Method Of Successive Approximations Research Articles

Boundary Layer Lines of Solutions to Singularly Perturbate Equations With a Saddle Point

In this paper, we study the asymptotic behavior of solutions of singularly perturbed equations in complex domains. The considered equations have a saddle point. The main goal is to prove the existence of boundary-layer lines, boundary-layer regions and regular, singular regions, identifying their features in comparison with previous studies. Harmonic functions and their level lines are involved in solving the set problem. Using the level line, geometric constructions are carried out. The area under consideration is divided into parts and integration paths are chosen that ensure the convergence of some functions with respect to a small parameter. Using the method of successive approximations, the existence and boundedness of the solution of the equation is proved. The features of the boundary lines are revealed.

Inverse Problem for Viscoelastic System in a Vertically Layered Medium

In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables $x_1$ and $x_2$ of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables $x_1$, $x_2$, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem.

Joint Resource Management and Trajectory Optimization for UAV-Enabled Maritime Network.

Due to the lack of places to employ communication infrastructures, there are many coverage blind zones in maritime communication networks. Benefiting from the high flexibility and maneuverability, unmanned aerial vehicles (UAVs) have been proposed as a promising method to provide broadband maritime coverage for these blind zones. In this paper, a multi-UAV-enabled maritime communication model is proposed, where UAVs are deployed to provide the transmission service for maritime users. To improve the performance of the maritime communication systems, an optimization problem is formulated to maximize the minimum average throughput among all users by jointly optimizing the user association, power allocation, and UAV trajectory. To derive the solutions with a low computational complexity, we decompose this problem into three subproblems, namely user association optimization, power allocation optimization, and UAV trajectory optimization. Then, a joint iterative algorithm is developed to achieve the solutions based on the successive convex approximation and interior-point methods. Extensive simulation results validate the effectiveness of the proposed algorithm and demonstrate that UAVs can be used to enhance the maritime coverage.

ИССЛЕДОВАНИЕ ОБЛАСТИ ОПРЕДЕЛЕНИЯ ПАРАМЕТРОВ БАЗИСНОЙ ФУНКЦИИ ДВУХ АРГУМЕНТОВ ПРИ ПОСТРОЕНИИ Λ-ОРТОГОНАЛЬНОЙ БАЗИСНОЙ ФУНКЦИИ

The purpose of the research is to determine the area of setting the argument τ and its discretizationintervals when constructing an algorithm for adaptive nonlinearoptimal smoothing of multi-parameter data of trajectory measurements, which makes it possible tojointly implement the spatial and temporal redundancy of the data obtained. The research wascarried out by constructing a Λ-orthogonal basis function in order to obtain independent estimatesfor the coefficients of the smoothing polynomial. It is shown that it is advisable to solve the problemof determining the maximum likelihood estimate of the coefficient vector of the smoothingpolynomial by the method of successive approximations. When constructing a Λ-orthogonal basisfunction, the maximum likelihood estimate of the coefficient vector of the smoothing polynomial isachieved in 2-3 iterations. It follows from the research results presented in the paper that the accuracyindex Qт as a function of two arguments ( is the smallest value of the argument and Δτ isthe discretization interval of the argument τ) in a wide range of values of these arguments changesslightly, but increases sharply at . In this case, the values of these arguments should notexceed, respectively, the maximum and minimum possible numbers that can be written without lossof accuracy in the re grid of the computer used. With a uniform discretization step of the argument τ it is advisable to choose the argument in the middle part of the interval¸where andrespectively, are the minimum and maximum numbers that can be written into the bit grid of acomputer without loss of accuracy. In case of adverse conditions Approaching the edges of theinterval can lead to an increase in calculation errors in determining the secondary parameters ofthe position of the aircraft due to the fact that the main matrix of the system of equations becomesill-conditioned.

