Fixed-point Iteration Algorithm Research Articles

Tensor power flow formulations for multidimensional analyses in distribution systems

In this paper, we present two multidimensional power flow formulations based on a fixed-point iteration (FPI) algorithm to efficiently solve hundreds of thousands of Power flows (PFs) in distribution systems. The presented algorithms are the base for a new TensorPowerFlow (TPF) tool and shine for their simplicity, benefiting from multicore Central processing unit (CPU) and Graphics processing unit (GPU) parallelization. We also focus on the mathematical convergence properties of the algorithm, showing that its unique solution is at the practical operational point. The proof is validated using numerical simulations showing the robustness of the FPI algorithm compared to the classical Newton–Raphson (NR) approach. In the case study, a benchmark with different PF solution methods is performed, showing that for applications requiring a yearly simulation at 1-minute resolution, the computation time is decreased by a factor of 164, compared to the NR in its sparse formulation. Finally, a set of applications is described, highlighting the potential of the proposed formulations over a wide range of analyses in distribution systems.

Efficient Analysis of Overdispersed Data Using an Accurate Computation of the Dirichlet Multinomial Distribution.

Modeling count data using suitable statistical distributions has been instrumental for analyzing the patterns it conveys. However, failing to address critical aspects, like overdispersion, jeopardizes the effectiveness of such an analysis. In this paper, overdispersed count data is modeled using the Dirichlet Multinomial (DM) distribution by maximizing its likelihood using a fixed-point iteration algorithm. This is achieved by estimating the DM distribution parameters while comparing the recent Languasco-Migliardi (LM), and the Yu-Shaw (YS) procedures, which address the well-known computational difficulties of evaluating its log-likelihood. Experiments were conducted using multiple datasets from different domains spanning polls, images, and IoT network traffic. They all showed the superiority of the LM procedure as it succeeded at estimating the DM parameters at the designated level of accuracy in all experiments, while the YS procedure failed to produce sufficiently accurate results (or any results at all) in several experiments. Moreover, the LM procedure achieved a speedup that ranged from 2-fold to 20-fold over YS.

Revisiting trace norm minimization for tensor Tucker completion: A direct multilinear rank learning approach

To efficiently express tensor data using the Tucker format, a critical task is to minimize the multilinear rank such that the model would not be over-flexible and lead to overfitting. Due to the lack of rank minimization tools in tensor, existing works connect Tucker multilinear rank minimization to trace norm minimization of matrices unfolded from the tensor data. While these formulations try to exploit the common aim of identifying the low-dimensional structure of the tensor and matrix, this paper reveals that existing trace norm-based formulations in Tucker completion are inefficient in multilinear rank minimization. We further propose a new interpretation of Tucker format such that trace norm minimization is applied to the factor matrices of the equivalent representation, rather than some matrices unfolded from tensor data. Based on the newly established problem formulation, a fixed point iteration algorithm is proposed, and its convergence is proved. Numerical results are presented to show that the proposed algorithm exhibits significant improved performance in terms of multilinear rank learning and consequently tensor signal recovery accuracy, compared to existing trace norm based Tucker completion methods.

An efficient PGD solver for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based on the Hamiltonian formalism. The novelty of this work lies in the implementation of a solver that is halfway between Modal Decomposition and the conventional PGD framework, so as to accelerate the fixed-point iteration algorithm. Additional procedures such that Aitken’s delta-squared process and mode-orthogonalization are incorporated to ensure convergence and stability of the algorithm. Numerical results regarding the ROM accuracy, time complexity, and scalability are provided to demonstrate the performance of the new solver when applied to dynamic simulation of a three-dimensional structure.

Theoretical insights on the pre-image resolution in machine learning

While many nonlinear pattern recognition and data mining tasks rely on embedding the data into a latent space, one often needs to extract the patterns in the input space. Estimating the inverse of the nonlinear embedding is the so-called pre-image problem. Several strategies have been proposed to address the estimation of the pre-image; However, there are no theoretical results so far to understand the pre-image problem and its resolution. In this paper, we provide theoretical underpinnings of the resolution of the pre-image problem in Machine Learning. These theoretical results are on the gradient descent optimization, the fixed-point iteration algorithm and Newton’s method. We provide sufficient conditions on the convexity/nonconvexity of the pre-image problem. Moreover, we show that the fixed-point iteration is a Newton update and prove that it is a Majorize-Minimization (MM) algorithm where the surrogate function is a quadratic function. These theoretical results are derived for the wide classes of radial kernels and projective kernels. We also provide other insights by connecting the resolution of this problem to the gradient density estimation problem with the so-called mean shift algorithm.

