Method Of Successive Approximations Research Articles

Federated learning via over-the-air computation in IRS-assisted UAV communications

Intelligent reflective surface (IRS) and unmanned aerial vehicle (UAV) communication are two key technologies in the sixth generation of mobile communication (6G). In this paper, IRS is equipped on UAV to form aerial IRS, which can achieve 360° panoramic full-angle reflection and flexible deployment of IRS. In order to achieve high-quality and ubiquitous network coverage under data privacy and low latency requirements, we propose an Federated learning (FL) network via Over-the-Air computation (AirComp) in IRS-assisted UAV communications. Our goal is to minimize the worst-case mean square error (MSE) by jointly optimizing the IRS phase shift, denoising factor for noise suppression, the user’s transmission power, and UAV trajectory. Optimizing and quickly adjusting the UAV position and IRS phase shift, it flexibly assists the signal transmission between users and base stations (BS). In order to solve this complex non-convex problem, we propose a low-complexity iterative algorithm, which divides the original problem into four sub-problems, respectively using the semi-definite programming (SDP) method, slack variable introduction method, successive convex approximation (SCA) method to solve each sub-problem. Through the analysis of simulation results, our proposed design scheme is obviously better than other benchmark schemes.

Application of the Generalized Method of Eigenoscillations to the Solution of the Problems of Scattering on Nanostructures

We study the problem of scattering of electromagnetic waves on small-sized dielectric bodies by using the generalized method of eigenoscillations. The inhomogeneous problem is formulated on the basis of a version of the generalized method of eigenoscillations in which the dielectric permittivity plays the role of an eigenvalue. The corresponding homogeneous problem is solved with the use of Debye potentials. We obtain a system of linear algebraic equations for the unknown expansion coefficients. For the solution of this system, we use the method of successive approximations. The results of the performed numerical analyses confirm the efficiency of the proposed method.

Efficient Rate-Splitting Multiple Access for the Internet of Vehicles: Federated Edge Learning and Latency Minimization

Rate-Splitting Multiple Access (RSMA) has recently found favour in the multi-antenna-aided wireless downlink, as a benefit of relaxing the accuracy of Channel State Information at the Transmitter (CSIT), while in achieving high spectral efficiency and providing security guarantees. These benefits are particularly important in high-velocity vehicular platoons since their high Doppler affects the estimation accuracy of the CSIT. To tackle this challenge, we propose an RSMA-based Internet of Vehicles (IoV) solution that jointly considers platoon control and FEderated Edge Learning (FEEL) in the downlink. Specifically, the proposed framework is designed for transmitting the unicast control messages within the IoV platoon, as well as for privacy-preserving FEEL-aided downlink Non-Orthogonal Unicasting and Multicasting (NOUM). Given this sophisticated framework, a multi-objective optimization problem is formulated to minimize both the latency of the FEEL downlink and the deviation of the vehicles within the platoon. To efficiently solve this problem, a Block Coordinate Descent (BCD) framework is developed for decoupling the main multi-objective problem into two sub-problems. Then, for solving these non-convex sub-problems, a Successive Convex Approximation (SCA) and Model Predictive Control (MPC) method is developed for solving the FEEL-based downlink problem and platoon control problem, respectively. Our simulation results show that the proposed RSMA-based IoV system outperforms both the popular Multi-User Linear Precoding (MU–LP) and the conventional Non-Orthogonal Multiple Access (NOMA) system. Finally, the BCD framework is shown to generate near-optimal solutions at reduced complexity.

Energy Efficiency in Secure VLC Networks With SLIPT

This letter focuses on maximizing the energy efficiency (EE) for secure visible light communication (VLC) network with simultaneous lightwave information and power transfer (SLIPT) by optimizing the beamforming vectors and direct current (DC) bias under the constraints of the minimal rate requirement, the secure transmission requirement, the minimal energy harvesting (EH) requirement, and total power budget at the transmitters. To solve the non-convex optimization problem, we present a dual-layer iterative algorithm based on Dinkelbach’s and successive convex approximation (SCA) methods. An efficient algorithm is also designed to find a good initial point to accelerate the convergence. Simulation results show three typical phenomena: (1) the EE first remains unchanged and then decreases with the increasing EH requirement; (2) compared with the traditional sum-rate maximization design, our proposed approach achieves notable EE performance gain under the secure constraints; (3) the number of the authorized receivers, as well as the number of the unauthorized receivers, imposes significant impact on the EE performance.

