Nonlinear Problem Research Articles

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.

Continuous Switch Model and Heuristics for Mixed-Integer Nonlinear Problems in Power Systems

Continuous Switch Model and Heuristics for Mixed-Integer Nonlinear Problems in Power Systems

FORMULATION OF THE OPTIMIZATION PROBLEM FOR UNDERGROUND ENGINEERING NETWORK ROUTES

The article considers the implementation of methods for optimizing underground external engineering networks in order to reduce construction costs. For this purpose, the problem of optimizing the route of such networks is set in the form of a nonlinear programming problem. The route is modeled by a spatial polyline with rectilinear segments, and the coordinates of its vertices are considered design variables. The ground surface in the designated area is modeled as a continuous function of two coordinates in the plan using BIM technologies, which ensures the automated use of topographic information when calculating the volume of earthworks and checking compliance with constraints specified by standards requirements. The main result of the study is the formulation of the problem of optimizing the route of the engineering network taking into account the terrain.

Surface geometry inversion of transient electromagnetic data

We investigate an emerging method called surface geometry inversion (SGI) for the inversion of transient electromagnetic (TEM) data. Conventional minimum-structure inversion methods parameterize the earth model with many mesh cells within which the physical properties are constant, and construct a physical property model that is usually smoothly varying as well as fits the observations. With these smooth models, it is difficult to extract the interface between different geologic units, and it can be especially difficult to target drillholes for thin, plate-like targets, which are frequently encountered in mineral exploration. Our SGI parameterizes the model in terms of the coordinates of the nodes (vertices) used to connect together the surfaces that define the geologic interfaces. The algorithm then inverts for the locations of these nodes, which directly provides geometric information about the target. This can be more useful than a fuzzy image of conductivity, especially for an exploration project. A genetic algorithm (GA) is used to solve the nonlinear, overdetermined optimization problem. We use a finite-element solver to solve the TEM forward-modeling problem of each candidate model in the GA population. Because forward modeling is independent for each model, we implement a hybrid message passing interface (MPI) + OpenMP parallel method to improve computational efficiency. We investigate a new parameterization method specifically designed for thin, plate-like structures, which is more efficient and can effectively avoid self-intersection. We first illustrate the effectiveness of our SGI algorithm on a synthetic block model before testing the new parameterization method on a synthetic thin plate model. Finally, we apply our SGI to a real data set collected for the exploration of thin graphitic faults in a uranium exploration project in Canada. The constructed model from our SGI corresponds well with the drilling data.

A Bayesian optimization approach for data‐driven mixed‐integer nonlinear programming problems

AbstractThe optimization of process systems within the field of Chemical Engineering often confronts uncertainties that can exert significant influences on the performance and dependability of the obtained solutions. This research endeavors to investigate the application of Bayesian optimization in the realm of constrained mixed integer nonlinear problems. A comparative analysis was conducted, exploring different surrogate models, and evaluating diverse kernel functions and acquisition functions. Furthermore, a sampling strategy was devised to assess the enhancement achieved by the acquisition function. The findings of this article reveal the superior performance of sparse Gaussian processes in conjunction with computationally inexpensive acquisition functions, thereby highlighting their suitability for addressing mixed integer nonlinear programming problems characterized by noisy functions and stochastic behavior. Consequently, this article presents a computationally efficient approach to effectively tackle the challenges associated with data‐driven mixed integer nonlinear programming problems within the domain of Process System Engineering.

A novel seasonal grey prediction model with time-lag and interactive effects for forecasting the photovoltaic power generation

Electricity is one of the most important energy sources for the progress of human society. Photovoltaic power generation (PPG) has great potential as a renewable energy source to support sustainable development. However, the unpredictability of photovoltaics because of the uncertain weather conditions, time lags and the complex interactions among related factors may result in instabilities in PPG, which limits its applications in the electric power system. Therefore, accurately identifying the related factors and representing the interactions are the key issues in predicting PPG. For this purpose, a novel seasonal grey prediction model with time-lag and interactive effects, denoted as SGMLI(1,n), is developed to predict the capacity of photovoltaic power generation. More specifically, a time-lag driving term, a periodic driving term, and an interaction driving term are proposed to represent the lag relationship, periodicity, and the interactive effects between two factors, respectively. Utilizing the least square method and the Whale Optimization Algorithm (WOA), the optimal solution to the nonlinear programming problem is estimated. For elaboration and verification, six grey forecasting models, two statistical techniques, and two machine learning models, are employed for comparison with the SGMLI(1,n) model. Experimental results reveal that the SGMLI(1,n) outperforms benchmark models in forecasting PPG.

Feasibility of acousto-electric tomography

In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

Pure quasi-P-wave modeling and imaging using an approximated space-domain pseudo-differential operator in the VTI media

Incorporating anisotropy in seismic modeling and imaging is important to produce correct locations of subsurface reflectors. Traditional wave equations for quasi-P-wave in the transverse isotropic media either suffer from S-wave artifacts or require complicated and expensive computation strategies. To mitigate this issue, we present a novel pure quasi-P-wave equation with an approximated space-domain pseudo-differential operator in the vertical transverse isotropic (VTI) media. For the pure quasi-P-wave equation, we first simplify it to an elliptical anisotropy equation with an additional pseudo-differential correction term. Then, we directly approximate the pseudo-differential term with a space-domain convolution operator that is calculated by solving a nonlinear inverse problem. Phase-velocity analysis and numerical modeling show that the new space-domain pseudo-differential operator has a good accuracy to describe wave propagation in the VTI media. In addition, it is more suitable for the parallel computation with domain-decomposition than the Fourier transform that is necessary for solving traditional pseudo-differential operators. Finally, we apply the proposed quasi-P-wave propagator to reverse-time migration to correct the anisotropic effects in seismic imaging. Numerical experiments for benchmark models and a land survey demonstrate the feasibility and adaptability of the proposed method.