A Fast and Cost-Effective Calibration Strategy of Inter-Stage Residual Amplification Errors for Cyclic-Pipelined ADCs

Due to nonideal residue amplification, the limited resolution of pipelined analog-to-digital converters (ADCs) has become a popular research topic for ADC designers. High-gain and high-speed amplifiers usually consume too much power for a decent ADC. Hence, this paper proposes a fast and cost-effective foreground calibration strategy for cyclic-pipelined ADCs. The calibration strategy compensates for the gain error due to inter-stage residual amplification, which alleviates the DC gain requirement for internal amplifiers. Unlike other digital calibrations, the proposed scheme is implemented with a cyclic-pipelined structure, and only one parameter needs to be calibrated, whose value can be feasibly calculated by the Fix-Point Iteration algorithm. The proposed calibration scheme is implemented in an area-efficient 16-bit, 2 MS/s cyclic-pipelined ADC, fabricated in 180 nm CMOS technology. The ADC is designed and realized by cycling a 6-bit sub-ADC four times with 1-bit redundancy each time. The calibration algorithm manages to recover the sampled data to 93.85 dB spurious free dynamic range (SFDR) even with a 57.8 dB-DC-gain amplifier. The total power consumption of ADC is 17.92 mW and it occupies an active area of 1.8 mm2.

Maximum correntropy unscented filter based on unbiased minimum-variance estimation for a class of nonlinear systems

Introduction: The unscented Kalman filter based on unbiased minimum-variance (UKF-UMV) estimation is usually used to handle the state estimation problem of nonlinear systems with an unknown input. When the nonlinear system is disturbed by non-Gaussian noise, the performance of UKF-UMV will seriously deteriorate.Methods: A maximum correntropy unscented filter based on the unbiased minimum variance (MCUF-UMV) estimation method is proposed on the basis of the UKF-UMV without the need for estimation of an unknown input and uses the maximum correntropy criterion (MCC) and fixed-point iterative algorithm for state estimation.Results: When the measurement noise of the nonlinear system is non-Gaussian noise, the algorithm performs well.Discussion: Our proposed algorithm also does not require estimation of an unknown input, and there is no prior knowledge available about the unknown input or any prior assumptions. The unknown input can be any signal. Finally, a simulation example is used to demonstrate the effectiveness and reliability of the algorithm.

Emergency control of firefighting vehicles based on distributed integer programming

Fire-related emergency vehicle scheduling has always been an important part of autonomous vehicle emergency management. To maintain the fire safety of buildings and reduce the transportation loss of emergency resources, this paper constructs a multi-objective integer planning model for fire emergency resource scheduling, which minimizes the time spent in the transportation of fire materials and considers the transportation needs of potential fire vehicles. The fixed-point iterative algorithm is realized and optimized, and distributed computing is adopted to improve the solving speed. In this paper, the comparison and simulation experiments with Branch and Bound, improved particle swarm optimization and other algorithms are carried out. The results show that the model and the algorithm can effectively deal with the fire emergency vehicle scheduling problem under different simulated fires and multiple fires occurring at the same time, and have certain advantages in numerical stability and convergence speed.

Dynamical behavior of q-deformed logistic map in superior orbit

In this paper, we study the q-deformed logistic map in Mann orbit (superior orbit) which is a two-step fixed-point iterative algorithm. The main aim of this paper is to investigate the whole dynamical behavior of the proposed map through various techniques such as fixed-point and stability approach, time-series analysis, bifurcation plot, Lyapunov exponent and cobweb diagram. We notice that the chaotic behavior of q-deformed logistic map can be controlled by choosing control parameters carefully. The convergence and stability range of the map can be increased substantially. Moreover, with the help of bifurcation diagrams, we prove that the stability performance of this map is larger than that of existing other one dimensional chaotic maps. This map may have better applications than that of classical logistic map in various situations as its stability performance is larger. Mathematics Subject Classification (2010): 34H10, 37M10, 37B25, 37F45. Received 09 April 2021; Accepted 08 October 2021