Average Rate Maximization for Mobile STAR-RIS-Aided NOMA System

Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has been regarded as a revolutionary technology for the sixth generation (6G) mobile networks. This letter investigates the transmission scheme for mobile STAR-RIS-assisted non-orthogonal multiple access (NOMA) system. Specifically, an uplink average rate maximization problem is solved by jointly optimizing the phase shifts, the channel assignment, and the STAR-RIS’s location for mobile STAR-RIS-aided NOMA system. Particularly, a channel assignment scheme is developed to enhance the performance of NOMA system with multiple users. Furthermore, a successive convex approximation (SCA) method is invoked to address the STAR-RIS location design subproblem and a semi-definite programming (SDP) method is leveraged to solve the phase shift optimization subproblem. Simulation results show that the proposed mobile STAR-RIS-empowered NOMA method achieves better average rate performance than various benchmarking schemes.

Reconstruction of Two Functions in the Model of Vibrations of a String One End of Which Is Placed in a Moving Medium

The paper considers an inverse problem of determining the coefficients in the model of small transverse vibrations of a homogeneous finite string one end of which is placed in a moving medium and the other is free. The vibrations are simulated by a hyperbolic equation on an interval. One boundary condition has a nonclassical form. Additional data for solving the inverse problem are the values of the solution of the forward problem with a known fixed value of the spatial argument. In the inverse problem, it is required to determine the function in the nonclassical boundary condition and a functional factor on the right-hand side of the equation. Uniqueness and existence theorems for the inverse problem are proved. For the forward problem, conditions for unique solvability are established in a form that simplifies the analysis of the inverse problem. For the numerical solution of the inverse problem, an algorithm is proposed for the stage-by-stage separate reconstruction of the sought-for functions using the method of successive approximations for integral equations.

Numerical solution of the control problem on the depletion of gas reservoirs with weakly permeable top

The current stage in the development of mathematical and software support for the processes of designing the development of hydrocarbon fields is characterized not only by the improvement of the means of geological and hydrodynamic modeling of reservoir fluid filtration but also by the use of algorithms for optimizing the development of gas deposits. The paper considers the problem of optimal control of the depletion of a gas reservoir with a low-permeability top. Using the so-called Myatiev-Girinsky hydraulic scheme, a two-dimensional equation describing the unsteady gas flow in a reservoir with a jumper is averaged over the capacity of the productive reservoir. This comes down to a one-dimensional equation with an additional term, taking into account gas-dynamic relationships between the reservoir and the jumper. For the numerical solution of process control problems, a formula for the gradient of the functional characterizing the reservoir depletion is found, and the method of successive approximations based on Pontryagin’s maximum principle is applied. In this case, the direct and conjugate boundary value problems are solved by the method of straight lines, and the required flow rate, without taking it beyond the maximum and minimum possible, is found by the gradient projection method with a special choice of step. A brief block diagram of the algorithm for solving the problem is shown; on its basis, a computer program was compiled. The results of calculations are presented to identify the influence of the values of the complex communication parameter not only on the state of the object but also on the operating mode of the well. The expediency of using the presented optimization tool is dictated by an increase in the share of deposits

Decentralized ϵ‐Nash strategy for linear quadratic mean field games using a successive approximation approach

AbstractThis paper presents a successive approximation method for decentralized strategy design in the large‐scale linear quadratic (LQ) Gaussian game. The strategy consists of transforming the original mean field game (MFG) problem into solving decoupled ordinary differential equations (ODEs) whose numerical solutions are obtained by a new two‐loop iteration algorithm. It should be noted that we employ the augmented model technique and the LQ framework to derive these low‐dimensional solvable ODEs, which is the cornerstone of constructing the decentralized ‐Nash strategy. In addition, the quadratic ODEs contained therein are approximately solved for by a sequence of iterative linear ordinary differential equations (LODEs) with guaranteed convergence. A numerical example is given to show the effectiveness of the proposed algorithm.

