Applications of Mathematics最新文献

This study introduces an accelerated gradient descent method based on a non-monotone backtracking line search scheme. A simple adaptive quadratic model is enhanced by utilizing a real, positive definite scalar matrix derived from the Taylor expansion of the objective function, rather than relying on the exact Hessian. The global and superlinear convergence of the defined model is established under appropriate conditions. Numerical experiments on a set of standard unconstrained optimization problems and image restoration problems show that the new algorithm outperforms other comparable methods in terms of efficiency and robustness.

This paper addresses the identification of the leak bound function g in the Stokes system with threshold leak boundary conditions, where g varies spatially. The state problem is solved using the dual formulation of the algebraic system, and the resulting optimization problem is formulated as a nonsmooth optimization problem. We establish the existence of solutions for both the continuous and discrete formulations of the problem. The theoretical developments are complemented by numerical experiments, which compare the performance of the nonsmooth optimization approach with traditional regularization-based methods and global optimization techniques.

We propose and analyze three kinds of two-grid penalty Arrow-Hurwicz (A-H) iterative finite element methods for the stationary incompressible magnetohydrodynamic (MHD) equations, which adopt the existing A-H iterative method to obtain the coarse mesh solution, and then correct the solution by three different one-step schemes (Oseen type, Stokes type and Newton type) with the usual penalty method on the fine mesh. These methods combine the A-H iterative method, the penalty method and the two-grid strategy, maintaining the advantage of three methods and overcoming some of their limitations. Rigorous analysis of the optimal error estimate and stability for three methods are provided. Ample numerical experiments are reported to validate the theoretical results and the efficiency of the numerical schemes.

In the linear regression model, the standard coefficient of determination R² and its weighted counterpart are commonly used to assess the quality of the linear fit. However, both metrics are susceptible to the influence of outliers and heteroskedasticity within the dataset. This paper introduces a robust version of R², based on the least weighted squares (LWS) estimator, and examines its statistical properties in detail. We investigate the impact of data quantization on R² and its robust variants, and propose a hypothesis test for assessing the equality of expected values between two R² versions. Numerical experiments on 29 publicly available datasets reveal that confidence intervals for the LWS-based coefficient of determination are generally narrower than those for existing measures, especially in homoskedastic settings. In contrast, under heteroskedasticity, narrower intervals do not necessarily imply greater robustness, highlighting the nuanced behavior of these estimators. The comparison with the well-known least trimmed squares (LTS) estimator underscores the promise of the LWS approach, which exhibits favorable efficiency properties and more reliable interval estimation in many practical scenarios.

We study the self-consistent chemotaxis-fluid system with nonlinear resource consumption

under no-flux boundary conditions in a bounded domain Ω ⊂ ℝ³ with smooth boundary. It is proved that this system possesses a global weak solution provided m > 1 and (alpha>{4 over 3}).

A fractional model is developed to study the transmission dynamics of tuberculosis disease. The use of a fractional model provides a memory effect and long-term dynamics often observed in chronic infectious diseases such as tuberculosis, which is characterized by a prolonged incubation period and risks of reactivation. The basic reproduction number is computed and we derive the qualitative stability analysis of equilibria. A sensitivity analysis is conducted to assess the impact of the model parameters. Three control strategies are applied, namely treatment, vaccination, and infection rate management, to minimize the number of infected individuals. Numerical simulations are carried out to illustrate the theoretical results obtained.

The tensor eigenvalue problem has been widely studied in recent years. In this paper, several new properties of eigenvalues and determinants of tensors are explored. We also proposed a formula to compute the determinant of a tensor as a mimic of the matrix determinant. The Perron-Frobenius theorem, one of the most important results in non-negative matrix theory, is proposed for the class of non-negative tensors in the Einstein product framework. Further, the power method, a widely used matrix iterative method for finding the largest eigenvalue, is framed for tensors using the Einstein product. The proposed higher-order power method is applied to calculate the largest eigenvalue of the Laplacian tensors associated with hyper-stars and hyper-trees. The numerical results show that the higher-order power method with the Einstein product is stable.

We propose a weighted HS (Hestenes-Stiefel)-FR (Fletcher-Reeves) hybrid conjugate gradient method for unconstrained multiobjective optimization problem, in which a new positive coefficient of the multiobjective steepest descent direction is adaptively updated to keep its positiveness. The method takes advantage of a weighted hybrid of our modified HS and FR parameters and under the Armijo-type backtracking line search, it has global convergence to a Pareto critical point (point satisfying the first-order necessary condition for Pareto optimality) without convexity assumption on the objectives. Numerical experiments show that the practical performance of the method is competitive with the existing methods such as conjugate gradient method, steepest descent method, Newton method, and quasi-Newton method for unconstrained multiobjective optimization.

We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions, the so-called one-troughed travelling waves. The main goal of this paper is to show the existence of at least two different one-troughed travelling waves for particular wave speeds and input parameters of the studied problem. We present the upper bounds for the maximal number of one-troughed solutions together with a visualisation of obtained results and corresponding solutions. Finally, we list several open questions regarding this topic.

We represent the point clouds of objects and audio signals as manifolds of Gaussian Mixture Models, and analyze the shape variation and compare the audio patterns using three divergence measures, namely the Kullback-Leibler Divergence, Jensen-Shannon Divergence, and Modified Symmetric Kullback-Leibler Divergence. Experiments are conducted on basic geometric shapes, 3D human body shapes, animal shapes, point clouds of the same object produced from the dense point clouds in the PU-GAN (Point Cloud Upsampling Adversarial Network) dataset. Then, we present a method to generate a point cloud of an audio signal using the Short-Time Fourier Transform. The audio-derived point clouds represent frequency, time, and magnitude relationships, enabling analysis of speech and audio patterns. The results across all datasets show that the Modified Symmetric Kullback-Leibler Divergence provides the most distinct and stable comparison between different point clouds, demonstrating its robustness for point cloud comparison.