulted line and fault location identification problem. On the other hand, it readily determines the faulted line regardless of the fault type, fault resistance, and measurement errors. To show the effectiveness of the method, it is applied to the WSCC 9-bus and IEEE 118-bus test systems. Simulation results verify successful identification of the faulted BPZ as well as the faulted line within the faulted BPZ with limited measurement points.

Quality assessment for strain field determination based on the NISLT approach

By determining the state of strain, the status of a structure can be effectively evaluated. Strain field determination based on the non-iterative solution for linear transformations (NISLT) approach has been recently proposed to successfully identify rigid-body motion and natural deformation, which are coupled in the displacement fields in strain calculation. However, the NISLT approach cannot provide a direct measure of the quality of estimated parameters. This study has rigorously derived an analytical expression of the covariance matrix of linear transformation parameters using the NISLT approach. The derivation was used to determine the quality of the displacement field and can be used to estimate the quality of the determined parameters. The derivation was verified in numerical and laboratory experiments, which demonstrated that the estimated maximum principal strains differed from the originally assumed principal strain (real strain) within 1.96 times of the standard deviation (1.96σ). Therefore, the derived quality assessment of the NISLT approach provided a fair estimation of how accurate the strain would be under a certain displacement accuracy. The quality of the structure’s integrity monitoring system could thus be regulated, which is essential in the planning of structural monitoring tasks.

A spectral Phase–Amplitude method for propagating a wave function to large distances

The phase and amplitude (Ph–A) of a wave function vary slowly with distance, in contrast to the wave function that can be highly oscillatory. Hence the Ph–A representation of a wave function requires far fewer computational mesh points than the wave function itself. In 1930 Milne presented an equation for the phase and the amplitude functions (which is different from the one developed by Calogero), and in 1962 Seaton and Peach solved these equations iteratively. The objective of the present study is to implement Seaton and Peach’s iteration procedure with a spectral Chebyshev expansion method, and at the same time present a non-iterative analytic solution to an approximate version of the iterative equations. The iterations converge rapidly for the case of attractive potentials. Two numerical examples are given: (1) for a potential that decreases with distance as 1/r3, and (2) a Coulomb potential ∝1/r. In both cases the whole radial range of [0–2000] requires only between 25 and 100 mesh points and the corresponding accuracy is between 10−3 and 10−6. The 0th iteration (which is the WKB approximation) gives an accuracy of 10−2. This spectral method permits one to calculate a wave function out to large distances reliably and economically.

Iterative and Non-Iterative Solution of Planar Resection

Planar resection is a common surveying work. The paper introduces its frequently-used solution i.e. the iterative solution. However, the iterative solution needs linearization, initial value of parameter and iteration, unfortunately the initial value of parameter is hard to assess in advance, resultantly, the iteration will fail. Due to inexistence of the above problem, the non-iterative solution based on polynomial resultant or Groebner basis and Jacobi algorithm is presented. A numerical case is given to demonstrate the two solutions. It is suggested for the redundant observation cases that the non-iterative solution should be adopted to assess the initial value of parameter, and then the iterative solution should be employed to compute the final result.

Estimation Of Global Positioning System Measurement Errors For GAGAN Applications

