Nonlinear Problem Research Articles

Hybrid sea surface temperature inversion model for the South China sea based on IMLP and DBN

ABSTRACT Sea surface temperature (SST) is a key variable in the study of the global climate system and one of the important parameters in the process of air–sea interaction. Therefore, the demand for SST is developing towards high quality and high precision. In this paper, the ability of the multi-layer perceptron (MLP) optimized by the Aquila optimizer (AO) to solve nonlinear problems and the advantages of the deep belief network (DBN) to effectively process complex data are used to construct the IMLP-DBN inversion algorithm. The algorithm takes into account the influences of atmospheric conditions and satellite zenith angles. The data set is the infrared remote sensing data of the moderate resolution imaging spectroradiometer (MODIS) and the actual measurement data of buoys on sunny days and few clouds. Analysis of the inversion results shows that the root mean square error (RMSE) of the inversion value and the measured value is 0.14, and the sum of square errors (SSE) is 0.78. Compared with the MOD28 product data, the RMSE and SSE of the inversion are reduced by 66% and 24%, respectively.

A novel multilevel finite element method for a generalized nonlinear Schrödinger equation

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

Online actor‐critic learning control with self‐triggered mechanism for nonlinear regulation problems

Online actor‐critic learning control with self‐triggered mechanism for nonlinear regulation problems

Sharp boundary concentration for a two-dimensional nonlinear Neumann problem*

Sharp boundary concentration for a two-dimensional nonlinear Neumann problem*

Unified approach to nonlinear Caputo fractional derivative boundary value problems‎: ‎extending the upper and lower solutions method

Unified approach to nonlinear Caputo fractional derivative boundary value problems‎: ‎extending the upper and lower solutions method

Control of a Directional Downhole Drilling System Using a State Barrier Avoidance Based Method

Abstract Vibrations such as bit-bouncing, stick-slip, and whirl motion can significantly reduce downhole drilling?s performance and efficiency. These phenomena are likely to arise once the dynamical states of drilling fall into undesired operating regimes, as verified by both theoretical analyses and experimental studies. To effectively avoid such undesired operating conditions of a downhole drilling process, this paper introduces an advanced nonlinear state-constrained control method to formulate a setpoint tracking problem of high-order directional drilling dynamics under state constraints. These constraints are carefully defined using the field test results, and their shapes are in complex state space regions, causing additional complexity to the control design. Moreover, unlike vertical drilling, the directional drilling system requires modeling of coupled axial and torsional dynamics with a higher degree of freedom. In this study, the proposed method will tackle these challenges by converting the original high-order constrained control problem into a standard nonlinear control problem. Leveraging on the Linear Parameter Varying method that is applied to the converted system, we can actively prevent drilling from falling into the undesired regimes, to achieve the constrained control objective.

Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations

AbstractThe one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.

IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview

The trajectory estimation of a highly maneuvering target is a challenging problem and has practical applications. The interacting multiple model (IMM) filter is a well-established filtering algorithm for the trajectory estimation of maneuvering targets. In this study, we present an overview of IMM filtering algorithms for tracking a highly-maneuverable fighter aircraft using an air moving target indicator (AMTI) radar on another aircraft. This problem is a nonlinear filtering problem due to nonlinearities in the dynamic and measurement models. We first describe single-model nonlinear filtering algorithms: the extended Kalman filter (EKF), unscented Kalman filter (UKF), and cubature Kalman filter (CKF). Then, we summarize the IMM-based EKF (IMM-EKF), IMM-based UKF (IMM-UKF), and IMM-based CKF (CKF). In order to compare the state estimation accuracies of the IMM-based filters, we present a derivation of the posterior Cramér-Rao lower bound (PCRLB). We consider fighter aircraft traveling with accelerations 3g, 4g, 5g, and 6g and present numerical results for state estimation accuracy and computational cost under various operating conditions. Our results show that under normal operating conditions, the three IMM-based filters have nearly the same accuracy. This is due to the accuracy of the measurements of the AMTI radar and the high data rate.

