Nonlinear Problem Research Articles

Adaptive Pseudospectral Successive Convex Optimization for Six-Degree-of-Freedom Powered Descent Guidance

Guidance for six-degrees-of-freedom powered descent is more challenging due to its stronger nonlinearity compared to three-degrees-of-freedom. The standard convex programming algorithm has been difficult to effectively address this problem. To enhance the performance of successive convex programming in terms of both optimality and accuracy, an adaptive pseudospectral successive convex optimization algorithm is proposed in this paper. First, it transforms the nonlinear optimal control problem into a convex subproblem by integrating global pseudospectral discretization with local linearization. Second, a parameter-adaptive successive convexification algorithm is proposed, which adaptively adjusts the trust region size based on the update rate of the optimal trajectory. Last, the global error and local error are accurately calculated based on the pseudospectral method. To tackle the issue of excessively large local errors caused by nonlinearity, an adaptive grid method is proposed to refine the mesh grid. Simulation results demonstrate the efficacy of the proposed pseudospectral convex optimization method and adaptive mesh grid method in reducing local errors and improving solution accuracy. Furthermore, the proposed parameter-adaptive successive convex optimization algorithm exhibits adaptability and optimality across different initial conditions, while also improving solution accuracy.

Weighted sum method based multi-objective optimal power flow considering various objectives: an application of whale optimization algorithm

Nowadays, multi-objective optimization plays a vital role in solving optimal power flow problems. Multi-objective optimal power flow (MOOPF) is a nonlinear optimization problem aimed at optimizing control variables while balancing multiple objective functions and satisfying both equality and inequality constraints and addresses this by integrating two more objectives into a single objective using a weighting factor. In this paper this weighted sum type multi-objective technique has been used to formulate the objective function. The whale optimization algorithm (WOA) has been used to reduce the cost, emission, losses, and voltage stability by considering various multi objectives like fuel cost along with emission, fuel cost with losses, fuel cost with voltage stability, fuel cost with voltage deviation and finally fuel cost with emission, losses, voltage deviation. In this paper, the IEEE 30 bus structure has been used to analyze the effect of WOA on the improvement of system performance. Obtained results with WOA have been compared with other optimization techniques like ensemble constraint handling technique with differential evolution (ECHT-DE), the superiority of feasible differential evolution (SF-DE), moth swarm algorithm (MSA), and moth-flame optimization (MFO), available in the literature.

A divergence-free and [formula omitted]-conforming embedded-hybridized DG method for the incompressible resistive MHD equations

We present a divergence-free and Hdiv-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and Hdiv-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems.

Adaptive reverse task offloading in edge computing for AI processes

Nowadays, we witness the proliferation of edge IoT devices, ranging from smart cameras to autonomous vehicles, with increasing computing capabilities, used to implement AI-based services in users’ proximity, right at the edge. As these services are often computationally demanding, the popular paradigm of offloading their tasks to nearby cloud servers has gained much traction and been studied extensively. In this work, we propose a new paradigm that departs from the above typical edge computing offloading idea. Namely, we argue that it is possible to leverage these end nodes to assist larger nodes (e.g., cloudlets) in executing AI tasks. Indeed, as more and more end nodes are deployed, they create an abundance of idle computing capacity, which, when aggregated and exploited in a systematic fashion, can be proved beneficial. We introduce the idea of reverse offloading and study a scenario where a powerful node splits an AI task into a group of subtasks and assigns them to a set of nearby edge IoT nodes. The goal of each node is to minimize the overall execution time, which is constrained by the slowest subtask, while adhering to predetermined energy consumption and AI performance constraints. This is a challenging MINLP (Mixed Integer Non-Linear Problem) optimization problem that we tackle with a novel approach through our newly introduced EAI-ARO (Edge AI-Adaptive Reverse Offloading) algorithm. Furthermore, a demonstration of the efficacy of our reverse offloading proposal using an edge computing testbed and a representative AI service is performed. The findings suggest that our method optimizes the system’s performance significantly when compared with a greedy and a baseline task offloading algorithm.

