Method Of Successive Approximations Research Articles

TRANSFER OF LOADS FROM THREE HETEROGENEOUS ELASTIC STRINGERS TO AN INFINITE SHEET THROUGH ADHESIVE LAYERS

This paper considers the problem for an elastic infinite plate (sheet), which on parallel finite parts of its upper surface is strengthened by three finite stringers, two of which are located on the same line, having different elastic properties. The stringers are deformed under the action of horizontal forces. The interaction between infinite sheet and stringers takes place through thin elastic adhesive layers having other physical-mechanical properties and geometric configuration. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on three finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated. Further, for various values of changing characteristic parameters of the problem the multiple numerical results and its analysis are presented.

Sparse successive approximation for nonlinear [formula omitted] and [formula omitted] optimal control problems under residual errors

Successive approximation techniques are effective approaches to solve the Hamilton–Jacobi–Bellman (HJB)/Hamilton–Jacobi–Isaacs (HJI) equations in nonlinear H2 and H∞ optimal control problems (OCPs), but residual errors in the solving process may destroy its convergence property, and related numerical methods also pose computational burden and difficulties. In this paper, the HJB/HJI partial differential equations (PDEs) for infinite-horizon nonlinear H2 and H∞ OCPs are handled in a unified formulation, and a sparse successive approximation method is proposed. Taking advantage of successive approximation techniques, the nonlinear HJB/HJI PDEs are transformed into sequences of easily solvable linear PDEs, to which the solutions can be computed point-wise by handling simple initial value problems. Extra constraints are also incorporated in the solving process to guarantee the convergence under residual errors. The sparse grid based collocation points and basis functions are then employed to enable efficient numerical implementation. The performance of the proposed method is also numerically demonstrated in simulations.

Параметры опорного давления около очистной выработки, отрабатывающей угольный пласт с непрочными породными слоями его кровли и почвы

The problem of the stressed state of a carboniferous massif containing a coal seam worked out by a working belongs to the class of flat elastically plastic problems. To construct the stress field in the marginal extremely stressed zone of the seam, the Coulomb–Mohr criteria for the transition of the seam to the limiting state, as well as the methods of bulk medium mechanics developed by V.V. Sokolovsky, were used. If the rock layers of the soil and the roof of the coal seam are not strong enough, then the limiting state occurs first in it, and then at some distance from its edge and in the layers of rocks of the massif. It occurs when the reduced stress in the rock layers reaches the value of the reduced stress in the seam corresponding to the strength limit of the layers for uniaxial compression. After that, the seam and rock layers work together. Based on the constructed stress field of the seam, the problem of the stress state of the array is reduced to the problem of elasticity theory and solved by the method of boundary elements and successive approximations. According to the results of the computational experiment, the parameters of the reference pressure of the reservoir being worked out, depending on the width of the working, were obtained. Comparative estimates are given with similar parameters when mining a seam in the soil and roof of which there are strong layers of rocks.

Joint Trajectory Design and Resource Allocation for Secure Air–Ground Integrated IoT Networks

We investigate in this paper joint trajectory design and resource allocation for secure air-ground integrated Internet of Things (IoT) networks with unmanned aerial vehicle (UAV) jamming and device to device (D2D) enhancement. By jointly optimizing ground user scheduling, UAV flight trajectory and transmit power, we are able to maximize the minimum system secrecy rate of UAV and D2D communications. The formulated optimization problems of the two typical network scenarios, i.e., with and without UAV jamming, are challenging to be solved due to the corresponding non-smooth and non-concave objective functions. Therefore, we propose alternating iterative algorithms to solve the problems by employing the successive convex approximation and block coordinate descent methods. Extensive results indicate that the proposed joint optimization schemes can effectively improve the secrecy communication performance under different spatial distributions of ground users.