Energy Management System for the Optimal Operation of PV Generators in Distribution Systems Using the Antlion Optimizer: A Colombian Urban and Rural Case Study

This paper presents an Energy Management System (EMS) for solving the problem regarding the optimal daily operation of Photovoltaic (PV) distributed generators in Alternate Current (AC) distribution grids. To this effect, a nonlinear programming problem (NLP) was formulated which considered the improvement of economic (investment and maintenance costs), technical (energy losses), and environmental (CO2 emission) grid indices as objective functions, considering all technical and operating constraints for the operation of AC networks with the presence of PV sources. To solve this mathematical formulation, a master–slave methodology was implemented, whose master stage employed the antlion optimizer to find the power dispatch of PV sources in each period of time considered (24 h). In the slave stage, an hourly power flow based on the successive approximations method was used in order to obtain the values of the objective functions and constraints associated with each possible PV power configuration proposed by the master stage. To evaluate the effectiveness and robustness of the proposed methodology, two test scenarios were used, which included three installed PV sources in an urban and a rural network, considering the PV power generation and demand located reported for Medellín and Capurganá, respectively. These systems correspond to connected and standalone grids located in two different regions of Colombia. Furthermore, the proposed methodology was compared with three optimization methodologies reported in the literature: the Chu and Beasley genetic algorithm, the particle swarm optimization algorithm, and the vortex search optimization algorithm. Simulation results were obtained via the MATLAB software for both test scenarios with all the optimization methodologies. It was demonstrated that the proposed methodology yields the best results in terms of solution quality and repeatability, with shorter processing times.

Secure UAV Relay Communication via Power Allocation and Trajectory Planning

Benefiting from high mobility and maneuverability, unmanned aerial vehicle (UAV) can be flexibly deployed in various wireless networks to improve coverage and connectivity. In this article, the system with a UAV, which relays confidential messages between a couple of unconnected ground nodes with the presence of an eavesdropper is considered. In order to maximize the overall end-to-end secure throughput in a given time period, the joint power allocation and trajectory planning are investigated. The formulated optimization problem is nonconvex, which is difficult to obtain the optimal solution with acceptable complexity. Therefore, we propose an iterative algorithm to achieve a suboptimal solution approximately. In the proposed approach, the joint optimization problem is firstly divided into two subproblems, i.e., the power allocation subproblem and the trajectory planning subproblem. The power is optimized by using the method of successive convex approximation with fixed trajectory, and the trajectory was optimized by successive convex approximation with fixed transmit power. Finally, the power and trajectory are optimized alternately. Since the two subproblems are approximated as convex problems, they can be solved efficiently with acceptable complexity. The simulation results show that remarkable gain in terms of secure throughput can be achieved by the proposed joint optimization algorithm.

Energy-Efficient Trajectory Design for UAV-Aided Maritime Data Collection in Wind

Unmanned aerial vehicles (UAVs), especially fixed-wing ones that withstand strong winds, have great potential for oceanic exploration and research. This paper studies a UAV-aided maritime data collection system with a fixed-wing UAV dispatched to collect data from marine buoys. We aim to minimize the UAV’s energy consumption in completing the task by jointly optimizing the communication time scheduling among the buoys and the UAV’s flight trajectory subject to wind effect. The conventional successive convex approximation (SCA) method can provide efficient sub-optimal solutions for collecting small/moderate data volume, whereas the solution heavily relies on trajectory initialization and has not explicitly considered wind effect, while the computational/trajectory complexity both become prohibitive for the task with large data volume. To this end, we propose a new cyclical trajectory design framework with tailored initialization algorithm that can handle arbitrary data volume efficiently, as well as a hybrid offline-online (HO2) design that leverages convex stochastic programming (CSP) offline based on wind statistics, and refines the solution by adapting online to real-time wind velocity. Numerical results show that our optimized trajectory can better adapt to various setups with different target data volume and buoys’ topology as well as various wind speed/direction/variance compared with benchmark schemes. In particular, our proposed HO2 design can effectively adapt to random wind variations with feasible and robust online operation, and proactively exploit the wind for further energy savings in both single-buoy and multi-buoy scenarios.

Multi-UAV Trajectory Control, Resource Allocation, and NOMA User Pairing for Uplink Energy Minimization