A probabilistic integration of LSTM and Gaussian process regression for uncertainty-aware reservoir water level predictions

ABSTRACT Reservoir-level forecasting, while being crucial for optimal operation, is challenged by complex physical processes and changing climate conditions. Machine learning approaches offer deterministic predictions but often neglect system physics and uncertainty. This article presents a probabilistic data-driven approach combining long short-term memory (LSTM) and Gaussian process regression (GPR) to provide both point forecasts and uncertainty estimates. The hybrid model leverages LSTM’s fitting capabilities with GPR’s robust Bayesian frameworks for uncertainty estimation in non-linear problems, offering accurate predictions without extensive high-fidelity modelling, and avoiding frequent training and parameter optimization. Evaluation with real reservoir data from India shows the model’s superiority over the vanilla LSTM for both univariate and multivariate scenarios. The proposed model achieved a Nash-Sutcliffe efficiency of 0.97 to 0.98, a mean biased error of −0.5634 to −1.0314 for 10-day forecasts, and a continuous ranked probability score of 5.80 and 1.87 for the Bhakra and Pong reservoirs, respectively.

Asymptotic behavior of a nonlinear viscoelastic problems with Tresca friction law in a thin domain

Asymptotic behavior of a nonlinear viscoelastic problems with Tresca friction law in a thin domain

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

Abstract In this paper, we study a nonlinear fluid-structure interaction problem between a ‘square-root’ viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions in both directions, while both the beam and the fluid are under the effect of time-periodic forces. By using a decoupling approach, at least one time-periodic weak solution to this problem is constructed which has a bounded energy and a fixed prescribed mass. The lack of a priori energy bounds is overcome by a series of estimates based on a careful choice of parameters. The most challenging one is the pressure estimate, which is obtained by utilizing the specific periodic geometry and the Bogovskiǐ operator on a fixed domain that has a uniform constant. With uniform estimates and improved regularity of the beam as in (Muha and Schwarzacher 2023 Ann. Inst. Henri Poin. Anal. Non Lineaire 39 1369–412), the time-periodic solution is constructed by a series of limit procedures, following the finite-dimensional time-space construction from (Feireisl et al 2012 Arch. Rational Mech. Anal. 204 74586).

Dynamic description of soliton solutions of nonlinear schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Binary Observation‐Based FIR System Identification Against Replay Attacks

ABSTRACTIn the context of the increasing security issues of cyber‐physical systems (CPSs), this paper addresses the parameter identification of binary observation‐based finite impulse response (FIR) systems under replay attacks, overcoming the problem of high nonlinearity of quantized systems and greater data sparsity caused by replay attacks. For the attacker, based on the energy‐constrained condition, an optimization attack model is established to maximize the absolute error of identification, giving the method of obtaining the optimal attack strategy. Following the defender, the identifiability of unknown parameters is discussed and a robust defense scheme is proposed. This scheme involves a joint identification strategy for both the attack strategy and unknown parameters. By enhancing the excitability of system inputs, consistent identification is ensured despite replay attacks. An algorithm for the accomplishment of the joint identification strategy is presented based on the grid search method. Rationality of the method is confirmed with performing numerical simulations.

Mechanism and Control Strategies for Current Sharing in Multi-Chip Parallel Automotive Power Modules

Multi-chip parallel power modules are highly favored in applications requiring high capacity and high switching frequency. However, the dynamic current imbalance between parallel chips caused by asymmetric layouts limits the available capacity. This paper presents a method to optimize dynamic current distribution by adjusting the lengths and connection points of bond wires. For the first time, a response surface model and nonlinear constraint optimization algorithm are introduced, along with parameter analysis based on finite element methods, to establish the response surface models for the parasitic inductance of bond wires and DBC (direct bonded copper). By leveraging the optimization goals for parasitic inductance and the analytical expressions of all response surfaces, the dynamic current sharing issue was transformed into a nonlinear constrained optimization problem. The solution to this optimization problem identified the optimal connection points for the bond wires, enhancing dynamic current sharing performance. Simulations and experiments were conducted, revealing that the optimized automotive-grade module exhibited a significant reduction in current differences between parallel branches, from 41.7% to 5.03% compared with the original design. This indicated that the proposed optimization scheme for adjusting bond wire connection points could significantly mitigate current disparities, thereby markedly improving current distribution uniformity.