Cooperative design of feeder bus and bike-sharing systems

Feeder services, which provide travellers effective access from local areas to trunk-line systems, are essential, especially when the long-haul trunk routes are designed with large spacing between stations. Apart from the short-haul bus, the development of bike-sharing systems has led to a powerful supplementary feeder service. Most research on feeder services focuses on the separate design of bike and bus systems, but their joint design deserves further study. This research proposes a bi-level mixed-integer programming model to simultaneously design the locations of feeder buses and shared bike stations as well as the bus routes. A modified genetic algorithm that encapsulates the fixed-point iteration and Frank-Wolfe algorithms is developed to solve the model for large network applications. Numerical examples are given to compare the joint design with the separate design. The key factors that affect the design outcomes are investigated. The effects of joint design on operators and travellers are also discussed.

A faster fixed point iterative algorithm and its application to optimization problems

<p>In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $ (\alpha, \beta) $-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $ (\alpha, \beta) $-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.</p>

Algebraic method for multisensor data fusion.

In this contribution, we use Gaussian posterior probability densities to characterize local estimates from distributed sensors, and assume that they all belong to the Riemannian manifold of Gaussian distributions. Our starting point is to introduce a proper Lie algebraic structure for the Gaussian submanifold with a fixed mean vector, and then the average dissimilarity between the fused density and local posterior densities can be measured by the norm of a Lie algebraic vector. Under Gaussian assumptions, a geodesic projection based algebraic fusion method is proposed to achieve the fused density by taking the norm as the loss. It provides a robust fixed point iterative algorithm for the mean fusion with theoretical convergence, and gives an analytical form for the fused covariance matrix. The effectiveness of the proposed fusion method is illustrated by numerical examples.

A Modified Fixed-Point Iteration Algorithm for Magnetic Field Computation With Hysteresis Models

A Modified Fixed-Point Iteration Algorithm for Magnetic Field Computation With Hysteresis Models

Clustering and Beamwidth Optimization for UAV-Assisted Wireless Communication

With the development of wireless communication technology, unmanned aerial vehicles (UAV) are now widely used in many complex communication scenarios. When a UAV serves as an aerial base station for urban and rural ground users or marine users, it is necessary to consider the clustering of ground users and the energy efficiency of the UAV since the users are usually randomly distributed. For the scenario with randomly distributed ground users and different densities of ground users in urban and rural areas, a clustering and beamwidth optimization method for UAV-assisted wireless communication is proposed. Firstly, the energy efficiency expression of a UAV serving ground users was derived in a downlink wireless communication system assisted by a UAV. Secondly, based on the geographical location information of non-uniformly distributed users, an improved k-means method is proposed to cluster ground users, ensuring that the number of users in each cluster is within an appropriate range. Then, based on the clustering results, a fixed-point iteration (FPI) algorithm was proposed to design the optimal beamwidth of UAVs and improve their energy efficiency. Finally, the superiority of the proposed algorithm in improving energy efficiency was verified through simulation analysis, and the impact of parameters such as the cluster number and transmission power on system energy efficiency was also analyzed.

An Analytical Model for Weak Avalanche Induced Kink Effect of PDSOI nMOSFETs

Silicon-on-insulator (SOI) devices and circuits are widely used in semiconductor industry for its superior natures on integration density, isolations degree. Inefficient body contact of partially-depleted SOI (PDSOI) devices cannot evacuate excess carriers which transported from the weak avalanche multiplication of pinch-off region at high drain voltage bias. These accumulated carriers elevate the internal body potential and trigger electrical effects namely the body bias effect of threshold voltage and the turn-on of parasitic bipolar junction transistor (BJT). A comprehensive analytical model which include all these mechanisms are proposed to describe this kink effect. This model is constructed by an equivalent circuit and five strong nonlinearly coupled equations to describe the electrical behaviors. To solve these equations, a C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$++$</tex-math> </inline-formula> Qt-based graphical interface software is developed for the parameter extraction of the model. The fixed point iteration algorithm are adopted for the solution of equations. The effectiveness and accuracy of this model are verified by the T-gate shaped PDSOI nMOSFETs of 130 nm manufacturing process node. It can predict the electrical behaviors of kink effect. This is meaningful for the circuit simulation of PDSOI platform.