A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

AbstractThis article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs

Abstract In 1996 Seo proved that two appropriate pairs of current and voltage data measured on the surface of a planar homogeneous object are sufficient to determine a conductive polygonal inclusion with known deviating conductivity. Here we show that the corresponding linearized forward map is injective, and from this we deduce Lipschitz stability of the solution of the original nonlinear inverse problem. We also treat the case of an insulating polygonal inclusion, in which case a single pair of Cauchy data is already sufficient for the same purpose.

Dynamic uncertainty propagation analysis framework for nonlinear control problem based on manifold learning and optimal polynomial method and its application on active suspension system

A novel uncertainty propagation method based on manifold learning and optimal polynomial model is proposed to quantify and analyze the dynamic uncertainties of the nonlinear feedback control system. To deal with the multiple objectives and constraints of the nonlinear control problem, a barrier Lyapunov-function-based nonlinear filtered backstepping controller is developed and applied to active suspension system to obtain superior ride comfort performance under limited structural constraints. Considering the errors in the production, manufacturing, and assembly, the probability density function is employed to quantify the structural parameter uncertainties in the nonlinear control system. Moreover, to reveal the dynamic propagation mechanism of uncertainties in the system with nonlinearity, the manifold learning method is proposed to reduce the dimensionality of the dynamic system to avoid the complexity of uncertainty propagation. Simultaneously, data-driven optimal polynomial model is utilized to accurately approximate the internal mechanism of nonlinear filtered backstepping control system. Based on that, the response uncertainties of the nonlinear control system are accurately and quickly quantified through dynamic moment information and uncertain fluctuation space. Finally, an active suspension system with nonlinearities and uncertainties is developed to verify the effectiveness of the controller with improved ride comfort and better handling safety and the superiority of the framework in terms of efficiency and accuracy of the dynamic uncertainty propagation analysis for nonlinear control problem.

A Novel Pelican Bird Optimization Algorithm based Optimal Allocation of Combined DGs and EVCSs for Improving Voltage Stability of RDS

Objectives: The main objective of the proposed investigation is to enhance the voltage stability of Radial Distribution System (RDS). It is achieved by proper allocation between combined Electric Vehicle Charging Stations (EVCSs) and Distributed Generators (DGs) in RDS. Here, three objective functions such as Real Power loss, Average Voltage Devotion Index (AVDI) and Voltage Stability Index (VSI) are considered. Methods: The multi-objective optimization problem is implemented using a novel and an effective Pelican Optimization Algorithm (POA) to attain the best compromised solutions. POA is a nature inspired parameter less optimization tool and it is well suited for solution of complex, non-linear voltage stability problem. It determines the most excellent site and size of the EVCSs and DG units. The control parameter of POA considered are population size as 40, the number of variables as 8 and the maximum number of iterations as 100. MATLAB 14.0 platform is utilized for the projected problem. Findings: The bench mark test system of IEEE-33 node system is considered to authenticate the presentation of planned POA methodology. The outcomes such as minimized power loss and AVDI, improved voltage at each bus and VSI are presented. The percentage of loss reduction 39.30% is enhanced, when compared with the base case approach and 5.42 % more improved than the other existing methods like TLBO and HHO approaches. Similarly, the VSI 24.96 % is improved, when compared with the base case method and 1.58 % better than the other obtainable methods. Novelty: The POA is a newly urbanized and well-designed optimization algorithm. It efficiently reaches the global optimal solutions with less time duration. The comparative study with the existing algorithm of TLBO and HHO is considered to prove the performance of Projected POA approach. Keywords: Radial distribution system, Distributed Generators, Electric Vehicles, Charging Stations, Voltage stability, Network loss reduction, Pelican Optimization Algorithm

Influence of Rotation and Viscosity on Parallel Rolls of Electrically Conducting Fluid