QHDOPT: A Software for Nonlinear Optimization with Quantum Hamiltonian Descent

We develop an open-source, end-to-end software (named QHDOPT), which can solve nonlinear optimization problems using the quantum Hamiltonian descent (QHD) algorithm. QHDOPT offers an accessible interface and automatically maps tasks to various supported quantum backends (i.e., quantum hardware machines). These features enable users, even those without prior knowledge or experience in quantum computing, to utilize the power of existing quantum devices for nonlinear and nonconvex optimization tasks. In its intermediate compilation layer, QHDOPT employs SimuQ, an efficient interface for Hamiltonian-oriented programming, to facilitate multiple algorithmic specifications and ensure compatible cross-hardware deployment. The detailed documentation of QHDOPT is available at https://github.com/jiaqileng/QHDOPT . History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the U.S. Department of Energy’s Advanced Research Projects Agency–Energy [Grant DE-SC0020273], the Alfred P. Sloan Foundation, the Simons Foundation [Simons Investigator Award 825053], the Simons Quantum Postdoctoral Fellowship, the National Science Foundation [Grants CCF-1816695, CCF-1942837, and ECCS-2045978], the Unitary Fund, and the Air Force Office of Scientific Research [Grant FA95502110051]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0587 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0587 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Local regularity for nonlocal double phase equations in the Heisenberg group

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$ . This solves a problem raised by Palatucci and Piccinini et al. in 2022 and 2023 for the nonlinear integro-differential problems in Heisenberg setting. Our proof of the a priori estimates bases on De Giorgi–Nash–Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev–Poincaré type inequality in local domain, which may be of independent interest and potential applications.

Convexity for a parabolic fully nonlinear free boundary problem with singular term

Convexity for a parabolic fully nonlinear free boundary problem with singular term

Recent Trends in Proxy Model Development for Well Placement Optimization Employing Machine Learning Techniques

Well placement optimization refers to the identification of optimal locations for wells (producers and injectors) to maximize net present value (NPV) and oil recovery. It is a complex challenge in all phases of production (primary, secondary and tertiary) of a reservoir. Reservoir simulation is primarily used to solve this intricate task by analyzing numerous scenarios with varied well locations to determine the optimum location that maximizes the targeted objective functions (e.g., NPV and oil recovery). Proxy models are a computationally less expensive alternative to traditional reservoir simulation techniques since they approximate complex simulations with simpler models. Previous review papers have focused on analyzing various optimization algorithms and techniques for well placement. This article explores various types of proxy models that are the most suitable for well placement optimization due their discrete and nonlinear natures and focuses on recent advances in the area. Proxy models in this article are sub-divided into two primary classes, namely data-driven models and reduced order models (ROMs). The data-driven models include statistical- and machine learning (ML)-based approximations of nonlinear problems. The second class, i.e., a ROM, uses proper orthogonal decomposition (POD) methods to reduce the dimensionality of the problem. This paper introduces various subcategories within these two proxy model classes and presents the successful applications from the well placement optimization literature. Finally, the potential of integrating a data-driven approach with ROM techniques to develop more computationally efficient proxy models for well placement optimization is also discussed. This article is intended to serve as a comprehensive review of the latest proxy model techniques for the well placement optimization problem. In conclusion, while proxy models have their own challenges, their ability to significantly reduce the complexity of the well placement optimization process for huge reservoir simulation areas makes them extremely appealing. With active research and development occurring in this area, proxy models are poised to play an increasingly central role in oil and gas well placement optimization.

Computational study of thin films made from the ferroelectric materials with Paul Painlevé approach and expansion and variational methods

In this paper, the thin-film ferroelectric material equation which enables a propagation of solitary polarization in thin-film ferroelectric materials, and it also can be described using the nonlinear evolution equations. Ferroelectrics are dielectric materials explain wave propagation nonlinear behaviors. Thin films made from the ferroelectric materials are used in various modern electronics devices. The Paul-Painlevé approach is adopted for the first time to solve these nonlinear thin-film equation analytically. To investigate the characterizations of new waves, the solitary wave dynamics of the thin-film ferroelectric material equation are obtained using the standard \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an (\\phi /2)$$\\end{document}-expansion technique and generalized G-expansion method. The bright and periodic solutions are obtained by semi-inverse variational principle scheme. Many alternative responses are achieved utilizing various formulaes; each of these solutions is shown by a distinct graph. The validity of such methods and solutions are demonstrated by assessing how well the relevant techniques and solutions match up. Three novel analytical and numerical techniques provide new, dependable approaches for determining and estimating responses. The effect of the free variables on the behavior of reached solutions to a few of graphs on the exact solutions is also explored depending upon the nature of nonlinearities. The simulations, which are exhibited in both two-dimensional (2D) and three-dimensional (3D), depict the behavior of a solitary solution in both the natural and digital worlds. These findings demonstrate that this strategy is the most effective way to solve nonlinear mathematical physics problems.