In this work, we study the joint optimization of multiple unmanned aerial vehicles (UAVs)’ trajectories, power allocation, user-UAV association, and user pairing for UAV-assisted wireless networks employing the nonorthogonal multiple access (NOMA) for uplink communications. The design aims to minimize the total energy consumption of ground users while guaranteeing to successfully transmit their required amount of data to the UAV-mounted base stations. The underlying problem is a mixed-integer nonlinear program (MINLP), which is difficult to solve optimally. To tackle this problem, we derive the optimal power allocation as a function of other variables, which is used to transform the optimization problem into an equivalent form. We then propose an iterative algorithm to solve the resulting optimization problem by using the block coordinate descent (BCD) method where three subproblems are solved in each iteration and this process is repeated until convergence. Specifically, given the UAVs’ trajectories and data rates, we solve the NOMA user pairing, and user-UAV association subproblem optimally by exploiting its special structure. Then, we describe how to optimize the users’ data rates and tackle the UAV trajectory optimization in the second and third subproblems, respectively, by using the successive convex approximation (SCA) method. Numerical results show that our proposed algorithm can provide efficient active-inactive schedules (by setting user’s transmit powers to zero), and lower energy consumption compared to an existing baseline, and an OMA-based resource allocation and UAV-trajectory optimization strategy.

Optimal Resource Management for NOMA-Based Visible Light Communication Systems With Shot Noise

In this paper, the problem of power allocation is investigated for downlink visible light communication (VLC) systems with non-orthogonal multiple access. In the system, both input-dependent shot noise and input-independent Gaussian noise are considered due to the properties of practical VLC channels. We consider two scenarios for the transmitter with a single light emitting diode (LED) and multiple LEDs, respectively. For single-LED scenarios, the problem is posed as a joint optimization problem of alternating current and direct current power control, whose goal is to maximize the minimum signal-to-interference-plus-noise-ratio (SINR) of all users. Although the original problem is non-convex, a geometric programming based algorithm is proposed to acquire the global optimal solution. For multi-LED scenarios, the transmit and dimming beamforming vectors are jointly optimized to maximize the minimum SINR of all users. A joint optimization algorithm that alternately optimizes transmit and dimming beamforming vectors is proposed. Particularly, the transmit beamforming problem is converted into a second-order cone programming problem through the successive convex approximation method. The dimming beamforming problem is transformed into a feasibility problem utilizing a low-complexity bisection algorithm. Numerical results reveal the superior performance of the proposed algorithm compared with its orthogonal multiple access counterpart and conventional schemes without shot noise.

Secure Satellite Transmission With Active Reconfigurable Intelligent Surface

The long distance characteristic of satellite transmission makes it difficult to achieve secure communication due to the highly correlated channels. This letter investigates secure satellite multicast transmission in the presence of an eavesdropper (Eve), where the direct links from the satellite to its users and Eve are blocked by obstacles, and an active reconfigurable intelligent surface (RIS) is adopted to rebuild the satellite downlinks and eliminate the “double fading” effect. By assuming imperfect wiretap channel state information is available, our objective is to maximize the minimal secrecy energy efficiency while guaranteeing the transmit power budgets at the satellite and RIS. In order to solve the above intractable optimization problem, an alternating optimization scheme is proposed, where the successive convex approximation method and Nemirovski Lemma are utilized to obtain the beamforming weight vectors with a fixed reflective factor while the optimization of the reflective factor is solved using the S-Lemma with fixed beamforming weight vectors, iteratively. Finally, our simulation results demonstrate that the proposed scheme outperforms the existing benchmark schemes.

Active Reconfigurable Intelligent Surface for Mobile Edge Computing

This letter investigates an active reconfigurable intelligent surface (RIS)-aided mobile edge computing (MEC) system. Compared with passive RIS, the active RIS is equipped with active reflective amplifier, which can effectively circumvent the “double path loss” attenuation. We propose a joint computing and communication design to minimize the maximum computational latency (MCL), subject to both the phase shift constraints and the edge computing capability constraints. Specifically, the original problem is decoupled into four subproblems, and then the block coordinate descent (BCD) method and the successive convex approximation (SCA) method are applied to alternately optimize the subproblems. The simulation results show that with the same power budget, the performance gain achieved by the active RIS is much larger than that by the passive RIS.