Calibration of Cyclic-Pipelined ADCs Using CMOS for Area-Efficiency

One of the principal obstacles in the development of pipeline analog-to-digital converters (ADCs) is the imprecision associated with residue amplification. Operational amplifiers (Op-Amps) that possess high gain and speed are recognized for their excessive power consumption, making them unsuitable for employment in proficient analog-to-digital converters (ADCs). The study presents a new method for foreground calibration that addresses amplification differences in cyclic-pipelined ADCs, reducing the need for internal amplifier DC gain. The calibration technique was applied to a cyclic-pipelined ADC with a sampling rate of 2 MS/s and 16-bit resolution. The design of this ADC was optimized for area efficiency, and its fabrication utilized 180 nm CMOS technology. The analog-to-digital converter (ADC) used a 5-bit resolution sub-ADC performing 4 cycles to reduce potential errors. Each cycle contained one bit of redundancy. A fixed-point iterative algorithm was used to find the exact gain for each amplifier. Simulation data shows a SINAD of 100. 6 dB, despite a 57 dB DC gain amplifier. The ADC's active area is 1. 8 mm2 at 30 consumption. 43 mW.

An optimization-based supervised learning algorithm for PXRD phase fraction estimation

In powder diffraction data analysis, phase identification is the process of determining the crystalline phases in a sample using its characteristic Bragg peaks. For multiphasic spectra, we must also determine the relative weight fraction of each phase in the sample. Machine learning algorithms (e.g., Artificial Neural Networks) have been applied to perform such difficult tasks in powder diffraction analysis, but typically require a significant number of training samples for acceptable performance. We have developed an approach that performs well even with a small number of training samples. We apply a fixed-point iteration algorithm on the labelled training samples to estimate monophasic spectra. Then, given an unknown sample spectrum, we again use a fixed-point iteration algorithm to determine the weighted combination of monophase spectra that best approximates the unknown sample spectrum. These weights are the desired phase fractions for the sample. Our approach gave the best accuracy and lowest mean absolute error when compared with several machine learning algorithms, including a single hidden-layer neural network.

A bi-level framework for interactive competition and CPPs-based demand response

This paper proposes a multi-leader–follower framework for interactive competition among participants. To do so, a Stackelberg game model is formulated to take into account the price competition between leaders and captive power plants (CPPs). The retailer and the distribution company (DISCO) adopt a rival strategy to determine sale prices, aiming to maximize respective profit in a competitive market. DISCO pricing bound is set during the interaction process. The response at the lower level also affects hierarchical prices and CPP revenue. CPPs enact demand response viable through load shifting and peer-to-peer (P2P) energy trading. A quadratic penalty function is employed in the model to reduce the peak load demand and reshape the load profile, which brings the load consumption curve close to the forecast value. A general Nash bargaining game is utilized to incentivize P2P energy sharing with neighbors. To solve the multi-leader–follower model, a fixed-point iteration algorithm is proposed, which enables the model to reach convergence. Simulations are conducted to demonstrate the price–demand relationship and hierarchical interaction.

The matrix splitting fixed point iterative algorithms for solving absolute value equations

This paper describes two new iterative algorithms for determining absolute value equations. The algorithms are based on a splitting of the coefficient matrix. Moreover, we analyze the convergence effects of the presented algorithms via some theorems. Eventually, numerical tests are provided to confirm the credibility of our procedures.

Signal-to-Interference-Plus-Noise Ratio Based Optimization for Sound Zone Control

Sound zone control is posed as an optimization problem where finite impulse response control filters are jointly optimized for low transmit power, while maintaining a sufficiently high signal-to-interference-plus-noise ratio (SINR) in all zones. This problem statement in particular allows the consideration of sound zone control under the additional influence of external noise, which is rarely taken into account and indeed necessitates an alternative problem statement. In addition, the spectral characteristics of the audio signals are taken into account, both for the transmit power minimization and computing the SINR. The optimization problem is shown to be solved optimally by semidefinite relaxation, but the computational cost is high. An alternate method is proposed, inspired by a duality between transmit and receive beamforming commonly exploited in digital communications. A virtual receive optimization problem for sound zone control is derived, and it is shown to have the same optimal solution as the original transmit optimization problem. The receive optimization problem is solved efficiently through a fixed point iteration algorithm. The two proposed methods are compared against acoustic contrast controlin a simulated reverberant environment, where it is shown that the proposed methods require less transmit power to produce similar SINRs.