Newton Algorithm Research Articles

Loading–shape interaction analysis of membrane structures subjected to deformation dependent hydrostatic loadings

Due to their flexibility, membrane structures loaded and/or supported by gas and/or fluid undergo large deformations when subjected to external loads, for example when a relatively flat membrane deflects under accumulative ponding. The most noteworthy feature during the deformation process is the loading–shape interaction since changes in the form of the structure induce changes in the loading profiles, and vice versa. Description of the hydrostatic loading is complex since the discontinuity of structural domain and fluid domain occurs with the deformation of membrane. In this paper, an interactive model dealing with accurate hydrostatic load description and nonlinear finite element analysis is employed to analysis the structural response. The hydrostatic loading is described in vectors by fluid level and loading direction in current mesh of the membrane. In case to conserve a specific volume, the free surface height is calculated with a Newton algorithm. Equivalent nodal forces evaluation on different elements are deduced especially when they are partially or fully loaded by judging the spatial relation between the element and the fluid domain, and integrating with shape functions and area coordinates in a local coordinate system. In case of multiple fluid domains, the hydrostatic load description procedure is proposed to be conducted for each fluid domain with care in identifying respective loading direction and then sum them up to get the total hydrostatic loading forces. Two numerical examples, (a) single fluid domain case: a flat membrane subjected to accumulative water loading, and (b) multiple fluid domains case: an air-inflated dam partially filled with water and loaded by lateral pressures of water, are presented to demonstrate correctness, performance and effectiveness of the proposed method in large displacement analysis of flexible membrane structures subjected to deformation dependent hydrostatic loading.

Based on the difference of Newton’s method integrated energy system distributed collaborative optimization

With the integration of renewable energy into the grid, the traditional power system stability faced by huge challenges, and the development of integrated energy system, it is of essence to improve the coupling of multiple integrated energy systems of different types, management in the integrated energy system and reduce the pressure of communication and computing, in this paper, we construct a distributed Newton algorithm based on Newton’s method to accelerate the solving speed, which decreases the times of iterations to reduce the pressure of communication and calculation, saving the cost of operation. Besides, privacy protection is particularly important for a distributed control system, under the premise that calculation speed is guaranteed, meanwhile, privacy protection of all agents in an integrated energy system is also critical. This study uses annular directed distributed algorithm to enhance the privacy of integrated distributed energy systems in the intelligent body, so as to fully ensure the privacy safety of all agents in the system. Moreover, the forementioned difference Newton algorithm in this study avoid the behavior of Zeno, greatly accelerating the speed of iteration and finding the best energy market price,. At the same time, the privacy safety of all agentsin the distributed energy system are ensured. Finally, a distributed integrated energy system based on the algorithm proposed by this study has went through theoretical proof and simulation experiment, whose result shows the validity of the algorithm.

On Searching All Solutions of Microwave Filter Synthesis Based on Interval Arithmetic

In this letter, the numerical interval Newton algorithm (NINA) is introduced into coupling matrix (CM) synthesis (reconfiguration) for the first time. In particular, all existing real solutions to microwave filter CM can be found with mathematical certainty, which provides designers with the flexibility to select the most suitable configuration. Also, CM limits can be directly imposed in the synthesis process to only compute solutions in the specified range. This synthesis process is independent of initial variable values, and the global convergence is theoretically guaranteed. Moreover, NINA is a deterministic method that can verify the existence or nonexistence, uniqueness or non-uniqueness of a given CM topology with the desired filtering performance. Furthermore, the processes dealing with both well-determined and over-determined systems are discussed to cover the synthesis of CMs with diverse configurations. One practical sixth-order dielectric filter with parasitic couplings is exemplified to validate the effectiveness of the presented synthesis approach.

