Minimization Of Function Research Articles

Dynamics and optimal control for a spatial heterogeneity model describing respiratory infectious diseases affected by air pollution

Air pollution has a serious impact on the spread of respiratory infectious diseases. However, the concentration of air pollution is not the same in different places. In order to characterize the spatial heterogeneity and the diffusion of pollutants, this paper proposes a spatial heterogeneity model, where the transmission rate is closely related to air pollution. The dynamics are established in terms of the basic reproduction number R0: if R0≤1, the disease-free equilibrium is globally asymptotically stable; if R0>1, there exists at least one endemic equilibrium and the disease is uniformly persistent. Our findings suggest that spatial heterogeneity, increased pollutants emission, and limited movement of infected individuals will all contribute to the spread of the respiratory infectious diseases. Subsequently, the corresponding strategies are introduced into the model to control the prevalence of the diseases. Under the minimization of the objective function, the necessary condition for the optimal control is acquired with the help of Pontryagin’s maximum principle. Numerical simulations are applied to validate the theoretical results of threshold dynamics and to demonstrate the influences of inflow and clearance rates of pollutants on threshold dynamics and optimal control.

An extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

An extrapolation approach within the Wiener path integral (WPI) technique is developed for determining the stochastic response of diverse nonlinear dynamical systems. Specifically, the WPI technique treats the system response joint transition probability density function (PDF) as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Further, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited herein. Specifically, it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently the PDF values at neighboring points without the need for considering additional BVPs. Notably, the herein developed approach enhances significantly the computational efficiency of the WPI technique without, practically, affecting the associated degree of accuracy. Two numerical examples relating to a Duffing nonlinear oscillator subjected to combined stochastic and deterministic periodic loading, and to an oscillator with asymmetric nonlinearities and fractional derivative elements are considered to demonstrate the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

Satellite image recognition using ensemble neural networks and difference gradient positive-negative momentum

The modern machine learning theory finds application in many areas of human activity. One of the most dispersed tasks is pattern recognition on satellite images. It is difficult for a person to recognize a large number of images in a short time. It made the researchers develop the automation process, such as neural network engagement. The loss function minimization and ensemble learning raise the pattern recognition accuracy. We propose the robust difference gradient positive-negative momentum optimization algorithm that achieves the global minimum of the loss function with higher accuracy and fewer iterations than known analogs. Such an optimization algorithm contains the generalized average moving estimation approach and more effective learning rate control by additional parameters. The proposed optimizer has the regret-bound rate estimation, belonging to OT, and converges to the global minimum. However, the main problems in optimization theory are vanishing and blowing gradient values, where the standard gradient-based algorithms fail to achieve the required objective function value. The vanishing and blowing gradient problems meet in Rastrigin and Rosebrock test functions, where the proposed optimization algorithm attains the global extreme in the shortest number of iterations and has a more stable convergence process than state-of-the-art methods. Afterward, there are trained deep convolutional neural networks with different optimizers on satellite images from the University of California merced dataset containing 21 object classes, where the proposed algorithm gives the highest accuracy. There is a suggested ensemble-learning model consisting of 4 networks with different optimizers. The prediction results receive weight coefficients distributed according to the majority voting and ensemble neural network retrains with the higher pattern recognition accuracy. The suggested ensemble-learning model with the developed optimizer raised the accuracy by 1 %–4 % percentage points.

Influence of the radius of Monson’s sphere and excursive occlusal contacts on masticatory function of dentate subjects

ObjectiveTo determine the influence of the radius of Monson’s sphere, the number of posterior laterotrusive, mediotrusive, and protrusive contacts, and the chewing rate on food comminution. DesignSixty healthy dentate subjects, aged 21.22 ± 2.30 years, were selected. The three-dimensional coordinates of the cusp tips of the lower canine, premolar, and molar teeth were identified from the subjects’ digital models. Monson’s sphere was designed using the simplex method for function minimisation by adjusting the coordinates on its surface. The contacts were verified using 12 µm metal strips in jaw excursions at 0.5, 1.0, 2.0, and 3.0 mm. The masticatory performance and efficiency, swallowing threshold, and chewing rate were assessed through particle size fractionation. Data were analysed with multiple linear regression (α = 0.05). ResultsThe sphere’s radius, laterotrusive and protrusive contacts at 0.5 mm, and chewing rate were found to be negative predictor variables for masticatory performance until 20 chewing cycles (R2 = 0.429). For 40 cycles, the radius and total contacts (0.5 mm) were also explanatory factors (R2 = 0.223). Only the radius (R2 = 0.176) and the chewing rate (R2 = 0.082) were found to be significant for 60 cycles and swallowing threshold, respectively. Masticatory efficiency was influenced by masticatory performance until 40 and 60 cycles, as well as the radius and total contacts at 2.0 and 3.0 mm (R2 = 0.958). ConclusionA larger radius of Monson’s sphere and a greater number of posterior excursive contacts were found to be related to better masticatory function.

