Journal of Computational Mathematics and Data Science最新文献

In this paper, we consider the localized method of approximate particular solutions (LMAPS) for solving a two-dimensional distributive optimal control problem governed by elliptic partial differential equations. Both radial basis functions and polynomial basis functions (RBFs) are used in the LMAPS discretization, while the leave-one-out cross-validation is adopted for the selection of the shape parameter appeared in RBFs. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

Ozone is an active gas in the atmosphere. Its content is quite low, but it plays an important role in protecting the health of human beings and other living things on earth. Ozone circulates in the atmosphere, and its total distribution and variation trend are related to geographical position. In this paper, we collected global Ozone tendency data and investigated the changes in global distribution of Ozone in 2018 using $k$-means clustering algorithm. We observed that (1) the global Ozone tendency can be broadly divided into four regions; (2) the data with a large variation range of total Ozone tendency is mainly concentrated near the sea–land boundary, and their distribution is similar to the coastline contour to some extent; (3) after clustering, the concentration area of the data with great changes in the total Ozone tendency is roughly x-shaped distribution, and the acute angle between the data and the latitude line is between $25°$ and $45°$. Our findings can contribute to a clearer understanding and analysis of the tendency of global Ozone change and help mitigate the Ozone hole problem in different regions.

A novel algorithm to compute the convex hull of any given hyperdimensional data set is presented. This algorithm has lower memory requirements than state of the art software, and runtimes which are typically much faster than conventional programs and algorithms which do the same. A discussion is presented which examines the large importance that convex hull computations serve in creating general surrogate models from data sets, and their importance to machine learning algorithms. In addition to the deep reaching applications in many fields, this algorithm can be used to help solve design problems, specifically those in preliminary design when surrogate models are used to perform rapid design trades. The algorithm is presented, in addition to algorithms which compute volumes and facilitate understanding of hyperdimensional spaces which cannot be easily visualized. This paper concludes with the presentation of a representative design problem containing similar dimensionality and numbers of points as a standard engineering preliminary design problem. The minimum number of points needed for the interpolation of a general surrogate model during design and analysis is then discussed, including the proposal of a new metric.

The recent investigations ensure that, the effect of an endoscope on the peristaltic flow is very important for medical diagnosis and it has many clinical applications such as gastric juice motion in the small intestine when an endoscope is inserted through it. In the current article, the influence of magnetohydrodynamic (MHD) on the peristaltic propulsion of non-Newtonian fluid (considered as couple stress fluid) in a tube consisting of endoscope has been considered. The couple stress fluid occupies the space between two co-axial inclined tubes. The inner tube is uniformly circular and rigid while the outer tube considered as sinusoidal wave. The fluid motion is discussed in a wave frame which is moving with the constant velocity. The governing equations of two-dimensional flow have been abridged under the lubrication approach. Analytical solutions have been obtained for the velocity and pressure gradient with the help of modified Bessel functions. Numerical integration is used to evaluate the pressure difference and friction forces. The effect of emerging flow parameters on the velocity, frictional forces, pressure difference, pressure gradient and trapping phenomenon have been discussed. It is noted that, the magnetic force resists the flow and pumping rate in the peristaltic flow enhances from the horizontal to vertical tube. The present study has a wide range of applications in bio-medical engineering like the transport phenomenon in peristaltic micro pumps.

Strongly interacting electrons in solids are generically described by Hubbard-type models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effort.

Recently some neural networks have been proposed for computing approximate solutions to partial differential equations. For second order elliptic or parabolic PDEs, this is possible by using penalized Least squares formulations of PDEs. In this article, for some second order elliptic PDEs we propose a theoretical setting, and we investigate the abstract convergence results between the solution and the computed one with neural networks. These results are obtained by minimizing appropriate loss functions made of a least squares formulation of the PDE augmented with a penalization term for accounting the Dirichlet boundary conditions. More precisely, it is shown that the error has two components, one due to the neural network and one due to the way the boundary conditions are imposed (via a penalization technic). The interplay between the two errors shows that the accuracy of the neural network has to be chosen accordingly with the accuracy of the boundary conditions.