Nonlinear Problem Research Articles

Designing employee benefits to optimize turnover: A prescriptive analytics approach

Employee turnover significantly impacts organizations, particularly those with substantial investments in training their workforce. To mitigate these effects, we propose a Prescriptive Human Resources Analytics approach that optimizes employee benefits to minimize total costs, focusing on turnover management The methodology models employee decision-making using a discrete choice model, with parameters estimated through maximum likelihood. We solve the resulting nonlinear optimization problem with a heuristic tailored to the problem’s complexity. We applied this methodology to a hospital case study, which was used to enhance the transportation system as an employee benefit, considering the associated turnover costs. The results demonstrate that our approach can reduce total costs, optimize the usage level of the designed benefits, and increase employee satisfaction. This research provides a robust framework for data-driven decision-making in HR, offering practical tools for improving employee retention strategies.

Blow up of solutions to nonlinear Neumann boundary value problem for Schrödinger equations

Blow up of solutions to nonlinear Neumann boundary value problem for Schrödinger equations

Branching evolution for unknown objective optimization in biclustering

Biclusters hold significant importance in microarray analysis. Given the EA algorithm’s efficacy in tackling nonlinear problems, it has become a prevalent choice for evolutionary biclustering in microarray analysis. However, in conventional approaches, the objective of bicluster volume remains elusive, as it heavily relies on the yet-to-be-discovered real bicluster. This discrepancy introduces a novel research problem termed “unknown objective optimization” in our study. To address this issue, our paper introduces an innovative branching evolution strategy within a multi-objective framework. This strategy aims to resolve the challenge of unknown objectives. Throughout the biclustering search process, we meticulously observe the evolution of optimal bicluster individuals. Stability in both mean squared residue (MSR) and volume suggests a high likelihood of reaching an optimal solution, whether local or global. If a global optimal solution is attained at the end of the final evolution, our initial assumption is validated; otherwise, it necessitates an update. The proposed branching strategy is subsequently implemented to bifurcate the original evolution into two branches. One continues the original evolution with an unknown objective of bicluster volume, while the other pursues a new evolution with an estimated objective of bicluster volume. Our algorithm’s performance is assessed through comparisons with various traditional and evolutionary biclustering algorithms. The experimental results affirm its enhanced efficacy on both synthetic datasets and real gene datasets.

Parameter-adaptive variational autoencoder for linear/nonlinear blind source separation

AbstractBlind source separation (BSS) serves as an important technique in the field of structural health monitoring (SHM), particularly for solving modal decomposition tasks. This study proposes a novel approach to both linear and nonlinear BSS problems in the Variational Autoencoder (VAE) framework, where the encoding and decoding processes of VAE are interpreted as procedures for inferring sources from observations and remixing these sources, respectively. In this way, the distribution of latent variables inferred by VAE is equivalent to the distribution of sources. We make improvements to the vanilla VAE to augment its effectiveness for BSS. First, we substitute standard normal distributions with trainable Gaussian processes (GP) as priors for latent variables and implement an exponential function as the activation function for adaptive parameters in the GP kernel functions. While the form of the priors is set as GP, the parameters of their kernel functions are not fixed but automatically converge to suitable values during the model training process. Additionally, a hyperparameter is introduced to balance the terms in the loss function. The proposed method is referred to as parameter-adaptive VAE (PAVAE). Then, upon different assumptions of the variances of sources, the proposed PAVAE is branched into two types: homoscedastic PAVAE (Ho-PAVAE) and heteroscedastic PAVAE (He-PAVAE). Through numerical and laboratory experiments, we demonstrate the effectiveness of the proposed method in solving BSS problems and their potential to underpin future research in SHM.

Fast Nonlinear Model Predictive Control Combining Online Trajectory Optimization and Value Function Regression

ABSTRACTModel predictive control (MPC) is a well‐developed method capable of handling complex control tasks. Implementation of an MPC requires solution of a deterministic finite horizon nonlinear optimal control (OC) problem. OC problems can be solved globally, explicitly expressing the optimal policy (and value function) as a function of the present state, or, locally, using online trajectory optimization, generating a solution that is only valid for the present state. For nonlinear problems however, neither is possible analytically, and the latter, finding a local solution online, is usually preferred. The difficulty with online trajectory optimization is that the solution must be available within a single sampling period. The main parameter affecting the computational demand of MPC is the length of the prediction horizon. The goal in this work is to reduce the length of the prediction horizon of a model predictive controller (MPC) to reduce computation time whilst preserving optimality guarantees. To this end, we propose approximation of the trajectory optimization problem for MPC by learning a finite horizon value function. The approximated value function is inserted into a truncated trajectory optimization problem so that the MPC can be attained with a reduced prediction horizon, ergo reduced computational load. By sampling the value function in a goal oriented way, we show that an effective approximate value function can be found by including both the function value and the gradients of the value function. The result is an accurate approximate MPC which leverages learning methodologies to reduce computational cost while still accounting for constraints.