Problem of plane stress state of two-layer body in dynamic elastic-plastic formulation

Composite materials are widely used in industry and everyday life. Many different methods are used to calculate and develop composite materials. Many methods of calculation and design of such materials are successfully used. In this article, for the design of composite and reinforced materials, a technique for solving dynamic contact problems in more precise an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. The problem of a plane stress state (PStS) of a beam made from the composite reinforced double-layered material is being solved in dynamic elastic-plastic mathematical model. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. This composite base is rigidly attached to an absolutely hard half-space. Rigid adhesion of the layers to each other is assumed. Glass is a very strong and very fragile material at the same time. The fragility of glass is due to the fact that there are many microcracks on the surface, and when a load is applied to the glass surface, these microcracks begin to grow and lead to the destruction of glass products. If we glue or immobilize the tops of microcracks on the surface, we will get a strong reinforced armed material that will be lighter, stronger and not subject to degradation of material properties such as aging, corrosion and creep. The impact process was modelled as a non-stationary plane stress state problem with an even distributed load in the contact area, which changes according to a linear law. The fields of the Odquist parameter and normal stresses were studied and compared to corresponding results of plane strain (PSS) problem with the same material of layers, same their thickness and size of the contact area. The upper reinforcing layer of metal or steel can be applied to the glass surface so that metal or steel atoms penetrate deeply, fill microcracks and bind their tops. The top layer can be quite thin.

Plane Strain and Stress States of Two-Layer Composite Reinforced Body in Dynamic Elastic-Plastic Formulation

For the design of composite and reinforced materials, a technique for solving dynamic contact problems in more precise an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. In contrast to the traditional plane strain, when one normal stress is equal to a certain constant value, for a more accurate description of the deformation of the sample, taking into account the possible increase in longitudinal elongation, we present this normal stress as a function that depends on the parameters that describe the bending of a prismatic body that is in a plain strain state. The problems of a plane strain and stress states of a beam made from the composite reinforced double-layer material is being solved. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. Glass is a non-crystalline, often transparent amorphous solid that has widespread practical and technological use in the modern industry. Glass has high strength and is not affected by the processes of aging of the material, corrosion, and creep. In addition, this material is cheap and widely available. Glass can be strengthened, for example, in a melt quenching process. The reinforced composite beam is rigidly linked to an absolutely solid base and on which an absolutely solid impactor acts from above in the centre on a small area of initial contact.

Orthogonalization of the measuring basis of triaxial vibration transducers by the method of successive approximations

The transversal sensitivity of tri-axial vibration transducers, which have found vast application in various vibration measuring and material testing systems, decreases the accuracy of measurements. We present a solution to the problem of eliminating the error attributed to the presence of the transverse sensitivity. The measurement error depends on many factors including the ratio of vibration components along the sensitivity axes, the width of the vibration spectrum, and the presence of intrinsic resonances of the vibration transducers in the main and transverse directions. The developed method provides almost complete compensation of the error. Analysis of the components of sensitivity vectors along the measuring directions showed that the transversal sensitivity vectors are in fact the parasitic penetrations from other directions, and can be decomposed along the measuring axes. To orthogonalize the sensitivity vectors during calibration, the sensitivity vectors should be rotated until they coincide with the orthogonal directions. A measuring basis with zero transverse sensitivities is thus obtained, and the sensitivity matrix is reduced to a diagonal form. With this approach the fundamentals and practical algorithm using successive approximations of triaxial transducer calibration with orthogonalization are outlined. Each orthogonalization step is illustrated by the current sensitivity matrix, showing the progress of transversal sensitivity reduction. The results of the developed tri-axial transducer coupled to a special preamplifier-converter with transversal sensitivities reduced to nearly zero values using the developed method are presented. Since calibration is carried out without dismantling of a vibration transducer, it can be carried out not only during its manufacture, but also during verification upon operation, which increases the service life. The use of the described technique not only improves the metrological parameters of vibration transducers, but also provides a significant reduction of the requirements for the accuracy of manufacturing their measuring system, thus reducing the production costs. The presented method is promising for the developing new precise multi-axis vibration transducers and measuring systems on their base.