A UAV-Assisted Secure Communication System by Jointly Optimizing Transmit Power and Trajectory in the Internet of Things

Secure communication technology has attracted more and more attention to improve the security of the Internet of Things (IoT). In this article, we discuss the maximum secrecy capacity problem for a mobile unmanned aerial vehicle (UAV) assisted secure communication in IoT, where the UAV relays the covert information from the transmitter (Alice) to the legal receiver (Bob) without detection by an eavesdropper (Eve) though sharing the spectrum with the ground device-to-device (D2D) communications. Our design aims to maximize the sum secrecy capacity Bob received by optimizing the transmit power and the UAV’s trajectory jointly while guaranteeing that the normal communication behavior between ground D2D pairs is not affected. Therefore, we propose a jointly optimizing the transmit power and UAV trajectory algorithm (JOTUTA) based on the block coordinate descent (BCD) method. Specifically, we first transform the non-convex problem of transmit power based on a fixed UAV’s trajectory into a convex problem by D.C. (difference of two convex functions) programming. Secondly, we solve the problem of UAV’s trajectory optimization with given transmit power by the successive convex approximation (SCA) method. Experiments results show the effectiveness of JOTUTA and the advantages of improving the secrecy capacity are verified compared with existing work.

Optimal Suppression of Oscillations in the Problem of a Spin-Up of a Two-Mass System

We consider a controlled mechanical system of many bodies, consisting of a load-bearing disk that rotates around its axis fixed in space, and a carried disk attached to it using weightless elastic elements. The presented bodies are in the same plane. The problem of minimizing the amplitude of radial oscillations is studied. To solve this problem over a sufficiently large interval, two numerical methods are used: the method of successive approximations in the control space and Newton’s method. The properties of the phase trajectories of the system are studied depending on the initial states of the disks. Various disk spin-up modes are detected. Using the smoothing procedure for optimal control, a continuous control is constructed that reduces the amplitude of radial oscillations.

Weibull parameter estimation and reliability analysis with small samples based on successive approximation method

Weibull parameter estimation and reliability analysis with small samples based on successive approximation method

A Review of Analytical Algorithms for Approximate Solutions of Ordinary Differential Equation

This review article aims to provide an in-depth analysis of a wide range of analytical methods used for solving ordinary differential equations (ODEs). ODEs are fundamental in numerous scientific and engineering disciplines, making the development, and understanding of effective solution techniques crucial. We explore various approaches, including the Adomian decomposition method, homotopy perturbation method, homotopy analysis method, variational iteration method, Daftardar-Jafari method, successive approximation method, power series method, and modified Adomian decomposition method. Each method is discussed in terms of its principles, applications, advantages, limitations, and computational considerations. This comprehensive overview will serve as a valuable resource for researchers, practitioners, and students interested in solving ODEs.

A comparative study to solve fractional initial value problems in discrete domain

Abstract In this article, we used some methods to solve Riemann-type fractional difference equations. Firstly, we use a method that is a composite method based on the successive approximation method with the Sumudu transform. Secondly, we use a method that is a composite method that consists of the homotopic perturbation method with the Sumudu transform. It was found that the results obtained from these combined methods are identical when compared. We also provided theories and illustrative examples to support the research findings.

Математическое моделирование прогноза урожайности сельскохозяйственных культур на основе данных полевого мониторинга и дистанционного зондирования

A mathematical model for forecasting crop yields based on field monitoring data and remote sensing of the Earth has been constructed. The model includes the following main values: vegetation indices NDVI, total solar radiation flux at the lower boundary of the atmosphere, efficiency of using photosynthetically active solar radiation, biomass respiration costs. After the parameterization of the mathematical model using observational data, the number of uncertain (calibration) coefficients of the model decreases from 8 to 2. These coefficients are determined by the method of successive approximations when comparing the results of calculations with observational data on the yield of a particular agricultural crop (ACC) in a given field. It is shown that the values of these coefficients strongly depend on the choice of optimal conditions for the growth of ACC. To reduce the yield forecast error, an approach based on the numerical integration of the total energy flux density by methods of the second and fourth orders of accuracy is proposed. When using numerical integration methods of a high order of accuracy, the yield forecast error decreases on average by 20% compared to the widely used model for calculating biomass growth, which has the first order of accuracy.