An inverse source problem for a pseudoparabolic equation with memory

<abstract><p>This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine <italic>lsqnonlin</italic>. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.</p></abstract>

A game approach to multi-dimensional opinion dynamics in social networks with stubborn strategist agents

In a social network, individuals express their opinions on several interdependent topics, and therefore the evolution of their opinions on these topics is also mutually dependent. In this work, we propose a differential game model for the multi-dimensional opinion formation of a social network whose population of agents interacts according to a communication graph. Each individual’s opinion evolves according to an aggregation of disagreements between the agent’s opinions and its graph neighbors on multiple interdependent topics exposed to an unknown extraneous disturbance. For a social network with strategist agents, the opinions evolve over time with respect to the minimization of a quadratic cost function that solely represents each individual’s motives against the disturbance. We find the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure. Moreover, we propose a distributed implementation of the Nash/worst-case equilibrium solution. We examine the non-distributed and proposed distributed open-loop Nash/worst-case strategies on a small social network with strategist agents in a two-dimensional opinion space. Then we compare the evolved opinions based on the Nash/worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

Computer-Aided Intra-Operatory Positioning of an MRgHIFU Applicator Dedicated to Abdominal Thermal Therapy Using Particle Swarm Optimization.

Transducer positioning for liver ablation by magnetic resonance-guided high-intensity focused ultrasound (MRgHIFU) is challenging due to the presence of air-filled organs or bones on the beam path. This paper presents a software tool developed to optimize the positioning of a HIFU transducer dedicated to abdominal thermal therapy, to maximize the treatment's efficiency while minimizing the near-field risk. A software tool was developed to determine the theoretical optimal position (TOP) of the transducer based on the minimization of a cost function using the particle swarm optimization (PSO). After an initialization phase and a manual segmentation of the abdomen of 5 pigs, the program randomly generates particles with 2 degrees of freedom and iteratively minimizes the cost function of the particles considering 3 parameters weighted according to their criticality. New particles are generated around the best position obtained at the previous step and the process is repeated until the optimal position of the transducer is reached. MR imaging data from in vivo HIFU ablation in pig livers was used for ground truth comparison between the TOP and the experimental position (EP). As compared to the manual EP, the rotation difference with the TOP was on average -3.1 ± 7.1° and the distance difference was on average -7.1 ± 5.4 mm. The computational time to suggest the TOP was 20s. The software tool is modulable and demonstrated consistency and robustness when repeating the calculation and changing the initial position of the transducer.

Deterministic Gradient-Descent Learning of Linear Regressions: Adaptive Algorithms, Convergence Analysis and Noise Compensation.

Weight learning forms a basis for the machine learning and numerous algorithms have been adopted up to date. Most of the algorithms were either developed in the stochastic framework or aimed at minimization of loss or regret functions. Asymptotic convergence of weight learning, vital for good output prediction, was seldom guaranteed for online applications. Since linear regression is the most fundamental component in machine learning, we focus on this model in this paper. Aiming at online applications, a deterministic analysis method is developed based on LaSalle's invariance principle. Convergence conditions are derived for both the first-order and the second-order learning algorithms, without resorting to any stochastic argument. Moreover, the deterministic approach makes it easy to analyze the noise influence. Specifically, adaptive hyperparameters are derived in this framework and their tuning rules disclosed for the compensation of measurement noise. Comparison with four most popular algorithms validates that this approach has a higher learning capability and is quite promising in enhancing the weight learning performance.