Sum Rate Optimization for Multiple Access in Multi-FD-UAV-Assisted NOMA-Enabled Backscatter Communication Network

With the rapid development of the Internet of Things (IoT) network, research on low-power and energy-saving devices has attracted extensive attention from both academia and the industry. Although the backscatter devices (BDs) that utilize the environmental power to activate circuits and transmit signals are a promising technology to be deployed as IoT nodes, it is challenging to design a flexible data backhaul scheme for massive BDs. Therefore, in this paper, we consider an unmanned-aerial-vehicle (UAV)-assisted backscatter communication network, where BDs are served by multiple full-duplex (FD) UAVs with the non-orthogonal multiple access (NOMA) schemes and modulate their signals on the downlink signals, which are generated by the UAVs to serve the coexisting regular user equipments (UEs). To maximize the sum rate of the considered system, we construct an optimization problem to optimize the reflection coefficient of BDs, the downlink and the backhaul transmission power, and the trajectory of UAVs jointly. Since the formulated problem is a non-convex optimization problem and is difficult to solve directly, we decouple the original problem into three sub-problems and solve them with the successive convex approximation (SCA) method, thereby addressing the original problem by a block coordinate descent (BCD)-based iterative algorithm. The simulation results show that, compared with the benchmark schemes, the proposed algorithm can obtain the highest system sum rate and utilize limited time-frequency resources more efficiently.

Motion Planning in UAV-Aided Data Collection with Dynamic Jamming

Unmanned-aerial-vehicle (UAV)-aided data collection for Internet of Things applications has attracted increasing attention. This paper investigates motion planning for UAV collecting low-power ground sensor node (SN) data in a dynamic jamming environment. We targeted minimizing the flight energy consumption via optimization of the UAV trajectory while considering the indispensable constraints which cover the collection data demodulation threshold, obstacle avoidance, data collection volume, and motion principle. Firstly, we formulate the UAV-aided data collection problem as an energy consumption minimization problem. To solve this nonconvex optimization problem, we rewrite the original problem by introducing relaxation variables and constructing equivalence constraints to obtain a new relaxation convex problem, which can be solved iteratively using the successive convex approximation (SCA) method. However, SCA is susceptible to initial values, especially in dynamic environments where fixed initial values may lead to a wide range of results, making it difficult to obtain a truly optimal solution to the optimization problem. To solve the initial value problem in dynamic environments, we further propose a communication-flight-corridor(CFC)-based initial path generation method to improve the reliability and convergence speed of the SCA method by constructing reliable communication regions and resilient secure paths in real time. Finally, simulation results validate the performance of the proposed algorithm compared to the benchmark algorithms under different parameter configurations.

Average AoI minimization for data collection in UAV-enabled IoT backscatter communication systems with the finite blocklength regime