An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models

In this paper, we propose and analyze a variant of Lagrange multiplier approach with a penalty term for phase-field models. More precisely, we develop two types of linear second-order decoupled and unconditionally energy stable time-stepping schemes for phase-field models based on the Crank-Nicolson (CN) and the second order backward differentiation formulations (BDF2), where the scalar multiplier variable is evaluated by solving a nonlinear algebraic equation. Comparing with the Lagrange multiplier method (Cheng et al. (2020) [6]) for phase-field models, a scalar penalty term is first introduced to improve the time-step size constraint for the convergence of Newton's iterative algorithm for the nonlinear algebraic equation, which allows us to obtain robust and efficient numerical schemes for phase-field models. Moreover, we only use first order time-stepping scheme to approximate the scalar multiplier variable, which will not affect the second order accuracy of the unknown phase function with some special Lagrange multiplier functions. Meanwhile, it plays an essential role in the derivation of energy stability of the proposed BDF2-based scheme on the nonuniform temporal mesh. In addition, the energy stability of the CN scheme is rigorously established. Furthermore, we deduce that the proposed BDF2 scheme is energy stable with the adjacent time-step ratios no more than 4.8645. Finally, a series of numerical tests are presented to numerically validate the efficiency and stability of our proposed Lagrange multiplier approach.

Efficient segment-update block LMS-Newton algorithm for active control of road noise

For active control of road noise inside a vehicle, many reference signals are required in adaptive control systems, resulting in slow convergence speeds and heavy computational loads. In this paper, a means of reducing the computational complexity of adaptive control algorithms is investigated based on the fast-converging least-mean-squares Newton (LMS-Newton) algorithm. By analyzing the autocorrelation matrix of reference signals filtered by secondary paths, it is found that the dimensions of the autocorrelation matrix of filtered reference signals do not necessarily have to equal the dimensions of the adaptive control filters, and instead the dimensions can be determined from the correlation between reference signals filtered by secondary paths. Then a computationally efficient segment-update block algorithm is proposed that splits adaptive control filters into several segments and the coefficients of each segment are updated independently. Theoretical analysis shows that the proposed segment-update block algorithm has similar convergence speed and noise-reduction performance as the traditional LMS-Newton algorithm when a segment of appropriate length is chosen, but the computational complexity is reduced dramatically because the inverse autocorrelation matrix is replaced by a smaller matrix. The performance of the proposed algorithm is verified through simulation experiments with different recorded road noises and secondary paths, and the results are compared with those of some typical algorithms. The simulation results showed that the proposed algorithm achieved a similar noise-reduction performance with an approximately 8.9-times reduction of computational complexity compared with the traditional LMS-Newton algorithm.

Identification of twin-shaft gas turbine based on hybrid decoupled state multiple model approach

Monitoring control of industrial systems is essential for the good productivity and safety of installations and operators, with better performance that must be guaranteed. This is often challenging due to the nonlinearities and dynamic complexities of these systems, adding operating constraints and instability. Hence, the multi-models constitute then an adapted tool for the modeling of the nonlinear systems to characterize their dynamic behaviors. Indeed, this work proposes the implementation of a hybrid identification approach of the operating variables of a gas turbine, thus making it possible to interconnect the various linear sub-models with decoupled states in order to generate the global output of their nonlinear model, from the exploitation in real time of the turbine's input/output data. However, the suggested decoupled-state multi-model approach offers an interesting alternative to the optimization procedure of the estimated turbine parameters. By using gradient and Gauss–Newton algorithms, improved by genetic algorithms combined with NSGA-II in hybrid form, in order to converge toward the best solutions with an optimal cost function, the obtained implementation results show that this approach allows the convergence of the estimated turbine variables and describes its behavior in real time, with the guaranteed efficiency of the proposed decoupled state multi-model method.

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

Optimal design of chemoepitaxial guideposts for the directed self-assembly of block copolymer systems using an inexact Newton algorithm

Directed self-assembly (DSA) of block copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP melts to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can be formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies.In this paper, we adopt a phase field model given by a nonlocal Cahn–Hilliard partial differential equation (PDE) based on the minimization of the Ohta–Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. We demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.

