Nonlinear Problem Research Articles

A Novel Fixed-Point Iteration Approach for Solving Troesch’s Problem

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green’s function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces.

Nonlinear resonant responses of a novel FGM sandwich cylindrical shell structure for supersonic flight vehicles based on 1:1 internal resonance between conjugate modes

A new design scheme for the functionally graded material (FGM) sandwich cylindrical shell structure made of ceramic-FGM-carbon fiber composites is proposed, which is geared towards supersonic flight vehicles and has high specific stiffness and high-temperature resistance. We focus on analyzing the nonlinear resonant responses of the novel FGM sandwich cylindrical shell subjected to external excitation and aerodynamic force. Firstly, the mechanical properties of the novel FGM sandwich material are calculated, and then the nonlinear dynamic model of the FGM sandwich cylindrical shell is established based on the first order shear deformation theory (FSDT) and Hamilton's principle. Due to the axisymmetric property of the cylindrical shell, a 1:1 internal resonance between the driven and companion modes always exists in the perfect cylindrical shell. Taking this into consideration, the nonlinear resonant response problems of the FGM sandwich cylindrical shell are numerically simulated by combining the Galerkin method and the pseudo-arc length continuation method. Finally, the effects of complex parameters such as external excitation, aerodynamic force, aspect ratio, gradient index, and skin-core-skin ratio on the nonlinear resonant responses of the FGM sandwich cylindrical shell are investigated.

On fuzzy fractional differential inclusion driven by variational–hemivariational inequality in Banach spaces

The aim of this paper is to examine an evolution problem (FFDIVHVI) involving a fuzzy fractional differential inclusion and a variational–hemivariational inequality (VHVI) in Banach spaces. First, we show a uniqueness and existence theorem for VHVI under the theory of monotone operators and the surjectivity theorem. Then, by utilizing fixed point theorem for multivalued contraction mapping and fuzzy set theory, we establish the existence result for FFDIVHVI. In addition, it is proven that the collection of all mild trajectories of FFDIVHVI exhibits compactness. Finally, we illustrate the applicability of the abstract theory by a nonlinear quasistatic thermoelastic frictional contact problem for which we provide existence results.

A stochastic approach to delays optimization for narrowband transmit beam pattern in medical ultrasound

In ultrasound imaging, state-of-the-art methods for transmit beam pattern (TBP) optimization have well-known drawbacks like non-uniform beam width over depth, significant side lobes, and quick energy drop-out beyond the focal region. To overcome these limitations, we develop a TBP optimization approach by focusing on its narrowband approximation and considering transmit delays as free variables instead of linked to specific focal depths. We formulate a non-linear Least Squares problem to minimize the difference between the TBP corresponding to a set of delays and the desired one, modeled as a 2D rectangle elongated along the beam axis. Three metrics are defined to quantitatively evaluate the results. Results obtained by synthetic simulations show that the main lobe width is considerably more intense (+31.5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+31.5\\%$$\\end{document} on average) and uniform (+28.7%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+28.7\\%$$\\end{document} on average) over the whole depth range compared to classical focalized Beam Patterns. The application of the method to elastography shows improvements in the ultrasound energy concentration along a desired axis.

Third moment method for reliability analysis with uncertain moments characterized as interval variables

Traditional reliability analysis aims to compute the failure probability based on probability distribution functions, which are constructed using the moments of random parameters. In practice, however, appropriate samples may be insufficient to obtain deterministic values of the moments of all random variables and the exact value of failure probability cannot be obtained. To be consistent with the reality, the uncertainties in moments can be measured as interval variables, and then the bounds of failure probability should be evaluated. In this study, an idealized case is considered, where there is at most one imprecise moment associated with any given input random variable. A third moment method is proposed with uncertain moments measured as interval variables, and is named as TMI method. The proposed TMI method is straightforward including only four steps. Firstly, the derivative of performance function to random variables having uncertain moments is calculated, with the random variables set to be their mean values. Secondly, the values of uncertain moments for computing the bounds of failure probability are determined. Then, with inverse normal transformation defined based on the moments, the performance function at the bounds in Gaussian space is directly constructed. Finally, bounds of failure probability can be evaluated by two times of classical reliability analysis corresponding to the constructed performance functions. The application of TMI method is validated by numerical examples, including high-dimensional and strong nonlinear problems.