Making Likelihood Calculations Fast: Automatic Differentiation Applied to RooFit

With the growing datasets of current and next-generation HighEnergy and Nuclear Physics (HEP/NP) experiments, statistical analysis has become more computationally demanding. These increasing demands elicit improvements and modernizations in existing statistical analysis software. One way to address these issues is to improve parameter estimation performance and numeric stability using Automatic Differentiation (AD). AD’s computational efficiency and accuracy are superior to the preexisting numerical differentiation techniques, and it offers significant performance gains when calculating the derivatives of functions with a large number of inputs, making it particularly appealing for statistical models with many parameters. For such models, many HEP/NP experiments use RooFit, a toolkit for statistical modeling and fitting that is part of ROOT. In this paper, we report on the effort to support the AD of RooFit likelihood functions. Our approach is to extend RooFit with a tool that generates overheadfree C++ code for a full likelihood function built from RooFit functional models. Gradients are then generated using Clad, a compiler-based source-codetransformation AD tool, using this C++ code. We present our results from applying AD to the entire minimization pipeline and profile likelihood calculations of several RooFit and HistFactory models at the LHC-experiment scale. We show significant reductions in calculation time and memory usage for the minimization of such likelihood functions. We also elaborate on this approach’s current limitations and explain our plans for the future.

A Customized Augmented Lagrangian Method for Block-Structured Integer Programming.

Integer programming with block structures has received considerable attention recently and is widely used in many practical applications such as train timetabling and vehicle routing problems. It is known to be NP-hard due to the presence of integer variables. We define a novel augmented Lagrangian function by directly penalizing the inequality constraints and establish the strong duality between the primal problem and the augmented Lagrangian dual problem. Then, a customized augmented Lagrangian method is proposed to address the block-structures. In particular, the minimization of the augmented Lagrangian function is decomposed into multiple subproblems by decoupling the linking constraints and these subproblems can be efficiently solved using the block coordinate descent method. We also establish the convergence property of the proposed method. To make the algorithm more practical, we further introduce several refinement techniques to identify high-quality feasible solutions. Numerical experiments on a few interesting scenarios show that our proposed algorithm often achieves a satisfactory solution and is quite effective.

An Algorithm for Global Minimization of Non-Convex Functions

An Algorithm for Global Minimization of Non-Convex Functions

Optimal velocity planning based on the solution of the Euler-Lagrange equations with a neural network based velocity regression

Trajectory optimization is a complex process that includes an infinite number of possibilities and combinations. This work focuses on a particular aspect of the trajectory optimization, related to the optimization of a velocity along a predefined path, with the aim of minimizing power consumption. To tackle the problem, a functional formulation and minimization strategy is developed, by means of the Euler-Lagrange equation. The minimization is later performed using a neural network approach. The strategy is deemed Lagrange-Net, as it is based on the minimization of the energy functional, by the means of Lagrange's equation and neural network approximations.

$\theta$-metric function in the problem of minimization of functionals

Изучаются аппроксимативные свойства множеств в зависимости от скорости изменения функции расстояния, где вместо метрики используется некоторый непрерывный функционал. В качестве примера приложения соответствующих утверждений мы доказываем неединственность приближения при помощи непрерывных функционалов специального вида в гильбертовых пространствах для невыпуклых множеств. Такого рода утверждения позволяют доказывать неединственность решений задач для уравнений градиентного типа. Библиография: 20 наименований.

$\theta$-metric function in the problem of minimization of functionals

We study approximative properties of sets as a function of the rate of variation of the distance function defined in terms of some continuous functional (in lieu of a metric). As an application, we prove non-uniqueness of approximation by non-convex subsets of Hilbert spaces with respect to special continuous functionals. Results of this kind are capable of proving non-uniqueness solvability for gradient-type equations.

Discrete Centralized AGC Using LQR-Based Cost Functional Minimization for Multi-Area Interconnected Power Systems

Discrete Centralized AGC Using LQR-Based Cost Functional Minimization for Multi-Area Interconnected Power Systems

The Role of Mathematics in Artificial Intelligence and Machine Learning

Mathematics serves as the foundational backbone of “artificial intelligence (AI) and machine learning (ML), providing the essential” tools and frameworks for developing sophisticated algorithms and models. the pivotal role of various mathematical disciplines, including linear algebra, calculus, probability theory, and optimization, in advancing AI and ML technologies. We begin by examining how linear algebra facilitates the manipulation and transformation of high-dimensional data, which is crucial for “techniques such as principal component analysis (PCA) and singular value decomposition (SVD)”. Next, we delve into the applications of calculus in training neural networks through gradient-based optimization methods, highlighting the importance of differentiation and integration in backpropagation and loss function minimization. the role of probability theory in handling uncertainty and making predictions, emphasizing its application in Bayesian networks, Markov decision processes, and probabilistic graphical models. Additionally, we discuss optimization techniques, both convex and non-convex, that are fundamental to finding optimal solutions in machine learning tasks, including support vector machines (SVMs) and deep learning architectures.