A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency

In this paper, a space-time absolute nodal coordinate formulation cable (SAC) element forming technique based on the Lagrange family of shape functions is proposed. Two distinct SAC elements, each with a distinct spatial shape function, have been generated by this method. Moreover, the external forces such as the bending moment and the air resistance formula have been accounted for. The Lagrange multiplier method, along with the concepts of replacement constraint and supplementary constraint, has been employed to provide a solution for the dynamics of constrained mechanical systems. Additionally, a constraint conversion strategy has been suggested. The solver has been constructed through Hamilton’s law of varying action. The space-time finite element method is used to solve dynamic problems, employing the Newton algorithm and quasi-Newton algorithm. The accuracy and efficiency of the solution has been verified by three simulations and one experiment. The circle-bending static simulation and the double-ended velocity impact dynamic simulation demonstrate the accuracy of the two elements. The correlation between statics and dynamics has been studied for different discretization methods and different solvers’ calculation accuracy and efficiency. Different modeling methods, time steps, order and the application of the quasi-Newton method all have a bearing on the efficiency of the solution. Finally, a comparison with an experiment in the free-pendulum simulation reveals the capability of this model to simulate dynamic problems with air resistance.

Newton time-extracting wavelet transform: An effective tool for characterizing frequency-varying signals with weakly-separated components and theoretical analysis

In this paper, we propose a high resolution time-frequency (TF) analysis method called Newton time-extracting wavelet transform (NTEWT), which is designed to analyze frequency-varying signals with fast varying group delay (GD). Firstly, we discuss the relationship among the fixed points of time reassignment operator, the ridge points of wavelet transform and GD of the signal. Combining the above relations and Newton algorithm, we propose a Newton GD estimator. By only retaining the TF information most related to frequency-varying features of the signal and removing the weakly-related TF coefficients, we further introduce the NTEWT, which can not only achieve a more concentrated TF representation, but also enable signal reconstruction. Meanwhile, we develop a theoretical analysis of NTEWT under the mathematical framework. Firstly, we introduce a precise mathematical definition of a class of weakly-separated frequency-varying chirp-like components, and we prove that Newton GD estimator can accurately estimate GD of arbitrary function in this class, and NTEWT does indeed succeed in decomposing these functions. Finally, we use numerical experiments to evaluate the performance of the proposed NTEWT in terms of TF concentration, GD estimation and signal reconstruction.

Error-Summation Enhanced Newton Algorithm for Model Predictive Control of Redundant Manipulators

Redundant manipulators have been investigated and employed in various fields, and its trajectory tracking is of much importance in the field of robotic control. In this article, a model predictive control (MPC) scheme for the trajectory tracking of redundant manipulators is constructed, which minimizes the tracking error, velocity norm, and acceleration norm simultaneously. The commonly used trajectory tracking schemes for redundant manipulators, such as the minimum-velocity-norm scheme and minimum-acceleration-norm scheme, handle joint limits at different levels by introducing additional parameters, which reduces the feasible region of decision variables. In contrast, the proposed scheme directly considers these limits at three different levels, without reducing the feasible region of decision variables. In addition, to compensate for the deficiencies of most existing algorithms in noise environments, an error-summation enhanced Newton (ESEN) algorithm is proposed for solving the MPC scheme. Through theoretical analysis, it is determined that the proposed ESEN algorithm has a small steady-state error under noise conditions. Finally, in contrast with the comparative trajectory tracking schemes, as determined through computer simulations and experiments, the proposed MPC scheme solved by the ESEN algorithm not only enables the redundant manipulator to perform the trajectory tracking task in excellent fashion, but also offers advantages of high efficiency, fast responsiveness, and strong noise tolerance.