Rayleigh–Bénard convection is a fundamental fluid dynamics phenomenon that significantly influences heat transfer in various natural and industrial processes, such as geophysical dynamics in the Earth’s liquid core and the performance of heat exchangers. Understanding the behavior of conductive fluids under the influence of heating, rotation, and magnetic fields is critical for improving thermal management systems. Utilizing the Boussinesq approximation, this study theoretically examines the nonlinear convection of a planar layer of conductive liquid that is heated from below and subjected to rotation about a vertical axis in the presence of a magnetic field. We focus on the onset of stationary convection as the temperature difference applied across the planar layer increases. Our theoretical approach investigates the formation of parallel rolls aligned with the magnetic field under free–free boundary conditions. To analyze the system of nonlinear equations, we expand the dependent variables in a series of orthogonal functions and express the coefficients of these functions as power series in a parameter ϵ. A solution for this nonlinear problem is derived through Fourier analysis of perturbations, extending to O(ϵ8), which allows for a detailed visualization of the parallel rolls. Graphical results are presented to explore the dependence of the Nusselt number on the Rayleigh number (R) and Ekman number (E). We observe that both the local Nusselt number and average Nusselt number increase as the Ekman number decreases. Furthermore, the flow appears to become more deformed as E decreases, suggesting an increased influence of external factors such as rotation. This deformation may enhance mixing within the fluid, thereby improving heat transfer between different regions.

Optimization-based design of structure and operation parameters simultaneously for real-world distillation column based on rate-based model

Simultaneously optimizing the structure and operating parameters of distillation columns based on rigorous rate-based models is critical for various process intensification technologies to achieve their intended purpose in real-world distillation columns. However, efficiently optimizing a distillation column based on rate-based models while considering the column hydraulics is far from practical. The biggest obstacle in using rigorous models for distillation columns in an equation-oriented environment is ensuring a robust solution for large-scale, strongly coupled nonlinear models, along with the simultaneous optimization of integer and continuous variables. In this study, a new optimization framework was introduced. It incorporated a pseudo-transient simulation method to ensure convergence of the simulation in equation-oriented environment. Additionally, an efficient relaxation method transformed the mixed integer nonlinear programming problem into a nonlinear programming problem, facilitating simultaneous optimization of both the integer variable representing column structure and the continuous variables governing column operation. Three examples, including a complex intensified catalytic distillation column, were optimized to demonstrate the efficiency of the proposed framework. Specifically, its advantages in terms of mathematical optimality and industrial feasibility were illustrated.

New disturbance observer-based speed estimator for induction motor

<p>This paper discusses a novel disturbance observer designed as an estimator to determine the rotor speed of an induction motor. This observer is a solution to obtain a simple structure with a small number of compact observer gains. Furthermore, the adaptation law is no longer required to estimate induction motor speed values. This is a machine model-based computation method that uses a stationary reference frame. The nonlinearity problem is solved using an additional state vector in the observer model, which is known as an extended state observer. This approach easily and systematically determines the observer gain by applying the linear quadratic regulator (LQR) method, thereby avoiding time-consuming trial errors. The proposed observer, which was presented in both continuous and discrete forms, was tested using a sensorless V/f- controlled induction motor. The simulation results show that the proposed observer can accurately estimate all states, namely, the rotor flux and stator current; therefore, the proposed estimator provides the speed and electromagnetic torque for a wide operational range of speeds and load torques. It was also shown that the proposed observer was robust to noise and uncertainty in induction motor parameters.</p>

Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.