Hazard evaluation of goaf based on DBO algorithm coupled with BP neural network

China is rich in mineral resources, and problems of goaf formed in the process of resource exploitation are serious obstacle to the development of China's economic, so it is of great significance for the assessment and management of goafs. This paper introduces emerging dung beetle optimizer (DBO) algorithm and establishes DBO-BP (back-propagation) model, at the same time, it is compared with a series of heuristic algorithms coupled with BP neural network models: PSO (particle swarm optimization) - BP model, WOA (whale optimization algorithm) - BP model, and SSA (sparrow search algorithm) - BP model. Then they are applied to evaluate the hazard of goafs, the result shows that the DBO-BP model gets the highest train set accuracy, which is at least 2.7 % higher than other models, while the DBO-BP model obtains the highest test set accuracy, meanwhile its effectiveness and stability have also been proven. Finally we apply the established DBO-BP model to evaluate the hazard of the tungsten mine goaf of Yaogangshan in Hunan Province, and its excellent practicability was confirmed. This paper may provide a reference for the solution of nonlinear engineering problems.

Robust Trajectory Planning of Gliding-Guided Projectiles with Weak Maneuverability

Due to constraints in launch platforms and cost, the maneuverability of gliding-guided projectiles is limited, necessitating a rational design of their trajectory schemes. To reduce the sensitivity of trajectory schemes to uncertainties while ensuring compatibility between flight schemes and guidance control systems and fully exploiting the control capability of the projectile, a closed-loop robust trajectory planning method is proposed. Models of major uncertain factors and state deviation at the control start point are established. Based on the NIPCE method, the stochastic dynamic model is transformed into a high-dimensional deterministic model with PCE coefficients as state variables, and the uncertainty propagation law is obtained. A PID algorithm is employed to design a tracking guidance law based on position error feedback, and open-loop and closed-loop robust trajectory planning models are established accordingly. The optimal control problem is solved by transforming it into a nonlinear programming problem using the direct shooting method. Our simulation results indicate that the NIPCE method can significantly improve the computational efficiency of uncertainty propagation while ensuring accuracy; compared with parallel MCS, the computation time is reduced by 96.8%. Open-loop robust planning can effectively mitigate the sensitivity of gliding trajectories to uncertainties (the standard deviations of terminal altitude and lateral deviations are reduced by 23.6% and 35.3%, respectively, compared to deterministic planning) but cannot completely eliminate terminal dispersion. Closed-loop robust planning effectively improves control effort consumption on the basis of open-loop planning.

Fast retrieval of XCO2 over east Asia based on Orbiting Carbon Observatory-2 (OCO-2) spectral measurements

Abstract. The increase in greenhouse gas concentrations, particularly CO2, has significant implications for global climate patterns and various aspects of human life. Spaceborne remote sensing satellites play a crucial role in high-resolution monitoring of atmospheric CO2. However, the next generation of greenhouse gas monitoring satellites is expected to face challenges, particularly in terms of computational efficiency in atmospheric CO2 retrieval and analysis. To address these challenges, this study focuses on improving the speed of retrieving the column-averaged dry-air mole fraction of carbon dioxide (XCO2) using spectral data from the Orbiting Carbon Observatory-2 (OCO-2) satellite while still maintaining retrieval accuracy. A novel approach based on neural network (NN) models is proposed to tackle the nonlinear inversion problems associated with XCO2 retrievals. The study employs a data-driven supervised learning method and explores two distinct training strategies. Firstly, training is conducted using experimental data obtained from the inversion of the operational optimization model, which is released as the OCO-2 satellite products. Secondly, training is performed using a simulated dataset generated by an accurate forward calculation model. The inversion performance and prediction performance of the machine learning model for XCO2 are compared, analyzed, and discussed for the observed region over east Asia. The results demonstrate that the model trained on simulated data accurately predicts XCO2 in the target area. Furthermore, when compared to OCO-2 satellite product data, the developed XCO2 retrieval model not only achieves rapid predictions (<1 ms) with good accuracy (1.8 ppm or approximately 0.45 %) but also effectively captures sudden increases in XCO2 plumes near industrial emission sources. The accuracy of the machine learning model retrieval results is validated against reliable data from Total Carbon Column Observing Network (TCCON) sites, demonstrating its ability to effectively capture CO2 seasonal variations and annual growth trends.