The Global Positioning System (GPS) has been in use for providing positioning, navigation and timing (PNT) services in many parts of the world. There are several errors that affect the positional accuracy of GPS. Prominent among them are ephemeris errors, satellite and receiver clock errors, multipath errors, signal propagation errors such as ionospheric delay, tropospheric delay, and instrumental biases of the satellite and receiver. In this paper, prominent estimation techniques to characterize various GPS measurement errors are reviewed. The GPS data in the Receiver INdependent EXchange (RINEX) format obtained from a Dual frequency GPS receiver is used in this analysis. Among all the errors, ionospheric delay is found to be the most dominant. However, these delay measurements are affected by the satellite and receiver instrumental biases. The instrumental biases exist as the signals at the two GPS frequencies experience different delays inside the satellite and receiver hardware. For estimation of the instrumental biases Extended Kalman filter(EKF) technique is adopted. The user equivalent range error (UERE) obtained due to all the error sources is of the order of few metres. After accounting for various errors, the estimation accuracy is significantly improved.The positional accuracy of the GPS Aided GEO Augmented Navigation (GAGAN) system is basically dependent on ranging errors and the satellite constellation geometry(1). The lineof-sight ionospheric measurements derived from the Global Positioning System (GPS) observables are corrupted by the instrumental biases present in both the GPS satellites and the receivers. The instrumental bias and Total Electron Content (TEC) results (Hyderabad GAGAN station (78.47°E, 17.45°N)) obtained using theExtended Kalman filter technique are presented in this paper. In this work the main focus is towards development of an improved error free approach for aircraft navigation. The existing navigation solutions are based on iterative algorithms and require an initial estimate of the receiver position for computing the navigation solution which is not possible in all conditions. Accurate, reliable and cost effective navigation solution algorithms that suit the Indian geographic area and area of application are proposed in this work. A non-iterative point solution approach algorithm computes the navigation solution based on the available satellite positions and pseudorange measurements(3). The improvement in GPS system has given a feasibility of automated operation of satellite based augmenting system in aircraft navigation can augment GAGAN and improves the positional accuracy of the user. In this paper a communication approach , high, accurate processing using EKF scheme is proposed for GPS aided geo augmented navigation (GAGAN) application. The effectiveness of the proposed approach for higher performance is evaluated under variable conditions.

Phase retrieval of images using Gaussian radial bases

Here, the possibility of a noniterative solution to the phase retrieval problem is explored. A new look is taken at the phase retrieval problem that reveals that knowledge of a diffraction pattern's frequency components is enough to recover the image without projective iterations. This occurs when the image is formed using Gaussian bases that give the convenience of a continuous Fourier transform existing in a compact form where square pixels do not. The Gaussian bases are appropriate when circular apertures are used to detect the diffraction pattern because of their optical transfer functions, as discussed briefly. An algorithm is derived that is capable of recovering an image formed by Gaussian bases from only the Fourier transform's modulus, without background constraints. A practical example is shown.

Simultaneous estimation of temperature-dependent volumetric heat capacity and thermal conductivity functions via neural networks

An artificial neural network based solution of the inverse heat conduction problem of simultaneous identification of the temperature-dependent volumetric heat capacity and thermal conductivity function of a solid material is presented in this paper. The inverse problem was defined according to the evaluation of the BICOND thermophysical property measurement method. The volumetric heat capacity and thermal conductivity vs. temperature functions are to be determined using the measured transient temperature histories of two sensors. In this study noiseless and noisy artificial measurements were generated by the numerical solution of the corresponding direct heat conduction problem. The inverse problem was solved by multi-layer feed-forward neural network trained by back-propagation algorithm (BP) and radial basis function (RBF) type neural network applying the whole history mapping approach. As suggested by Czél et al. [1], a novel (time vs. temperature) representation of the input data was applied. Numerical tests were performed to analyze the accuracy of the two network types with noiseless and noisy inputs. Moreover, it is not necessary to retrain the network when the temperature range of the measurement is changed. Based on the presented results it can be stated that feed-forward neural networks are powerful tools in non-iterative solution of function estimation inverse heat conduction problems and they are likely to be very effective in evaluation of real measured temperature histories to simultaneously determine the volumetric heat capacity and thermal conductivity as an arbitrary function of temperature.

LiDAR Point Cloud Registration by Image Detection Technique

In this letter, a novel approach that utilizes the spectrum information (i.e., images) provided in a modern light detection and ranging (LiDAR) sensor is proposed for the registration of multistation LiDAR data sets. First, the conjugate points in the images collected at varied LiDAR stations are extracted through the speedup robust feature technique. Then, by applying the image-object space mapping technique, the 3-D coordinates of the conjugate image points can be obtained. Those identified 3-D conjugate points are then fed into a registration model so that the transformation parameters can be immediately solved using the efficient noniterative solution to linear transformations technique. Based on numerical results from a case study, it has been demonstrated that, by implementing the proposed approach, a fully automatic and reliable registration of multistation LiDAR point clouds can be achieved without the need for any human intervention.