Space‐time phase‐field fracture optimal control computations

AbstractIn this work, we undertake additional computational performance studies of a recently developed space‐time phase‐field fracture optimal control framework. Therein, the phase‐field forward problem is formulated in a monolithic fashion. The optimal control problem is formulated with the help of the reduced approach in which the state variable is represented with a solution operator applied to the control. To this end, a Newton algorithm in the control variable is formulated for which auxiliary equations must be solved. Two numerical experiments demonstrate the capabilities of our framework.

A Sequential Generation Redispatch Algorithm to Ensure Power System Small Signal Stability under Low-Frequency Oscillations

Generation rescheduling is a major action against abnormal system conditions such as small signal instability. The operating point of the power system impacts the small signal stability; therefore, this article proposes a novel sequential generation rescheduling model as a preventive control against low-frequency fluctuations. An optimal power flow (OPF) problem is utilized to determine the base operating point. Then, in the process of improving the small signal stability, the modal analysis specifies the system stability status in each step, and the proposed second-order convex redispatch model defines the optimal direction of generation rescheduling. Small signal stability is considered a constraint in this model. Using a sensitivity analysis, system generators and system buses are divided into two increasing and decreasing groups, which are included in the model according to their contribution coefficients in damping improvement. Finally, the alternating current (AC) power flow analysis provides the subsequent operating points. The redispatch model is solved using the quadratic programming algorithm, and the Newton algorithm is used to manage the nonlinear and nonconvex characteristics of the power flow model. The proposed method is simulated over the IEEE 9, 39, and 118 bus test systems. It is shown that the proposed method increases the damping of the power system from the unstable state to the desired condition by controlling the operating point.

Privacy-preserving energy trading management in networked microgrids via data-driven robust optimization assisted by machine learning

Trading mechanism design, uncertainty treatment and privacy protection are the main issues in the energy management of networked microgrids (MGs). To address these issues in a comprehensive manner, this paper proposes a data-driven two-level transactive energy management framework, where the upper-level determines the optimal strategies of internal scheduling within MGs and external trading between MGs while the lower-level formulates a Nash bargaining game model for the fair allocation of trading benefits. The uncertainties of renewable energy sources are fully captured by an adjustable data-driven robust optimization approach with an uncertainty set constructed using the robust kernel density estimation (RKDE) as a machine learning technique. The resulting uncertainty set can provide robust scheduling and trading schemes even when the power generation data of wind turbines and photovoltaic systems are contaminated with anomalous samples, whereas conventional sets are not reliable in the case of contaminated data. To preserve the operational independence and information privacy of MGs, the proposed model is solved in a distributed manner by the alternating direction method of multipliers (ADMM) and the augmented Lagrange-based alternating direction inexact Newton (ALADIN) algorithms. ADMM is commonly used in previous studies, but it has low computational efficiency for handling the consensus process among a large number of MGs. This paper applies ADMM and ALADIN to enhance the applicability of the proposed model when the size and complexity of the networks increase. Numerical tests show the effectiveness of the proposed framework and solution methodology in terms of system cost, solution robustness, and convergence speed.

A Novel Method for Microphone Channel Frequency Response Calibration Based on Newton Algorithm

Microphone arrays are widely applied in practical applications. In an array, each channel conventionally contains an acoustic sensor and a following signal conditioning circuit (SCC). Frequency response mismatches among channels will adversely affect array’s overall performance. Existing calibration methods provide promising results; however, their accuracy varies considerably across full frequency band and the passband bandwidth shrinkage problem also arises. To address these problems, we propose a novel calibration method based on Newton algorithm. The proposed method cascades calibration filters after channel output signals allowing the frequency response calibration to be equivalent to the optimization of calibration filter coefficients. First, all uncalibrated channels’ bandwidths are analyzed using a chirp calibration signal, resulting in appropriate construction of the desired output. Next, a spectrum mean square error criterion between each channel output and the desired signal is utilized to formulate the frequency response mismatch. Finally, we employ Newton algorithm to design the calibration filter coefficients corresponding to each channel. Explicit mathematical derivation is provided. Since the cost function is quadratic, expressions of the constant Hessian matrix and its inverse are derived. Furthermore, the inverse Hessian can be efficiently computed using spectral coefficients of channel outputs. Simulation results reveal that the proposed method significantly outperforms the state-of-the-art calibration approaches in terms of calibration accuracy, calibrated passband bandwidth performance, and convergence rate. Real-world experiments also verify its effectiveness.