Exploring nanoparticle dynamics in binary chemical reactions within magnetized porous media: a computational analysis

Artificial Neural Networks are incredibly efficient at handling complicated and nonlinear mathematical problems, making them very useful for tackling these challenges. Artificial neural networks offer a special computational architecture that is extremely valuable in disciplines like biotechnology, biological computing, and computational fluid dynamics. The present work investigates the applicability of back-propagation artificial neural networks in conjunction with the Levenberg-Marquardt algorithm for evaluating heat transmission in hybrid nanofluids. This work focuses on the computational analysis of a MgO + GO/EG hybrid nanofluid’s steady mixed convection flow over an exponentially stretched sheet, considering multiple slip boundary conditions, thermal conductivity, heat generation, and thermal radiation. A nonlinear system of ordinary differential equations is produced from the basic associated partial differential system by performing the proper exponential similarities modifications. For generating benchmark datasets, the resulting ordinary differential equations are processed employing the bvp4c method. Considering benchmark datasets set aside for training (70%), testing (15%), and validation (15%), the Levenberg-Marquardt algorithm, which employs back-propagation in artificial neural networks, is implemented. The accuracy of the suggested strategy for handling nonlinear problems is verified utilizing mean squared error, error histograms, and regression analysis, which are all used to evaluate the methodology. Outstanding agreement is seen when ANN outputs are compared to numerical results. The flow properties, including temperature, velocity, and concentration profiles, are shown graphically and numerically. For practical purposes, it is therefore essential to analyze the flow and heat transfer in hybrid nanofluids over exponentially extending and shrinking surfaces under mixed convection and heat source scenarios. Hybrid nanofluid problems have a wide range of practical and industrial applications, such as medication delivery, manufacturing, microelectronics, nuclear plant cooling, and marine engineering.

A Nonlinear Approach in the Quantification of Numerical Uncertainty by High-Order Methods for Compressible Turbulence with Shocks

This is a comprehensive overview on our research work to link interdisciplinary modeling and simulation techniques to improve the predictability and reliability simulations (PARs) of compressible turbulence with shock waves for general audiences who are not familiar with our nonlinear approach. This focused nonlinear approach is to integrate our “nonlinear dynamical approach” with our “newly developed high order entropy-conserving, momentum-conserving and kinetic energy-preserving methods” in the quantification of numerical uncertainty in highly nonlinear flow simulations. The central issue is that the solution space of discrete genuinely nonlinear systems is much larger than that of the corresponding genuinely nonlinear continuous systems, thus obtaining numerical solutions that might not be solutions of the continuous systems. Traditional uncertainty quantification (UQ) approaches in numerical simulations commonly employ linearized analysis that might not provide the true behavior of genuinely nonlinear physical fluid flows. Due to the rapid development of high-performance computing, the last two decades have been an era when computation is ahead of analysis and when very large-scale practical computations are increasingly used in poorly understood multiscale data-limited complex nonlinear physical problems and non-traditional fields. This is compounded by the fact that the numerical schemes used in production computational fluid dynamics (CFD) computer codes often do not take into consideration the genuinely nonlinear behavior of numerical methods for more realistic modeling and simulations. Often, the numerical methods used might have been developed for weakly nonlinear flow or different flow types other than the flow being investigated. In addition, some of these methods are not discretely physics-preserving (structure-preserving); this includes but is not limited to entropy-conserving, momentum-conserving and kinetic energy-preserving methods. Employing theories of nonlinear dynamics to guide the construction of more appropriate, stable and accurate numerical methods could help, e.g., (a) delineate solutions of the discretized counterparts but not solutions of the governing equations; (b) prevent numerical chaos or numerical “turbulence” leading to FALSE predication of transition to turbulence; (c) provide more reliable numerical simulations of nonlinear fluid dynamical systems, especially by direct numerical simulations (DNS), large eddy simulations (LES) and implicit large eddy simulations (ILES) simulations; and (d) prevent incorrect computed shock speeds for problems containing stiff nonlinear source terms, if present. For computation intensive turbulent flows, the desirable methods should also be efficient and exhibit scalable parallelism for current high-performance computing. Selected numerical examples to illustrate the genuinely nonlinear behavior of numerical methods and our integrated approach to improve PARs are included.