Exploiting NOMA and RIS in Integrated Sensing and Communication

A novel integrated sensing and communication (ISAC) system is proposed, where a dual-functional base station is utilized to transmit the superimposed non-orthogonal multiple access (NOMA) communication signal for serving communication users and sensing targets simultaneously. Furthermore, a new reconfigurable intelligent surface (RIS)-aided-sensing structure, where a dedicated RIS is deployed to provide virtual line-of-sight (LoS) links for radar targets, is also proposed to address the significant path loss or blockage of LoS links for the sensing task. Based on this setup, the beampattern gain at the RIS for the radar target is derived and adopted as a sensing metric. The objective of this paper is to maximize the minimum beampattern gain by jointly optimizing active beamforming at the base station (BS), power allocation coefficients among NOMA users and passive beamforming at the RIS. To tackle the non-convexity of the formulated optimization problem, the beampattern gain and constraints are first transformed into more tractable forms. Then, an iterative block coordinate descent (IBCD) algorithm is proposed by employing successive convex approximation (SCA), Schur complement, semidefinite relaxation (SDR) and sequential rank-one constraint relaxation (SRCR) methods. To reduce the complexity of the proposed IBCD algorithm, a low-complexity iterative alternating optimization (IAO) algorithm is proposed. Particularly, the active beamforming is optimized by solving a semidefinite programming (SDP) problem and the closed-form solutions of the power allocation coefficients are derived. Numerical results show that: i) the proposed RIS-NOMA-ISAC system always outperforms the RIS-ISAC system without NOMA in beampattern gain and illumination power; ii) the low-complexity IAO algorithm achieves a comparable performance to that achieved by the IBCD algorithm. iii) high beampattern gain can be achieved by the proposed joint optimization algorithms in underloaded and overloaded communication scenarios.

Joint Optimization of Secure Over-the-Air Computation and Reliable Multicasting Assisted by a MIMO Untrusted Two-Way Relay

In this paper, we investigate joint optimization of secure AirComp and reliable multicasting assisted by a multiple-input multiple-output untrusted two-way relay, where artificial noise is employed at the access point (AP) to interfere the relay for ensuring secure AirComp. We aim to minimize the computation distortion at the AP by jointly designing the transmit variables of all the nodes and the aggregation variables at the AP and relay, under the secure AirComp constraint, the reliable multicasting constraint, and the transmit power constraints of all the nodes. We consider two scenarios that perfect and imperfect channel state information (CSI) are available. For the former, the formulated optimization problem is highly nonconvex, and we propose an effective block coordinate descent (BCD)-penalty successive convex approximation (penaltySCA) method to solve the nonconvex problem. For the latter, we model the imperfect CSI by using worst-case criterion and the formulated robust optimization problem is much more challenging than its counterpart with perfect CSI. To solve the robust problem effectively, we first transform it to a deterministic optimization problem by employing some powerful mathematical lemmas, and then apply the proposed BCD-penaltySCA method to solve the reformulated deterministic problem. The proposed methods are shown by simulations to significantly reduce the computation distortion compared with other benchmarks under considering secure AirComp and reliable multicasting.

Sensing-Efficient NOMA-Aided Integrated Sensing and Communication: A Joint Sensing Scheduling and Beamforming Optimization

Integrated sensing and communication (ISAC) system has been expected to play a vital role in future wireless networks and services. In this paper, we investigate a non-orthogonal multiple access (NOMA)-aided ISAC system in which the ISAC base station utilizes NOMA to serve multiple NOMA-users while performing radar sensing towards a group of sensing targets by leveraging the superimposed NOMA signals for NOMA-users. To investigate this problem, we formulate a joint optimization of the beamforming, the NOMA transmission duration and the sensing scheduling of sensing targets, with the objective of maximizing the sensing efficiency (i.e., the number of sensed targets within a time-interval) of ISAC system, while satisfying both the communication quality requirement of each NOMA-user and the sensing quality requirement of each selected targets. Despite that the formulated joint optimization problem is strictly non-convex, we propose a decomposition-based algorithm for solving the problem. Specifically, we exploit the successive convex approximation and penalty function method to equivalently transform the problem of optimizing the beamforming into a convex problem which can be addressed efficiently. Furthermore, with the optimal beamforming, we leverage the monotonic feature and adopt a bisection-search method to determine the NOMA transmission duration. Finally, we utilize an algorithm based on the cross-entropy learning to compute the optimal sensing scheduling. Numerical results verify the effectiveness of our proposed algorithms and show the performance advantage of our NOMA-aided ISAC system. Compared with some benchmark schemes, our NOMA-aided ISAC sensing scheduling scheme can achieve better performance in communication and sensing.