Broadband Impedance Matching and Residual Vibration Attenuation for Underwater Acoustic Tonpilz Transducers Based on Adaptive Gauss–Newton Algorithm

Broadband Impedance Matching and Residual Vibration Attenuation for Underwater Acoustic Tonpilz Transducers Based on Adaptive Gauss–Newton Algorithm

On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions

Most machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, with samples provided in a streaming fashion, we define general (weighted averaged) stochastic Newton algorithms, for which a theoretical analysis of their asymptotic efficiency is conducted. The corresponding implementations are shown not to require the inversion of a Hessian estimate at each iteration under a quite flexible framework that covers the case of linear, logistic or softmax regressions to name a few. Numerical experiments on simulated and real data give the empirical evidence of the pertinence of the proposed methods, which outperform popular competitors particularly in case of bad initializations.

Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of $$O(\epsilon ^{-3/2})$$ to obtain an $$(\epsilon ,\sqrt{\epsilon })$$ -second-order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $$\epsilon $$ and minimum eigenvalue of Riemannian Hessian lower bounded by $$-\sqrt{\epsilon }$$ , is established when the manifold is embedded in the Euclidean space. Furthermore, the paper proposes a computationally more appealing modification of the algorithm which only requires an inexact solution of the cubic regularized Newton subproblem with the same iteration complexity. The proposed algorithm is evaluated and compared with three other Riemannian second-order methods over two numerical studies on estimating the inverse scale matrix of the multivariate t-distribution on the manifold of symmetric positive definite matrices and estimating the parameter of a linear classifier on the sphere manifold.

Method of optimizing deterministic small free-form surfaces for off-axis optical systems with extremely low aberration

A mapping model of arbitrarily shaped surfaces and the system image quality is constructed to examine the optimization of the residual aberration of an off-axis optical system to construct an off-axis optical system with very small aberrations. First, orthogonal surfaces are chosen within the arbitrary aperture. The mapping relationship between the orthogonal surfaces and system wavefront aberration is then established. Finally, the surfaces required for optimization are acquired by solving the mapping relationship through singular value decomposition and the Gauss–Newton algorithm. The residual aberration of the system is optimized using free-form surfaces with small deviations. In this paper, the residual aberration of an off-axis optical system with a non-circular aperture is optimized by adopting the above method, with the residual aberration of the system reducing from 0.549 nm (root-mean-square, RMS) to 0.443 nm (RMS) after adding a small free-form surface to a single lens, and to 0.393 nm (RMS) after adding small free-form surfaces to two lenses. The optical system with a circular aperture is optimized and the residual aberration of the system is reduced from 2 nm (RMS) to 0.47 nm (RMS).

A Self-Calibrating Localization Solution for Sport Applications with UWB Technology

This study addressed the problem of localization in an ultrawide-band (UWB) network, where the positions of both the access points and the tags needed to be estimated. We considered a fully wireless UWB localization system, comprising both software and hardware, featuring easy plug-and-play usability for the consumer, primarily targeting sport and leisure applications. Anchor self-localization was addressed by two-way ranging, also embedding a Gauss-Newton algorithm for the estimation and compensation of antenna delays, and a modified isolation forest algorithm working with low-dimensional set of measurements for outlier identification and removal. This approach avoids time-consuming calibration procedures, and it enables accurate tag localization by the multilateration of time difference of arrival measurements. For the assessment of performance and the comparison of different algorithms, we considered an experimental campaign with data gathered by a proprietary UWB localization system.