A stochastic technique to solve interval non-linear programming problems using GH-difference

PurposeA stochastic technique for solving interval non-linear problems using generalized Hukuhara (GH)-difference is proposed. The non-linear programming problem in interval form is transformed into an equivalent non-linear programming problem with real coefficients by associating a Gaussian random variable to the interval and the six-sigma rule. The conceptualized idea eliminates the decision maker’s instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. To demonstrate a coherent understanding, numerical examples have been used.Design/methodology/approachA stochastic approach has been used to develop a solution technique for solving interval nonlinear programming problems which arise in the modeling of scientific and engineering problems under uncertain environments.FindingsThe proposed idea eliminates the decision maker’s instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. This method provides specific results rather than in the interval form, which are more practical and implementable by the decision maker.Originality/valueThis is to certify, that the research paper submitted is an outcome of original work. I have duly acknowledged all the sources from which the ideas and extracts have been taken. This article has not been submitted elsewhere for publication.

A Curved Surface Integral Method for Reliability Analysis of Multiple Failure Modes System with Non-Overlapping Failure Domains

Abstract In the study of reliability of systems with multiple failure modes, approximations can be obtained by calculating the probability of failure for each state function. The first-order reliability method and the second-order reliability method are effective, but they may introduce significant errors when dealing with certain nonlinear situations. Simulation methods such as line sampling method and response surface method can solve implicit function problems, but the large amount of calculation results in low efficiency. The curved surface integral method (CSI) has good accuracy in dealing with nonlinear problems. Therefore, a system reliability analysis method (CSIMMS) is proposed on the basis of CSI for solving multiple failure modes system reliability problems with non-overlapping failure domains. The order of magnitude of the failure probability is evaluated based on the reliability index and the degree of nonlinearity, ignoring the influence of low order of magnitude failure modes, and reducing the calculation of the system failure probability. Finally, CSIMMS and other methods are compared by three numerical examples, and the results show the stability and accuracy of the proposed method.

Research on optimization algorithms for localization and capacity determination of chargers considering the spatiotemporal distribution of electric vehicles

The optimized layout of electric vehicle (EV) chargers is not only crucial for users' convenience but also a key element in urban sustainable development, energy transition, and the promotion of new energy vehicles. In order to provide a basis for the problem of localization and capacity determination of chargers and compare the merits of several mainstream algorithms, this paper first establishes an optimization model with the objective of minimizing the total investment cost of all the chargers and the constraint of meeting the charging demands of all electric vehicles. Optimizations were performed using genetic algorithm (GA), surrogate optimization algorithm (SOA), and mixed integer linear programming (MILP) algorithm, respectively. In the case of using MILP, the original nonlinear optimization problem was transformed into a linear problem. In the planning of city-level EV chargers, MILP took 14182.57 s to calculate the minimum cost of 34.62 million yuan. After retaining only 10% of the original data amount, SOA took 87651.34 s to calculate the minimum cost of 3.01 million yuan. The results indicate that GA is prone to falling into local optima and is not suitable for large-scale optimization problems. SOA, on the other hand, requires significant memory consumption, so the issue of memory usage needs to be carefully considered when using it directly. Although MILP is only applicable to linear programming problems, it has the advantages of lower memory usage and higher reliability if the problem can be transformed into a linear one.