An optimized method for dose-effect prediction of traditional Chinese medicine based on 1D-ResCNN-PLS.

We introduce a one-dimensional (1D) residual convolutional neural network with Partial Least Squares (1D-ResCNN-PLS) to solve the covariance and nonlinearity problems in traditional Chinese medicine dose-effect relationship data. The model combines a 1D convolutional layer with a residual block to extract nonlinear features and employs PLS for prediction. Tested on the Ma Xing Shi Gan Decoction datasets, the model significantly outperformed conventional models, achieving high accuracies, sensitivities, specificities, and AUC values, with considerable reductions in mean square error. Our results confirm its effectiveness in nonlinear data processing and demonstrate potential for broader application across public datasets.

Particle Virtual Element Method (PVEM): an agglomeration technique for mesh optimization in explicit Lagrangian free-surface fluid modelling

Explicit solvers are commonly used for simulating fast dynamic and highly nonlinear engineering problems. However, these solvers are only conditionally stable, requiring very small time-step increments determined by the characteristic length of the smallest, and often most distorted, element in the mesh. In the Lagrangian description of fluid motion, the computational mesh quickly deteriorates. To circumvent this problem, the Particle Finite Element Method (PFEM) creates a new mesh (e.g., through a Delaunay tessellation, based on node positions) when the current one becomes overly distorted. A fast and efficient remeshing technique is therefore of pivotal importance for an effective PFEM implementation in explicit dynamics. Unfortunately, the 3D Delaunay tessellation does not guarantee well-shaped elements, often generating zero- or near-zero-volume elements (slivers), which drastically reduce the stable time-step size. Available mesh optimization techniques have limited applicability due to their high computational cost when runtime remeshing is required. An innovative possibility to overcome this problem is the use of the Virtual Element Method (VEM), a variant of the finite element method that can make use of polyhedral elements of arbitrary shapes and number of nodes. This paper presents the formulation of a 3D first-order Particle Virtual Element Method (PVEM) for weakly compressible flows. Starting from a tetrahedral mesh, poorly shaped elements, such as slivers, are agglomerated to form polyhedral Virtual Elements (VEs) with a controlled characteristic length. This approach ensures full control over the minimum time-step size in explicit dynamics simulations, maintaining stability throughout the entire analysis.

Differential evolution-based topology optimization using Karhunen-Loève expansion

Topology optimization is a widely adopted structural optimization approach that aims to maximize desired performance based on mathematical and physical principles. Gradient-based methods are common for obtaining optimal configurations; however, they struggle with problems involving numerous local optima and non-differentiable problems, which prevent the derivation of design sensitivity. To address these issues, this study explores the application of the Success History-Based Differential Evolution with Linear Population Size Reduction and Semi-Parameter Adaptation, which is regarded as LSHADE-SPA algorithm, known for its strong search capabilities. Nevertheless, its computational cost makes it impractical for topology optimization, especially as the number of design variables increases. To overcome this challenge, a novel structure representation method utilizing the Karhunen-Loève (KL) expansion is constructed. Applying truncated KL expansion instead of conventional density methods to represent structural configurations reduces the number of design variables. This article introduces a differential evolution-based topology optimization method utilizing truncated KL expansion and validates its effectiveness through applications in various structural and fluid problems. The method is particularly suited to highly nonlinear problems, such as negative Poisson's ratio structures and micro-mixer designs. The findings here indicate that differential evolution with truncated KL expansion can serve as an efficient and robust topology optimization method, proving particularly effective for solving optimization problems that are challenging for conventional gradient-based approaches. Notably, the proposed methodology not only delivers superior results but also alleviates computational burdens, thereby advancing the prospects of topology optimization in various engineering applications.

Analysis and evaluation of suitability of high-pressure dynamic constitutive model for concrete under blast and impact loading

Concrete demonstrates complex dynamic mechanical behaviors under the influence of blast or impact loads, and there are inherent limitations in experimental and theoretical methods when dealing with highly nonlinear problems. As computational technologies and mechanics continue to evolve, it is possible to conduct high-fidelity simulations of the transient response of concrete structures subjected to intense dynamic loads. Such simulations play a crucial role in revealing the propagation laws of stress waves, the progression of damage, the mechanisms of structural failure, and in conducting protective engineering design. An accurate concrete material model is essential for conducting numerical studies. This article reviews the development of high-pressure dynamic constitutive models for concrete in recent years from both experimental research and theoretical modeling perspectives, focusing on the analysis and evaluation of the modeling methods and main shortcomings of the equation of state(EOS), strength model, and damage model. Using single-element numerical simulations under a single loading path and numerical calculations of engineering cases under complex loading paths, a systematic analysis and comparison were conducted on the predictive capabilities of the HJC, RHT, KCC, and CSC models, as well as the newly developed Kong-Fang and Yan-Chen models. It pointed out the impact of high-pressure mechanical behavior of concrete and the cumulative effect of damage under hydrostatic pressure on the calculation results. Finally, a discussion was conducted on the inherent flaws, applicability, and research difficulties of local constitutive models of concrete in the finite element method. This provides a reference for the selection and research of constitutive models when conducting numerical analysis of the blast and impact resistance of concrete structures.