Optimal Coordination of Directional Overcurrent Relays in Microgrids Considering European and North American Curves

Protecting AC microgrids (MGs) is a challenging task due to their dual operating modes—grid-connected and islanded—which cause sudden variations in fault currents. Traditional protection methods may no longer ensure network security. This paper presents a novel approach to protection coordination in AC MGs using non-standard features of directional over-current relays (DOCRs). Three key optimization variables are considered: Time Multiplier Setting (TMS), the plug setting multiplier’s (PSM) maximum limit, and the standard characteristic curve (SCC). The proposed model is formulated as a mixed-integer nonlinear programming problem and solved using four metaheuristic techniques: the genetic algorithm (GA), Imperialist Competitive Algorithm (ICA), Harmonic Search (HS), and Firefly Algorithm (FA). Tests on a benchmark IEC MG with distributed generation and various operating modes demonstrate that this approach reduces coordination times compared to existing methods. This paper’s main contributions are threefold: (1) introducing a methodology for assessing the optimal performance of different standard curves in MG protection; (2) utilizing non-standard characteristics for optimal coordination of DOCRs; and (3) enabling the selection of curves from both North American and European standards. This approach improves trip time performance across multiple operating modes and topologies, enhancing the reliability and efficiency of MG protection systems.

Existence of weak solutions for nonlocal Dirichlet problems with variable-order fractional kernels

This paper is concerned with the existence and uniqueness of weak solutions for a class of linear and nonlinear nonlocal homogeneous Dirichlet boundary value problems with a truncated variable-order fractional kernel γ ( x , y ) . By the structural features of the kernel γ ( x , y ) , a new variable-order fractional Banach space [ W 0 α ( x , y ) , p ( Ω ′ ) ] d is introduced as the suitable solution space, and its some qualitative properties are established. Then under such a functional framework, based on the variational methods, we prove the existence results for the linear problem with a nonsymmetric kernel case and for the nonlinear problem with a symmetric kernel case, respectively.

A self-driven ESN-DSS approach for effective COVID-19 time series prediction and modelling.

Since the outbreak of the COVID-19 epidemic, it has posed a great crisis to the health and economy of the world. The objective is to provide a simple deep-learning approach for predicting, modelling, and evaluating the time evolutions of the COVID-19 epidemic. The Dove Swarm Search (DSS) algorithm is integrated with the echo state network (ESN) to optimize the weight. The ESN-DSS model is constructed to predict the evolution of the COVID-19 time series. Specifically, the self-driven ESN-DSS is created to form a closed feedback loop by replacing the input with the output. The prediction results, which involve COVID-19 temporal evolutions of multiple countries worldwide, indicate the excellent prediction performances of our model compared with several artificial intelligence prediction methods from the literature (e.g., recurrent neural network, long short-term memory, gated recurrent units, variational auto encoder) at the same time scale. Moreover, the model parameters of the self-driven ESN-DSS are determined which acts as a significant impact on the prediction performance. As a result, the network parameters are adjusted to improve the prediction accuracy. The prediction results can be used as proposals to help governments and medical institutions formulate pertinent precautionary measures to prevent further spread. In addition, this study is not only limited to COVID-19 time series forecasting but also applicable to other nonlinear time series prediction problems.

Spatial-time decay for nonlinear problems backward in time

Spatial-time decay for nonlinear problems backward in time

An Activity Network Design and Charging Facility Planning Model Considering the Influence of Uncertain Activities in a Game Framework

In the planning of public charging facilities and the charging activity network of users, there is a decision-making conflict among three stakeholders: the government, charging station enterprises, and electric vehicle users. Previous studies have described the tripartite game relationship in a relatively simplistic manner, and when designing charging facility planning schemes, they did not consider scenarios where users’ choice preferences undergo continuous random changes. In order to reduce the impacts of queuing phenomenon and resource idleness on the three participants, we introduce a bilateral matching algorithm combined with the dynamic Huff model as a strategy for EV charging selection in the passenger flow problem based on the three-dimensional activity network of time–space–energy of users. Meanwhile, the Dirichlet distribution is utilized to control the selection preferences on the user side, constructing uncertain scenarios for the choice of user charging activities. In this study, we establish a bilevel programming model that takes into account the uncertainty in social responsibility and user charging selection behavior. Solutions for the activity network and facility planning schemes can be derived based on the collaborative relationships among the three parties. The model employs a robust optimization method to collaboratively design the charging activity network and facility planning scheme. For this mixed-integer nonlinear multi-objective multi-constraint optimization problem, the model is solved by the NSGA-II algorithm, and the optimal compromise scheme is determined by using the EWM-TOPSIS comprehensive evaluation method for the Pareto solution set. Finally, the efficacy of the model and the solution algorithm is illustrated by a simulation example in a real urban space.