A Comprehensive Review on Discriminant Analysis for Addressing Challenges of Class-Level Limitations, Small Sample Size, and Robustness

The classical linear discriminant analysis (LDA) algorithm has three primary drawbacks, i.e., small sample size problem, sensitivity to noise and outliers, and inability to deal with multi-modal-class data. This paper reviews LDA technology and its variants, covering the taxonomy and characteristics of these technologies and comparing their innovations and developments in addressing these three shortcomings. Additionally, we describe the application areas and emphasize the kernel extensions of these technologies to solve nonlinear problems. Most importantly, this paper presents perspectives on future research directions and potential research areas in this field.

Dynamics modeling and nonlinear attitude controller design for a rocket-type unmanned aerial vehicle

This paper presents an altitude and attitude control system for a newly designed rocket-type unmanned aerial vehicle (UAV) propelled by a gimbal-based coaxial rotor system (GCRS) enabling thrust vector control (TVC). The GCRS is the only means of actuation available to control the UAV’s orientation, and the flight dynamics identify the primary control difficulty as the highly nonlinear and tightly coupled control distribution problem. To address this, the study presents detailed derivations of attitude flight dynamics and a control strategy to track the desired attitude trajectory. First, a Proportional-Integral-Derivative (PID) control algorithm is developed based on the formulation of linear matrix inequality (LMI) to ensure robust stability and performance. Second, an optimization algorithm using the Levenberg–Marquardt (LM) method is introduced to solve the nonlinear inverse mapping problem between the control law and the actual actuator outputs, addressing the nonlinear coupled control input distribution problem of the GCRS. In summary, the main contribution is the proposal of a new TVC UAV system based on GCRS. The PID control algorithm and LM algorithm were designed to solve the distribution problem of the actuation model and confirm altitude and attitude tracking missions. Finally, to validate the flight properties of the rocket-type UAV and the performance of the proposed control algorithm, several numerical simulations were conducted. The results indicate that the tightly coupled control input nonlinear inverse problem was successfully solved, and the proposed control algorithm achieved effective attitude stabilization even in the presence of disturbances.

Inverse a time-dependent potential problem of a generalized time-fractional super-diffusion equation with a nonlinear source from a nonlocal integral observation

This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function p(t), we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

Investigation of the highly complex nonlinear problems via modified energy balance method

In this paper, the modified energy balance method is proposed as an effective analytical approach to achieve more precise approximate results for a non-linear oscillator in a nano-electromechanical system (NEMS). The proposed solution is compared with previous approximate methods, including the Hamiltonian approach, the homotopy perturbation technique, and the numerical solution using Runge–Kutta 4th-order method. The results obtained from the modified energy balance method show excellent agreement with the numerical solutions. This technique is both efficient and precise in solving such nonlinear problems, offering significant advantages. The novelty of this paper lies in demonstrating the utility of the modified energy balance method for studying nano-electromechanical systems, enabling the construction of higher-order analytical solutions. This study allows for a thorough examination of the results obtained through these analytical solutions.

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

Research on laser communication ranging Integration and VLC nonlinear compensation technology

Abstract Visible light communication (VLC) is a new spectrum communication technology that provides a new direction to solve spectrum resource problems. The nonlinearity problem in VLC will lead to a significant reduction in the performance of the communication system. To solve this problem, this paper studies the integration of laser communication ranging and VLC nonlinear compensation technology based on analyzing the principles of ranging methods and factors affecting ranging performance. Aiming at the problem that the complexity of the Random Fourier Features Kernel adaptive filter (RFF-KAF) method will increase significantly as the dimension of the mapping space increases, we introduce some second-order Volterra series square terms into the input vector of RFF-KAF and propose an improved prediction method. Experimental simulation results show that the method proposed in this article can greatly improve the performance of the RFF-KAF method while maintaining the same mapping space dimension. Compared with obtaining similar performance by increasing the mapping space dimension, the cost of this method is greatly reduced.