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W2), which originates from the optimal transportation (OT) theory. The computation of the W2 distance on the manifold surface is boiled down to solving the generalized Monge-Ampère equation, whose solution is directly related to the gradient of the W2 distance. An efficient first-order method based on iteratively solving Poisson's equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W2 distance is developed to solve the three-dimensional electrical impedance tomography (EIT) problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in \mathbb{R}^{2} , describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, parameterized as the graph of a function f=f(t) , is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function f . Based on abstract parabolic theory, it is shown that the problem is well-posed in all subcritical spaces \mathrm{H}^{r}(\mathbb{S}) , with r\in(3/2,2) . Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

Existence and regularity of strict solutions for a class of fractional evolution equations

AbstractWe study the existence and Hölder regularity of solutions for fractional evolution equations of order . By means of an analytic resolvent, we construct an interpolation space, which can effectively lower the regularity of initial data. By virtue of the interpolation space and some properties of the analytic resolvent, we derive the existence and Hölder regularity of strict solutions for an inhomogeneous problem, as well as the existence and Hölder regularity of a nonlinear problem.

Resolution of a nonlinear elasticity problem subject to friction laws

Resolution of a nonlinear elasticity problem subject to friction laws

Adjoint-Based Calibration of Nonlinear Stochastic Differential Equations

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte–Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein–Uhlenbeck and a stochastic Prandtl–Tomlinson bath model are considered.

Method for prediction magnetic silencing of uncertain energy-saturated extended technical objects in prolate spheroidal coordinate system

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. References 48, figures 2.

IRROTATIONAL FLOW DUE TO FORCED OSCILLATIONS OF A BUBBLE

Abstract The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

An adaptive snow ablation-inspired particle swarm optimization with its application in geometric optimization

In response to the shortcomings of particle swarm optimization (PSO), such as low execution efficiency and difficulty in overcoming local optima, this paper proposes a multi-strategy PSO method incorporating snow ablation operation (SAO), known as SAO-MPSO. Firstly, Cubic initialization is performed on particles to obtain a good initial environment. Subsequently, SAO and PSO are combined in parallel, and a balanced search mechanism led by multiple sub-populations is devised, significantly improving the search efficiency of overall population. Finally, the degree day method of SAO is introduced, and particles are endowed with memory of environmental changes to prevent premature convergence of PSO, while balancing the exploration and exploitation (ENE) capabilities in later phases. All adaptive parameters are used throughout this method in place of fixed parameters to improve the robustness and adaptability. For a comprehensive analysis of SAO-MPSO, its good ENE ability is verified on CEC 2020 and CEC 2022 and this method is compared with existing improved PSO versions on both test sets. The results show that SAO-MPSO has certain advantages in the comparison of similar improved algorithms. In order to further validate the strength of SAO-MPSO in dealing with nonlinear optimization problems (OPs) with strong constraints, firstly, based on the ball Wang-Ball (BWB) curve, a combined BWB (CBWB) curve is constructed, and a construction method for CBWB curves that satisfy G1 and G2 continuity is derived. Then, with the energy minimization and scale parameters of the CBWB curve as the optimization objective and variables respectively, a shape optimization model that satisfies G2 continuity is established. Finally, three numerical optimization examples based on this model are solved using SAO-MPSO and compared with 10 other methods. The results show that the energy obtained by SAO-MPSO is the smallest, which verifies the effectiveness of this method applied to shape OPs of CBWB curve.

On the Numerical Solution of Linear and Non-Linear Dirichlet Boundary Value Problems of Ordinary Differential Equations: The Shooting Method

In this article, a numerical technique called shooting which entails the solution of initial value problems with the single-step fourth-order RungeKutta method together with the iterative root finding secant method is formulated for use on both linear and non-linear boundary value problems of ordinary differential equations with Dirichlet boundary conditions. Two examples are illustrated. One, the solution of the linear case with its analytic counterpart is compared, and two, the non-linear case. Graphical outputs of the solutions from two MATHEMATICA codes are presented.