Achieving Asymptotic Efficient Performance for Squared Range and Squared Range Difference Localizations

The estimation of a source location using directly the range or range difference measurements is difficult and requires numerical solution, which is caused by the highly non-linear relationship between the measurements and the unknown. We can obtain a computationally efficient and non-iterative algebraic solution by squaring the measurements first before solving for the unknown. However, a recent study has shown that such a solution is suboptimum in reaching the CRLB performance and the localization accuracy could be significantly worse in some localization geometries. This paper demonstrates that when range weighting factors are introduced to the squared measurements, the resulting solution will be able to reach the CRLB accuracy. Both the squared range and squared range difference cases are considered, and the mean-square error (MSE) and the bias of the resulting solutions are derived. The asymptotic efficiency of the proposed cost functions are proven theoretically and validated by simulations. The effects of range weighting factors on the localization performance under different sensor number, noise correlation, and localization geometry are examined. Introducing range weightings to the squared range measurements increases the bias but it is negligible in the MSE. Having range weightings in the squared range difference measurements improves both the MSE and bias.

Mixed integer (0–1) fractional programming for decision support in paper production industry

This paper presents an effective and efficient method for solving a special class of mixed integer fractional programming (FP) problems. We take a classical reformulation approach for continuous FP as a starting point and extend it for solving a more general class of mixed integer (0–1) fractional programming problems.To stress the practical relevance of the research we focus on a real-life application in paper production industry. The constantly advancing physical knowledge of large scale pulp and paper production did have a substantial impact on an existing DSS in which mixed integer (0–1) fractional programming is introduced. We show that the motivation to solve a real-life fractional programming problem can provide the basis for a new approach in a new context that has an added value of its own, even outside the given application area. We describe the main characteristics of the DSS, the necessity to develop a non-iterative solution procedure and demonstrate both the effectiveness and efficiency of the proposed approach from practical data sets.

Noniterative Datum Transformation Revisited with Two-Dimensional Affine Model as a Case Study

In geospatial applications, the datum transformation has been necessarily employed to transform the geospatial outcomes from the data-collection system to the user-interested system. Its key is to compute the transformation parameters that describe the geometric relation between two datum systems. The ordinary least-squares based transformation parameter estimation needs the iterative computations unless the initial values of parameters are approximate enough, which is usually time-consuming. Particularly with the development of (near) real-time data collection techniques, such iterative datum transformation method cannot meet the real-time applications. In this paper, we study the noniterative method in terms of the multivariate least-squares theory with two-dimensional empirical affine transformation as a case study. We address the noniterative transformation for the partially and fully error-affected affine models, respectively. The study indicates that the noniterative solution exists when the variance matrix of coordinate errors is structured as Q0⊗Q with Q the variance matrix of single point and Q0 the correlation matrix between points. The numerical examples show that the noniterative method can obtain the practically equivalent result with the ordinary method but improve the computation efficiency significantly. Therefore, the noniterative method is promising for the real-time datum transformation applications.

Accelerated Direct Solution of the Method-of-Moments Linear System

This paper addresses the direct (noniterative) solution of the method-of-moments (MoM) linear system, accelerated through block-wise compression of the MoM impedance matrix. Efficient matrix block compression is achieved using the adaptive cross-approximation (ACA) algorithm and the truncated singular value decomposition (SVD) postcompression. Subsequently, a matrix decomposition is applied that preserves the compression and allows for fast solution by backsubstitution. Although not as fast as some iterative methods for very large problems, accelerated direct solution has several desirable features, including: few problem-dependent parameters; fixed time solution avoiding convergence problems; and high efficiency for multiple excitation problems [e.g., monostatic radar cross section (RCS)]. Emphasis in this paper is on the multiscale compressed block decomposition (MS-CBD) algorithm, introduced by Heldring , which is numerically compared to alternative fast direct methods. A new concise proof is given for the N2 computational complexity of the MS-CBD. Some numerical results are presented, in particular, a monostatic RCS computation involving 1 043 577 unknowns and 1000 incident field directions, and an application of the MS-CBD to the volume integral equation (VIE) for inhomogeneous dielectrics.