Thanks to the autonomy of unmanned aerial vehicle (UAV), UAV-enabled data collection in Internet of Things (IoT) networks has become a key application for the next generation communication network. In this paper, we consider a scenario where an UAV is responsible for collecting data from sensor equipments (SEs) one by one and finally carrying the collected data to the computation center for processing. Different from the commonly used assumption that SEs always generate data at the beginning of each time slot, it is assumed that SEs can generate data at any instant during one time slot, which is more practical. Since SEs are energy limited, they upload data to the UAV by adopting backscatter communication technology to reduce energy consumption. Meanwhile, the updated information usually contains a small number of information bits but requires low latency and high reliability, thus the finite blocklength regime in ultra-reliable and low-latency communication is adopted for SEs’ data transmission to the UAV. To keep the freshness of the updated information, a joint resource allocation problem including data collection time allocation, transmission power and trajectory design of the UAV is formulated as an optimization problem to minimize the average age of information (AoI) of all SEs. The formulated problem mixes discrete and continuous variables, which makes it difficult to solve. Thereby, we decompose the optimization problem into data collection time minimization subproblem and UAV trajectory design subproblem, which are solved by the successive convex approximation method, and the backtracking algorithm and the genetic algorithm, respectively. Numerical results show that the backtracking-based algorithm that can obtain the optimal trajectory of the UAV gains the minimal average AoI, and the genetic-based algorithm achieves sub-optimal average AoI with much lower computational complexity. The results also demonstrate that the average AoI of the backtracking-based algorithm and the genetic-based algorithm is reduced by up to 54% and 46% compared with the greedy-based benchmark algorithm, respectively.

ЕSTIMАTIОN ОF STIFFNЕSS АND DISPLАСЕMЕNTS BY THЕ АNАLYTIСАL ЕXPRЕSS MЕTHОD DURING THЕ ОPЕRАTIОN ОF "SLIDING SHАFT-BЕАRING" СОNNЕСTIОNS.

Of all, when calculating structures in which a cantilever beam is used as an elastic element, it is not correct to solve this equation for the finite section by successive approximation of the transcendental equation. The exact solution of the differential equation leads to the transcendental equation in elliptic integrals of the 1st kind. From this equation, it is possible to determine the angle of rotation of the final section θk by the method of successive approximations, that is, in fact, by selection. For practical purposes, for example, when designing elastic elements of variable stiffness, it is necessary to have an explicit formula for the dependence of vertical displacement on the applied force. We differentiate the dependence of the reduced vertical displacement of the beam’s end on the reduced load. Then we will take the inverse value from it. As a result, we get a graph of the reduced stiffness, which fits almost perfectly into the square parabola y = 3 + x2. The internal energy of deformation of the previously unloaded and undeformed cantilever beam was determined. In order to verify this mathematical model, a comparison of the results of the nonlinear calculation with the results of the finite element calculations based on the traditional model and the one proposed in the work was performed. It was established that formulas (1) - (4) can be used when the displacement of the rod element does not exceed 80% of its length, which is a very significant geometric nonlinearity. The size of the stiffness matrix in the finite element method (FEM) is determined by the degree of discretization of the system. In practical tasks, the number of elements in the stiffness matrix is calculated by hundreds of thousands or millions. It is always a highly sparse matrix with a large number of zero elements.

On a Method for Optimizing Controlled Polynomial Systems with Constraints

A new optimization approach is considered in the class of polynomial in-state optimal control problems with constraints based on nonlocal control improvement conditions, which are constructed in the form of special fixed-point problems in the control space. The proposed method of successive approximations of control retains all constraints at each iteration and does not use the operation of parametric variation of control at each iteration, in contrast to known gradient methods. In addition, the initial approximation of the iterative process may not satisfy the constraints, which is a significant factor in increasing the efficiency of the approach. The comparative efficiency of the proposed method of fixed points in the considered class of problems is illustrated in a model example.

Mixed forms of free oscillations of a rectangular CFCF-plate

Mixed (symmetrically/antisymmetric) forms of natural oscillations of a thin rectangular plate of constant thickness, in which two parallel sides are rigidly pinched, and the other two are free (CFCF plate, C — clamped, F — free), are studied. When all the conditions of the boundary value problem are satisfied, a resolving infinite homogeneous system of linear algebraic equations with respect to unknown coefficients of the series is obtained using two hyperbolo-trigonometric series of the coordinate deflection function. Even functions on one coordinate and odd functions on another coordinate were used to obtain symmetric-antisymmetric waveforms. As a parameter, the resulting system contains the relative frequency of free oscillations. Nontrivial solutions of the reduced system were found by the method of successive approximations in combination with a search of the frequency parameter. Numerical results are obtained for the spectrum of the first six mixed (symmetric/antisymmetric — S-A and A-S) forms of free oscillations of a thin square CFCF plate of constant thickness. The natural frequencies were compared with the results of other authors and with known experimental values. The influence on the accuracy of the results of the number of members held in rows (the size of the reduced system) and the number of iterations is investigated. 3-D images of the found waveforms are presented. The results obtained can be used in the design of various sensors and sensors using the resonance phenomenon.