Securing Transmission for UAV Swarm-Enabled Communication Network

Deploying unmanned aerial vehicles (UAVs) in communication networks could achieve desirable degrees of freedom, but resulting from the high line-of-sight link attributes, security threats faced by UAV communications are serious. Taking that into consideration, we focus on a UAV swarm-enabled communication network in this article, where a fixed-wing UAV acts as a legitimate user and a neighboring UAV eavesdropper attempts to eavesdrop the confidential information forwarded by rotary-wing UAV relays. In particular, we jointly optimize transmit power on the base station and UAV relays, power allocation coefficient and beamforming on UAV relays, and trajectory of the fixed-wing UAV to maximize the average secrecy rate. Due to the nonconvexity of the optimization problem, the block coordinate descent method is utilized to solve it. Specifically, the overall optimizing problem is divided into four subproblems in which the transmit power, power allocation coefficient, beamforming vector, and the trajectory of user are optimized, respectively. By employing the successive convex approximate method, subproblems are transformed to convex problems which are numerically tractable. Numerical simulations validate the effectiveness of the proposed algorithm and show that compared with benchmark schemes, the considered transmission scheme with artificial noise and optimized network parameters can improve secrecy rate by about 13.8%.

Joint Power and Time Allocation in Hybrid NoMA/OMA IoT Networks for Two-Way Communications.

This article investigates two-way communications between an access point (AP) and multiple terminals in low-cost Internet of Things (IoT) networks. The main issues considered are the asymmetric transmission traffic on the uplink (UL) and downlink (DL), and the unbalanced receivers processing capability at the AP and the terminals. As a solution, a hybrid non-orthogonal multiple access/orthogonal multiple access (NoMA/OMA) scheme together with a joint power and time allocation method is proposed to address these issues. For the system design, we formulated the optimization problem with the aim of minimizing the system power and satisfying the UL and DL transmission rate constraints. Due to the coupling of power and time variables in the objective function and the multi-user interference (MUI) in the UL transmission rate constraints, the formulated problem is shown to be non-linear and non-convex and thus is hard to solve. To obtain a numerical, efficient solution, the original problem is first reformulated to be a convex one relying on the successive convex approximation (SCA) method, and then a numerical efficient solution is thus obtained by using an iterative routine. The proposed transmission scheme is shown to be not only physically feasible but also power-efficient.

Method of Sequential Approximation in Modelling the Processes of Heat Transfer and Gas Dynamics in Combustion Equipment

The behavior of the processes taking place in furnaces determines the efficiency of fuel chemical energy utilization, the quality of the final products and the environmental safety of the production. Mathematical models of the processes of gas dynamics and heat transfer in the working space of heating equipment are quite complex, and do not allow the establishment of a direct analytical relationship between the quality indicator of the process (F) and the influencing parameters (xi). To simplify the procedure for obtaining the values of the function F depending on the change of parameters xi, a method of successive approximation is presented in the article. The main idea of the method is that the representation of the function around a point from the domain of the function can be extended to the entire domain for many problems of mechanics. The relative error in the definition of the function acquires its maximum value at the border of the area, and a reasonable narrowing of it allows control of the size of the error. Thus, the advantages of using the method are obvious; it is able to provide approximation of the function in a multiplicative form with a controlled error. The distribution of the method to the field of heat transfer problems is presented in this paper. The successful implementation of this method for solving problems of this kind shows that the solution of practical problems may be generalized for the entire domain of the function, despite the fact that the errors of such a representation increase to 5–7% when approaching its limit, which, however, may be considered acceptable for engineering calculations.

ЗАДАЧИ УПРАВЛЕНИЯ ТЕХНОЛОГИЧЕСКИМИ ПРОЦЕССАМИ

Obtaining a quality end product is the goal of any manufacturing process control. In the formal description of control problems, this goal allows us to form the boundary conditions for solving the control task. An additional condition for choosing a solution is to ensure the reliability of the process. The starting conditions for solving the control problem are the components of the vector of initial state parameters, determined by the degree of adjustment of the initial parameters and process modes. Understanding the initial and boundary conditions of existence allows us to formulate a generalized formal problem of process control, which consists in finding a vector of control variables that provide a minimum of error, taking into account the influence of a vector of random external variables. The problem is solved by the method of successive approximations. The stages of primary and secondary optimization in the construction of control algorithms are considered. The possibility of reducing the complexity of control tasks is shown by refusing to control individual components of the state vector and transforming control to ensure a minimum of the generalized error measure. In this formulation, the problem of process control can be reduced to the problem of process regulation. A process control scheme is presented. It considers the separability of error minimization for each component of the state vector. The assertions are illustrated by the example of the problem of managing a digital production site for performing the technological operation of etching printed circuit boards. This operation most fully reflects the initial provisions on the continuity of the procedures performed during its implementation and the degradation of the original state vector.