Inverse Identification of Temperature-Dependent Volumetric Heat Capacity by Neural Networks

An artificial neural network (NN)-based solution of the inverse heat conduction problem of identifying the temperature-dependent volumetric heat capacity function of a solid material is presented in this paper. The inverse problem was defined according to the evaluation of the BICOND thermophysical property measurement method. The volumetric heat capacity versus temperature function is to be determined using the measured transient temperature history of a single sensor. In this study, noiseless and noisy artificial measurements were generated by the numerical solution of the corresponding direct heat conduction problem. The inverse problem was solved by back-propagation and radial basis function type neural networks applying the whole history mapping approach. The numerical tests included the comparison of two different data representations of the network inputs (i.e., temperature vs. time and time vs. temperature) and accuracy analysis of the two network types with noiseless and noisy inputs. Based on the results presented, it can be stated that feed-forward NNs are powerful tools in a non-iterative solution of function estimation inverse heat conduction problems and they are likely to be very effective in evaluation of real measured temperature histories to determine the volumetric heat capacity as an arbitrary function of temperature.

ASP&lt;i&gt;n&lt;/i&gt;P: An Accurate and Scalable Solution to the Perspective-&lt;i&gt;n&lt;/i&gt;-Point Problem

We propose an accurate and scalable solution to the perspective-n-point problem, referred to as ASPnP. Our main idea is to estimate the orientation and position parameters by directly minimizing a properly defined algebraic error. By using a novel quaternion representation of the rotation, our solution is immune to any parametrization degeneracy. To obtain the global optimum, we use the Gröbner basis technique to solve the polynomial system derived from the first-order optimality condition. The main advantages of our proposed solution lie in accuracy and scalability. Extensive experiment results, with both synthetic and real data, demonstrate that our proposed solution has better accuracy than the state-of-the-art noniterative solutions. More importantly, by exploiting vectorization operations, the computational cost of our ASPnP solution is almost constant, independent of the number of point correspondences n in the wide range from 4 to 1000. In our experiment settings, the ASPnP solution takes about 4 milliseconds, thus best suited for real-time applications with a drastically varying number of 3D-to-2D point correspondences.

Addressing the Inverse Problem of Far-Field Imaging: A Noniterative Exact Solution for Phase in Imaging

This talk describes an approach to a solution of inverse problems in far-field imaging with application to optical, electron/ion beam, x-ray and other imaging modalities. Two aspects of the inverse problem in optical imaging are phase and resolution. Whereas techniques such as STED, PALM, STORM etc address the latter, they lack phase information as does near-field optics. Furthermore, all of these techniques are serial. We introduce an approach based on a controllable nanoscopic far-field optical point source integrated into parallel imaging. In the first demonstration of this development we focus on the inverse problem of the phase of an image which has never been solved exactly and can only be approached through iterative methods with all their problems of nonconvergence, slow convergence, convergence to local minima, and stagnation. We now show that it is possible to obtain, with such a method, an exact solution to the inverse problem of phase both experimentally and theoretically. Our method is based on the breakthrough that crystallography experienced in phase retrieval for large molecular entities by Max Perutz's introduction of “heavy atoms” using the method of isomorphous replacement. Scanning probe microscopy and its full integration with optical microscopy allows us to apply these X-ray concepts to implement “heavy atom” restoration of phase in optical phase retrieval. In analogy to the heavy atom method, we acquire Fourier intensities in place of an X-ray diffraction pattern, and in place of the heavy atom, we utilize a nanometrically translatable point source of light, coherently related to the far-field illumination this leads to 3D optical imaging. The methodology has super-resolution potential, and thus, heavy atom restoration of phase with super-resolution (HARPS) has the potential to provide super-resolution 3D images in real time.