RNNPose: 6-DoF Object Pose Estimation via Recurrent Correspondence Field Estimation and Pose Optimization.

6-DoF object pose estimation from a monocular image is a challenging problem, where a post-refinement procedure is generally needed for high-precision estimation. In this paper, we propose a framework, dubbed RNNPose, based on a recurrent neural network (RNN) for object pose refinement, which is robust to erroneous initial poses and occlusions. During the recurrent iterations, object pose refinement is formulated as a non-linear least squares problem based on the estimated correspondence field (between a rendered image and the observed image). The problem is then solved by a differentiable Levenberg-Marquardt (LM) algorithm enabling end-to-end training. The correspondence field estimation and pose refinement are conducted alternately in each iteration to improve the object poses. Furthermore, to improve the robustness against occlusion, we introduce a consistency-check mechanism based on the learned descriptors of the 3D model and observed 2D images, which downweights the unreliable correspondences during pose optimization. We evaluate RNNPose on several public datasets, including LINEMOD, Occlusion-LINEMOD, YCB-Video and TLESS. We demonstrate state-of-the-art performance and strong robustness against severe clutter and occlusion in the scenes. Extensive experiments validate the effectiveness of our proposed method. Besides, the extended system based on RNNPose successfully generalizes to multi-instance scenarios and achieves top-tier performance on the TLESS dataset.

Methods and algorithms of swarm intelligence for the problems of nonlinear regression analysis and optimization of complex processes, objects, and systems: review and modification of methods and algorithms

The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problem that involves reducing computing costs, accelerating, and effectively searching for solutions to such problems. Since most serious problems involve the search for tens, hundreds, or thousands of optimal parameters of mathematical models, the search space for these parameters grows non-linearly. Currently, there are many modern methods and algorithms of swarm intelligence that solve today's scientific and applied problems, but they require modifications due to the large spaces of searching for optimal model parameters. Modern swarm intelligence has significant potential for application in the energy industry due to its ability to optimize and solve complex problems. It can be used to solve scientific and applied problems of optimizing energy consumption in buildings, industrial complexes, and urban systems, reducing energy losses, and increasing the efficiency of resource use, as well as for the construction of various elements of energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively applied to forecast energy production from renewable sources, such as solar and wind energy. This allows better management of energy sources and planning of their use. The relevance of modifications of methods and algorithms is due to the issues of speeding up their work when solving machine learning problems, in particular, in nonlinear regression models, classification, and clustering problems, where the number of observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal problems with and without restrictions, as well as problems of nonlinear regression analysis. The obtained modifications of the well-known classic effective methods and algorithms of swarm intelligence, which are present in the work, effectively solve complex scientific and applied tasks of designing complex objects and systems. A comparative analysis of methods and algorithms will be conducted in the next study on this topic. Keywords: optimization, swarm intelligence, mathematical modelling, nonlinear regression, complex objects and systems.

Mathematical analysis of non-linear boundary-value problems in reaction-diffusion model of chitosan-alginate microsphere using asymptotic methods

This article describes the non-linear reaction-diffusion model, which depicts the behaviour of hydrogen peroxide production and glucose oxidation in the chitosan-alginate microsphere. Analytical solutions of glucose, oxygen, gluconic acid and hydrogen peroxide concentrations in planar coordinates under steady-state circumstances are obtained for all reaction-diffusion parameters. The non-linear boundary value problem's approximate analytical expressions are obtained using asymptotic techniques based on the homotopy perturbation and Akbari-Ganji method. The exact and commonly used MATLAB program provided a numerical simulation. It is demonstrated that the resulting analytical expressions strongly agree with the numerical outcomes reported in the literature. Furthermore, determining hydrogen peroxide flux and pH profiles within the microspheres are also discussed. The theoretical findings make it possible to forecast and enhance the efficiency of enzyme kinetics.