Derivation of acoustical streaming equations for nonlinear and dispersive fluids

The equations of streaming generated by an acoustic mod propagating in a nonlinear dispersive medium (exhibiting absorption and dispersion of phase sound speed) are derived with an arbitrarily shaped incident acoustical field assumed. This field may be periodic or non-periodic. A general dispersion model represented by a convolution operator taking into account relaxation effects was taken into account. Making the assumption of a periodic acoustic field from the general streaming equation.The quasi-stationary flow is driven by a force given by the average value of the dispersion operator with respect to the velocity and acoustic pressure fields. In the spectral representation, it is given by the weighted spectral power density distribution of the acoustic field. The weight of the distribution is the dispersion coefficient - the eigenvalue of the dispersion operator. A new result also reveals the effect of the refractive index deviation on the driving force of streaming. The possibility of generalizing the description of streaming in the simplest case of a non-Newtonian fluid was analyzed. The Reiner-Revlin model of a simple liquid was assumed. It was also noted that the streaming model in the Maxwell liquid is analytically solvable. It was found that asymptotic states of streaming in this model and the Navier-Stokes model are identical.The derivations use new methods different from those used so far. They are based on the separation of nonlinear modes in the momentum transport equation and on the properties of the Gauss-Weierstrass function for the Fick diffusion operator. So far, the method of successive approximations has been used. The consistency of the obtained equations with the assumptions was checked. The obtained formulas generalize the known descriptions of the form of forces driving streaming and extend their application to the case of nonlinear propagation.

Energy-Efficient Cell-Free Network Assisted by Hybrid RISs

In this letter, we investigate a cell-free network aided by hybrid reconfigurable intelligent surfaces (RISs), which consists of a mixture of passive and active elements that are capable of amplifying and reflecting the incident signal. To maximize the energy efficiency (EE) of the system, we formulate a joint transmit beamforming and RIS coefficient optimization problem. To deal with the fractional objective function, Dinkelbach transform, Lagrangian dual reformulation, and quadratic transform are utilized, with a block coordinate descent (BCD) based algorithm proposed to decouple the variables. In addition, successive convex approximation (SCA) method is applied to iteratively to tackle the non-convexity of the sub-problems. Simulation results illustrate the effectiveness and convergence of the proposed algorithm through analyzing the EE and sum rate performance with varying parameter settings. The proposed hybrid RIS scheme can achieve 5% and 188% gain in EE compared with existing passive and active RISs. The fast convergence of the proposed algorithm is analyzed by numerical simulations as well.

Joint Resource Allocation and Trajectory Optimization for Completion Time Minimization for Energy-Constrained UAV Communications

An unmanned-aerial-vehicle (UAV)-aided communication scenario is studied in this research, where a UAV with constrained on-board battery capacity communicates with multiple ground users (GUs) during the flying process. We investigate the UAV trajectory as well as the resource allocation to minimize the time consumption of the specific task. The formulated problem is non-convex with innumerable variables. To make this problem tractable, a collection of line segments are utilized to represent the real trajectory through discretization. The reformulated problem is solved by applying the block coordinate descent (BCD) algorithm. To be specific, the reformulated problem is decoupled into two subproblems, i.e., resource allocation and trajectory optimization subproblems. The subproblems are both non-convex, which are efficiently resolved with the successive convex approximation (SCA) method. Then the result is obtained with an iterative algorithm. The simulation results reveal that, in comparison to benchmark schemes, the system performance has a significant improvement.