m-degree Center-connecting Line of Finite Points Set and Its Properties

In this paper, we first give the concept of m-degree center-connecting line in n-dimensional Euclidean space Enand investigate its several properties, then we obtain the length of m-degree center-connecting line formula in finite points set. As its application,we extend the Leibniz formula and length of medians formula in n-dimensional simplex to polytope.

IMAGE SUPER-RESOLUTION RECONSTRUCTION BY HUBER REGULARIZATION AND TAILORED FINITE POINT METHOD

In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.

A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM

In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.

Finite Point Method for the Simulation of Solidification and Heat Transfer in Continuous Casting Mold

A 2-D finite point meshless model was used to simulate the heat transfer and solidification of steel in continuous casting molds to illustrate its use in metallurgy. The latent heat of the pure metal was treated using the temperature recovery method and the latent heat of the alloy was treated using an appar- ent heat capacity method. The model was validated by calculating the classical Stefan moving boundary problem. Analysis of the solid shell growth and temperature distribution of a billet in a mold shows that the solution by the finite point meshless model is quite reasonable, which indicates that the model has potential in metallurgical engineering applications.

DISCRETE MAXIMUM PRINCIPLE AND CONVERGENCE OF POISSON PROBLEM FOR THE FINITE POINT METHOD

This paper makes some mathematical analyses for the finite point method based on directional difference. By virtue of the explicit expressions of numerical formulae using only five neighboring points for computing first-order and second-order directional differ- entials, a new methodology is presented to discretize the Laplacian operator defined on 2D scattered point distributions. Some sufficient conditions with very weak limitations are obtained, under which the resulted schemes are positive schemes. As a consequence, the discrete maximum principle is proved, and the first order convergent result of O（h） is achieved for the nodal solutions defined on scattered point distributions, which can be raised up to O（h2） on uniform point distributions.

THEORETICAL ANALYSES ON DISCRETE FORMULAE OF DIRECTIONAL DIFFERENTIALS IN THE FINITE POINT METHOD

For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1)a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2)various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neighboring points and making analysis;(3)the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of singularity.Finally,the theoretical analysis results are verified by numerical calculations.The results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.

A PARAMETER-UNIFORM TAILORED FINITE POINT METHOD FOR SINGULARLY PERTURBED LINEAR ODE SYSTEMS＊

In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordi- nary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.

A Method for Exact Calculation of the Discrepancy of Low-Dimensional Finite Point Sets(II)

In the present paper a general formula for exact calculation of the discrepancy of an arbitrary finite point set of dimension d≥2 is explicitly given only in terms of the components of the points.

A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers

In this paper we propose a development of the finite difference method,called the tailored finite point method,for solving steady magnetohydrodynamic(MHD)duct flow problems with a high Hartmann number.When the Hartmann number is large,the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer.Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy.However,the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh.Firstly,we devise the tailored finite point method for the scalar inhomogeneous convectiondiffusion problem,and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations.For each interior grid point of a given rectangular mesh,we construct a finite-point difference operator at that point with some nearby grid points,where the coefficients of the difference operator are tailored to some particular properties of the problem.Numerical examples are provided to show the high performance of the proposed method.

Tailored Finite Point Method for Numerical Solutions of Singular Perturbed Eigenvalue Problems

We propose two variants of tailored finite point(TFP)methods for discretizing two dimensional singular perturbed eigenvalue(SPE)problems.A continuation method and an iterative method are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE.We study the analytical solutions of two special cases of the SPE,and provide an asymptotic analysis for the solutions.The theoretical results are verified in the numerical experiments.The numerical results demonstrate that the proposed schemes effectively resolve the delta function like of the eigenfunctions on relatively coarse grid.

Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers

This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.

Extremum of second-order directional derivatives

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.

The Limit of Finite Sample Breakdown Point of Tukey’s Halfspace Median for General Data

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution （weak smoothness） and on data set （not necessary in general position）. The refined representation of Tukey＇s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.

Relations between Directional Derivatives for Smooth Functions in 2D and 3D Spaces

In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.