Joint Trajectory Design and Power Allocation for UAV Assisted Network With User Mobility

This paper investigates the unmanned aerial vehicle (UAV) assisted wireless communications where the UAV acts as the aerial base station to provide downlink data service to the ground users. We consider a typical hot-spot scenario in which the ground users move towards the target points of interests (PoIs) in groups. The UAV can also fly in air to cope with the movement of users. To maximize the system throughput, an optimization problem is formulated to jointly determine the subchannel assignment, the trajectory of UAV and its power allocation per time slot, where the navigation power of UAV, the co-channel interference as well as the quality of service (QoS) requirements of users are taken into account. As the problem is non-convex and hard to solve, we propose a cascade approach. First, we design a user clustering and pairing algorithm to accomplish the subchannel assignment. Then, we apply the block coordinate descent (BCD) based framework to decompose the remaining problem into the power allocation sub-problem and the trajectory design one. The sub-optimal solution can be gradually achieved through alternatively solving these two sub-problems. As for the power allocation sub-problem, we opt to the difference of convex (DC) functions form and the successive convex approximation (SCA) method to solve it. While for the trajectory design sub-problem, the auxiliary variables and SCA method are utilized. Vivid simulation results reveal that the proposed iterative algorithm can well satisfy the QoS of users and achieve significant performance gain compared with other benchmark schemes.

Dynamic NOMA-Based Computation Offloading in Vehicular Platoons

Both the Mobile edge computing (MEC)-based and fog computing (FC)-aided Internet of Vehicles (IoV) constitute promising paradigms of meeting the demands of low-latency pervasive computing. To this end, we construct a dynamic NOMA-based computation offloading scheme for vehicular platoons on highways, where the vehicles can offload their computing tasks to other platoon members. To cope with the rapidly fluctuating channel quality, we divide the timeline into successive time slots according to the channel's coherence time. Robust computing and offloading decisions are made for each time slot after taking the channel estimation errors into account. Considering a certain time slot, we first analytically characterize both the locally computed source data and the offloaded source data as well as the energy consumption of every vehicle in the platoons. We then formulate the problem of minimizing the long-term maximum task queue by optimizing the allocation of both the communication and computing resources. To solve the problem formulated, we design an online algorithm based on the classic Lyapunov optimization method and successive convex approximation (SCA) method. Finally, the numerical simulation results characterize the performance of our algorithm and demonstrate its advantages both over the local computing scheme and the orthogonal multiple access (OMA)-based offloading scheme.

Numerical solutions of the differential equations

In this paper, we propose to study the numerical integration of Cauchy's problem (PC-in short) for a system of differential equations of the first order by the method of successive approximations and the method of the polygonal line, the best known in practice. For example, the methods are applied to some particular functions. The novelty of this paper consists, mainly, in the translation of existing approximation methods into C++ code, in order to visualize the results including the graphic image of both approximate and exact values trajectories, for comparison.

Joint Optimization of Phase Shift Matrices and Trajectory for AF Relay-Based Cooperation Communication With RIS-Enabled UAV System

In the complicated or hard-to-reach communication areas, sometimes much higher average achievable rate (AAR) or wider coverage is in urgent and temporary need. To address this challenge, this article proposes a novel amplify-and-forward (AF) relay-based cooperation communication scheme with reconfigurable intelligent surface (RIS)-enabled unmanned aerial vehicle (UAV) enhancing. The UAV acts as a moving RIS in the sky, which provides additional links to enhance the performance of the AF relay-based cooperation on the ground. Aiming to maximize the AAR of the proposed scheme, we jointly optimize the trajectory of UAV and the phase shift matrices of RIS over total time frames. At first, we decompose the original intractable problem into two subproblems of the phase shift matrices optimization and trajectory optimization. Subsequently, the optimal phase shift matrices are achieved by aligning the phases of received signals. Since the trajectory optimization subproblem is still complicated and nonconvex, we apply the successive convex approximation (SCA) method to reformulate it into a convex problem and further solve it by iterative method. Simulation results demonstrate the superiorities of the proposed scheme as compared with the benchmark schemes and also reveal the effects of various factors on its performance.