Optimal Performance Trim Procedure for Multirotor Vertical Takeoff and Landing Vehicles

The present paper proposes a general approach capable of determining the set of trim commands, aerodynamic control surface actuation, and/or engine thrust regulation that allow specific trimmed steady-state flight conditions of nonconventional vertical takeoff and landing vehicles under given performance constraints. Exploiting the peculiarity of these vehicles to have redundant controls, namely more controls than those strictly required to define steady-state flight conditions, their trim problem is recast into a constrained minimization of a cost function. This approach is of particular interest in that it allows the identification of control settings for multirotor vehicles that, at the same time, guarantee trimmed flight conditions and optimization of selected target functions such as, for instance, performance or noise emission. The proposed approach is numerically applied to quadcopter and hexacopter configurations for three flight conditions (level forward flight, climb, and coordinate turn), and three strategies of trim by different trim variables (rotor angular velocity, blade collective pitch, and combination of angular velocity and blade collective pitch). The effect of the introduced cost function is investigated, comparing the outcomes of minimum control effort and minimum torque strategies. The results achieved demonstrate the capability of the proposed algorithm to exploit redundant controls of multirotor vehicles for identifying selected optimal trim operative conditions.

A variational-difference method for numerical simulation of equilibrium capillary surfaces

Objectives. A variational-difference method for numerical simulation of equilibrium capillary surfaces based on the minimization of the energy functional is proposed. As a test task a well-known axisymmetric hydrostatic problem on equilibrium shapes of a drop adjacent to a horizontal rotating plane under gravity is considered. The mathematical model of the problem is built on the basis of the variational principle: the shape of the drop satisfies the minimum total energy for a given volume. The problem of the functional minimization is reduced to a system of nonlinear equations using the finite element method. To solve the system a Newton's iterative method is applied.Methods. The variational-difference approach (the finite element method) is used. The finite linear functions are chosen as basic functions.Results. Equilibrium shapes of a drop on a rotating plane are constructed by the finite element method in a wide range of defining parameters: Bond number, rotational Weber number and wetting angle. The influence of these parameters on the shape of a drop is investigated. The numerical results are matched with the results obtained using the iterative-difference approach over the entire range of physical stability with respect to axisymmetric perturbations.Conclusion. The finite element method responds to the loss of stability of a drop with respect to axisymmetric perturbations. Therefore it can be used to study the stability of the equilibrium of axisymmetric capillary surfaces.

A Low-Complexity Double Vector Model Predictive Current Control for Permanent Magnet Synchronous Motors

Compared to the conventional finite control set model predictive control (FCS-MPC), the double vector model predictive current control (DVMPCC) for permanent magnet synchronous motors (PMSMs) has a better steady-state performance without significantly increasing the switching frequency. However, determining optimal vectors with their dwell times requires a high computational burden. A low-complexity DVMPCC in the steady state was proposed in this study to address this problem. Firstly, the operating state of the motor was judged according to the speed error. During steady-state operation, the first optimal active vector was selected from three candidate vectors adjacent or identical to the active vector applied in the previous control period, reducing the number of comparisons by half. Next, the second optimal vector was selected from the other two active vectors, and the zero vector, the second optimal vector with the duty cycle, was determined according to the deadbeat condition of the q-axis current and cost function minimization. Finally, simulation and experimental results proved that the proposed low-complexity DVMPCC for surface-mounted permanent magnet synchronous motors is practical and feasible.

Enhanced FS-PTC: Dynamic Weighting Factors for Optimal Flux and Torque Control

Recently, Finite State-Predictive Torque Control (FS-PTC) of induction motors has gained significant interest in high-performance motor drive applications. The effectiveness of FS-PTC relies on the successful minimization of a cost function achieved by selecting an appropriate voltage vector. Typically, the cost function for FS-PTC is composed of errors between the predicted and reference values of torque and flux; hence a weighting factor, \mathbit{\lambda}, is normally employed to establish different priorities between torque and flux. However, determining the optimal is a complex undertaking, since an incorrect or suboptimal choice can needlessly compromise torque or flux responses. This paper introduces an online tuning approach for the weighting factor, based on the dynamic change of flux error. Instead of using a fixed value, the weighting factor is dynamically adjusted using a simple P or PI controller. The proposed method's performance is evaluated in this paper, considering various configurations of the controller's settings. Simulation results demonstrate that the proposed technique enhances the overall torque and flux ripples across a broad range of operating speeds, surpassing the performance of the fixed value weighting factor technique.