Control strategy for trading off solar power and control input while rendezvous and docking

This paper presents a novel strategy for decoupling the attitude and orbit equations of a CubeSat for rendezvous and docking with an uncontrollable cooperative target. For computing a safe trajectory, the proposed control algorithm takes into account the solar energy received by the CubeSat. The CubeSat is equipped with a thruster for orbit maneuvers, magnetic coils for attitude control, and four fixed single-sided rectangular solar panels. The coupled dynamics arising from the dynamical model of the spacecraft is solved using a novel weighted vector-based approach. This paper presents a linearized nonlinear optimal control problem arising in small spacecrafts while trying to maximize solar energy input in the rendezvous and docking process. An intermediate orbit is defined and used to divide the problem into two different optimal control problems: rendezvous optimization and docking optimization problems. A geometrical approach based on the shape of the chaser and the target is contemplated for collision avoidance while docking. The proposed controller design is modified whenever the sunlight is obstructed by the Earth, and the maneuvering controls are redesigned accordingly for optimal rendezvous and docking. Numerical simulations have been carried out to show the efficacy of the proposed concept for steering the trade-off between solar energy and control input during the rendezvous and docking of CubeSat with a tumbling target.

Design cost minimization of a reinforced concrete column section using overnew swarm-based optimization algorithms

AbstractIt is very tiresome for a practiser to detect the best feasible sizing design of structural members including reinforced concrete columns that is a highly nonlinear and complicated structural engineering optimization problem. This is due to such a design is practically conducted via conventional trial-and-error computing methods in which resistance to external loads, cost efficiency, and aesthetic factors, etc. have to be considered. This study focuses on minimizing the design cost of primarily proposed reinforced concrete column design problem via three overnew swarm-based optimizers such as Coati Optimization Algorithm, Fox Optimizer and Pelican Optimization Algorithm (POA) that are firstly utilized for this purpose. In this regard, the type of steel rebar distribution, the characteristic strength of the concrete, the height and width of the column section, and the number and diameter of the rebars are treated as discrete design variables of the newly proposed complex reinforced concrete column design cost optimization problem. In solution, the design requirements specified in practice code provisions should also be met. Here, Turkish Building Earthquake Code 2018 specifications are considered as practice structural design constraints. Consequently, the algorithmic performances of three overnew swarm-based metaheuristic optimization algorithms are compared and evaluated in detail. Amongst them, the POA shows most fruitful algorithmic design solution performance.

Solution of a Scalar Two-Dimensional Nonlinear Problem of Diffraction on Objects of Arbitrary Shape

Solution of a Scalar Two-Dimensional Nonlinear Problem of Diffraction on Objects of Arbitrary Shape

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

Discontinuous Galerkin Method for Nonlinear Quasi-Static Poroelasticity Problems

Discontinuous Galerkin Method for Nonlinear Quasi-Static Poroelasticity Problems

Positive solutions for parametric equations with unbalanced growth and indefinite perturbation

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions.

Steady-state triad resonance between a surface gravity wave and two hydroacoustic waves based on the homotopy analysis method

Under water compressibility, resonant triads can occur within the family of acoustic-gravity waves. This work investigates steady-state triad resonance between a surface gravity wave and two hydroacoustic waves. The water-wave equations are solved as a nonlinear boundary value problem using the homotopy analysis method (HAM). Within the HAM framework, a potential singularity resulting from exact triad resonance is avoided by appropriately choosing the auxiliary linear operator. The resonant hydroacoustic wave component, along with the two primary waves (one hydroacoustic wave and one gravity wave), is considered to determine an initial guess for the velocity potential. Additionally, by selecting an optimal “convergence-control parameter,” the steady-state resonance between a surface gravity wave and two hydroacoustic waves is successfully obtained. It is found that steady-state resonant acoustic-gravity waves are ubiquitous under certain circumstances. The two primary wave components and the resonant hydroacoustic wave component take up most of the energy in the steady-state resonant acoustic-gravity wave system. The amplitude of the resonant hydroacoustic wave component is mainly determined by the primary hydroacoustic wave component, and the amplitudes of both hydroacoustic waves are approximately equal in all cases considered. In addition, when the overall amplitude of the wave system increases, both dimensionless angular frequencies decrease, indicating that the nonlinearity of the entire wave system becomes stronger with an increase in the wave system amplitude. The amplitude of the primary hydroacoustic wave has a relatively large effect on the system's nonlinearity. This work will enrich and deepen our understanding of acoustic-gravity waves.

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

A new optimal strategy for energy minimization in wireless sensor networks

In recent years, evolutionary and metaheuristic algorithms have emerged as crucial tools for optimization in the field of artificial intelligence. These algorithms have the potential to revolutionize various aspects of our lives by leveraging the multidisciplinary nature of wireless sensor networks (WSNs). This study aims to introduce genetic and simulated annealing algorithms as effective solutions for enhancing WSN performance. Our contribution entails two main phases. Firstly, we establish mathematical models and formulate objectives as a nonlinear constrained optimization problem. Secondly, we develop two algorithmic solutions to address the formulated optimization problem. The obtained results from multiple simulations demonstrate the positive impact of the proposed strategies on improving network performance in terms of energy consumption.

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.