Beamforming Design in Short-Packet Transmission for URLLC in Cell-Free Massive MIMO System

Low latency, high reliability, and high data rate are three performance indicators of Ultra-reliable and low latency communication (URLLC) systems, but they are not able to be met simultaneously. Based on the determined latency and reliability requirements, this paper investigates the problem of the weighted sum rate (WSR) in cell-free massive multi-input multi-output (MIMO) system with multiple access points (APs). User-centric access point selection is considered to further reduce the computational complexity for AP used with acceptable performance loss. The transmitting beamforming is designed to mitigate inter-user interference so as to maximize the WSR under the backhaul and transmitting power constraints. Since the proposed problem is non-convex, two algorithms are proposed to solve the problem under the general SINR regime and the high SINR regime, respectively. For the former, we first approximate the objective function with the difference of convex (DC) programming and reformulate the problem into a semi-definite relaxation (SDR) form. Afterward, auxiliary variables are introduced in accordance with the successive convex approximation (SCA) method to further transform the nonconvex problem into a convex one. In the end, the sub-optimal WSR is obtained by iteratively solving a final convex problem. For the latter, the well-known WMMSE method is applied to address the WSR problem. Numerical results demonstrate the efficiency of our proposed algorithm, e.g., the maximum performance gain of the proposed algorithm is about 25% compared to the classical MMSE method. And the user-centric approach achieves 8 times computational complexity reduction with 4% performance loss.

IRS-Assisted Secure UAV Communication System for Multiuser With Hardware Impairments

Perfect hardware conditions are often difficult to meet in practical networks due to the inevitable hardware impairments (HIs). The secure multiuser communication of an intelligent reflecting surface (IRS)-aided unmanned aerial vehicle (UAV) system with HIs at the transceiver and the IRS is studied in this article. Aiming for maximizing the average sum secrecy rate (ASSR) of the UAV system, we jointly optimize the transmit precoding (TPC) matrices, IRS phase shift matrices, and UAV trajectory. To deal with the nonconvexity, we break down this problem into three subproblems and solve them using an alternating optimization (AO) algorithm. Specifically, the semidefinite programming method is used to solve the TPC and phase shift matrices, while the successive convex approximation method is used to obtain the UAV trajectory. In addition, a low complexity mathod on the basis of the Riemannian Manifold Gradient is proposed to solve the IRS phase shift matrix. Simulation results verify the effectiveness of the algorithm. Our robust algorithm can improve ASSR by 22% compared to that without IRS. Furthermore, the nonrobust design without considering HIs will cause a loss of secrecy performance.

Further investigation on fixed point theorems via C-class functions in extended b-metric spaces

The purpose of this research project is to develop new theories, discuss, and extend some recent common fixed point results established when the underlying ambient space is an extended b-metric space and the contraction condition involves a new class of ψ-φ-C-contraction type mappings where ψ is the altering distance function and φ is the ultra-altering distance function. The unique fixed point theorems for such mappings in the setting of ψ-φ-complete metric spaces are proven. We also prove the fixed point theorem in partially ordered metric spaces. Moreover, some examples supporting the main results are given. Our results extend and generalize corresponding results in the literature. The start of the development of the theory of fixed points is tied to the end of the 19th century. The method of successive approximations is used in order to prove the solution's existence and uniqueness at the beginning of differential and integral equations. This branch of nonlinear analysis has been developed through various classes of spaces, such as metric spaces, topological spaces, probabilistic metric spaces, fuzzy metric spaces, and others. In developing the theory of fixed points, achievements are applied in various sciences, such as optimization, economics, and approximation theory. A very important step in the development of fixed point theory was taken by A.H. Ansari through the introduction of a C-class function. Using C-class functions, we generalize some known fixed point results, and Kamran et al. introduced a new intuitive concept of distance measure to extend the notion of b-metric space by further weakening the triangle inequality.