Boundary Value Problems for a Mixed Fourth-order Parabolic-Hyperbolic Equation With Discontinuous Gluing Conditions

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

Using (Direct Computation, Variation Iteration, Successive Approximation and Regularization) Methods to Solve Linear Fredholm Integral Equation and Comparison of These Methods

Integral equation is the equation in which the unknown function to be determined, appears under integral sign as it presented in introduction. I discussed about linear Fredholm integral equation in which it is one kind of integral equation and solved this equation by different methods (Direct Computation, Variational Iteration, Successive approximation and the Regularization methods and comparison of these methods in order to solve linear Fredholm integral equation). This paper has three parts: First part: I introduced the Fredholm integral equation Second part is methods that is written above and on third part, I solved one example by these different methods and compare the methods.

Joint optimization of production lot-sizing and condition-based maintenance in an imperfect production process with dependent indicators

ABSTRACT This paper addresses the integrated optimization of the economic manufacturing quantity and the condition-based maintenance policy of a deteriorating manufacturing system. The considered facility produces a single type of product and is inspected at the end of each production run. Upon inspection, two dependent indicators are revealed: one implies whether the production process is in control or not and the other one represents the deterioration level of the facility. The dependence between the two indicators is that whenever the facility deterioration exceeds a pre-determined level, the system becomes more vulnerable such that the production process may switch to the out-of-control state. In the out-of-control state, a proportion of defective items is fabricated. Considering the inter-dependencies between production process and machine degradation, this paper develops an integrated production and maintenance model to minimize the overall cost. The expected cost rate in the long run is taken as the objective function to assess the proposed model, where the joint optimization of the lot-sizing and the maintenance policy is developed. The problem is formulated in the context of a semi-Markov decision process and solved with the successive approximation method. A numerical example is given to illustrate the applicability of the proposed model.

Joint multiple resource allocation for offloading cost minimization in IRS-assisted MEC networks with NOMA

Effective resource allocation can exploit the advantage of intelligent reflective surface (IRS) assisted mobile edge computing (MEC) fully. However, it is challenging to balance the limited energy of MTs and the strict delay requirement of their tasks. In this paper, in order to tackle the challenge, we jointly optimize the offloading delay and energy consumption of mobile terminals (MTs) to realize the delay-energy tradeoff in an IRS-assisted MEC network, in which non-orthogonal multiple access (NOMA) and multiantenna are applied to improve spectral efficiency. To achieve the optimal delay-energy tradeoff, an offloading cost minimization model is proposed, in which the edge computing resource allocation, signal detecting vector, uplink transmission power, and IRS phase shift coefficient are needed to be jointly optimized. The optimization of the model is a multi-level fractional problem in complex fields with some coupled high dimension variables. To solve the intractable problem, we decouple the original problem into a computing subproblem and a wireless transmission subproblem based on the uncoupled relationship between different variable types. The computing subproblem is proved convex and the closed-form solution is obtained for the edge computing resource allocation. Further, the wireless transmission subproblem is solved iteratively through decoupling the residual variables. In each iteration, the closed-form solution of residual variables is obtained through different successive convex approximation (SCA) methods. We verify the proposed algorithm can converge to an optimum with polynomial complexity. Simulation results indicate the proposed method achieves average saved costs of 65.64%, 11.24%, and 9.49% over three benchmark methods respectively.

Power Allocation of Superimposed Pilots for URLLC With Short-Packet Transmission in IIoT

Intelligent manufacturing in future factories requires ultra-reliable and low-latency communications (URLLC) between central controllers and actuators (e.g., robots). This letter studies a superimposed pilot (SP)-aided uplink transmission system in industrial scenarios, where each actuator sends short-packet messages containing a superimposition of pilot and data signals to the central controller. A practical setting is considered where the blocklength is finite and imperfect pilot removal (IPPR) is available. Then, a closed-form lower bound of the achievable rate with a maximum ratio combining receiver is derived, which can be easily extended to the special cases of perfect pilot removal (PPR) and regular pilot (RP). Based on the derived lower bound, the pilot and data power allocation is optimized by exploiting successive convex approximation method to transform the weighted sum rate maximization problem into a series of geometric programs. Simulation results reveal that the SP-aided transmission outperforms the RP-aided transmission in both the PPR and IPPR cases.