Fuel-optimal acquisition and control of a cartwheel formation in Earth displaced heliocentric orbit

An optimization approach for cartwheel formation acquisition and maintenance in an Earth Displaced heliocentric orbit is presented. This work considers non-gravitational perturbations such as solar radiation pressure, thus extending the studies previously performed for the mission LISA. The problem is tackled as a Nonlinear Programming problem using a multiple shooting method. The optimization process is performed in two steps: first, the orbital elements of each satellite in heliocentric orbit are optimized to guarantee the stability during the science phase hence easing maintenance of the cartwheel formation in presence of orbital perturbations. Then, the obtained initial states are propagated to obtain a set of target orbital states which become the final target of a second optimization covering the transfer phase from Earth. For the science phase optimization presents two alternative cost functions are introduced, one based on the arm-length evolution and one on the arm-length-rate evolution. The performance of each cost function is analysed for different initial displacement angles: for target arm-lengths below 2.5 million kilometers the arm-length cost function provides the best results while no significant difference between the two optimized solutions is observed above this value. The transfer phase optimization presents two different approaches, one considering an injection on a trajectory more favourable for one of the three spacecraft and one considering an injection on an intermediate trajectory which minimizes the overall acquisition cost of all spacecraft. The proposed optimization approach performance are studied on a set of test cases covering both transfer and science phase, showing that stable configuration conditions can be found even in presence of orbital perturbations and that the multiple injection transfer is capable of providing a more homogeneous fuel consumption among the three spacecraft.

Binary Ant Colony Optimization Algorithm in Learning Random Satisfiability Logic for Discrete Hopfield Neural Network

This study introduced a novel ant colony optimization algorithm that implements the population selection strategy of the Estimation of Distribution Algorithm and a new pheromone updating formula. It aimed to optimize the performance of G-type random high-order satisfiability logic structures embedded in Discrete Hopfield Neural Networks, thereby enhancing the efficiency of the Hopfield Neural Network learning algorithm. Through comparative analysis with other metaheuristic algorithms, our model demonstrated superior performance in terms of global convergence, time complexity, and algorithm complexity. Additionally, we evaluated the learning phase, retrieval phase, and similarity analysis using various ratios of literals and clauses. It was shown that our proposed model exhibits stronger search ability compared to other metaheuristic algorithms and Exhaustive Search. Our model enhanced the efficiency of the learning phase, resulting in the number of global solutions accounting for 100%, and significantly improved the global solution diversity. These advancements contributed to the efficiency of the model in convergence, rendering it applicable to a wide range of nonlinear classification and prediction problems.

Optimal control of Volterra integral equations of third kind using Krall-Laguerre Polynomials

A novel method has been introduced to dissolve optimal control problems in systems governed by third-kind Volterra integral equations. This method utilizes Krall-Laguerre Polynomials as a basis function for expanding functions. By transforming the optimal control problem governed by third-kind Volterra integral equations (OCVIE3k) into a nonlinear programming problem (NLP), the solution process is significantly simplified. This approach involves converting the original problem into a more manageable form, which can be solved using established optimization techniques. The effectiveness and reliability of this proposed method are evaluated by comparing its outcomes to exact solutions when available.

End-to-end, decision-based, cardinality-constrained portfolio optimization

Portfolios employing a (factor) risk model are usually constructed using a two step process: first, the risk model parameters are estimated, then the portfolio is constructed. Recent works have shown that this decoupled approach may be improved using an integrated framework that takes the downstream portfolio optimization into account during parameter estimation. In this work we implement an integrated, end-to-end, predict-&-optimize framework to the cardinality-constrained portfolio optimization problem. To the best of our knowledge, we are the first to implement the framework to a nonlinear mixed integer programming problem. Since the feasible region of the problem is discontinuous, we are unable to directly differentiate through it. Thus, we compare three different continuous relaxations of increasing tightness to the problem which are placed as an implicit layers in a neural network. The parameters of the factor model governing the problem’s covariance matrix structure are learned using a loss function that directly corresponds to the decision quality made based on the factor model’s predictions. Using real world financial data, our proposed end-to-end, decision based model is compared to two decoupled alternatives. Results show significant improvements over the traditional decoupled approaches across all cardinality sizes and model variations while highlighting the need of additional research into the interplay between experimental design, problem size and structure, and relaxation tightness in a combinatorial setting.