Further study of alternating iterations for rectangular matrices

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.     

基于改进NAS-RIF的三维显微图像盲复原

对二维NAS-RIF算法进行了改进,并将它用于三维显微图像复原,分别用模拟样本和真实生物样本进行了实验。针对原算法对噪声有放大作用,提出了基于中值滤波去噪的改进。在支持域的确定、最小点的搜索上进行改进,提高复原质量。改进了原算法仅适用于均匀背景的不足,扩大了应用范围。实验结果表明,该算法在三维显微图像盲复原中具有较好的效果。     

On the equivalence between strong solvability and strict semimonotonicity for some systems involving Z-functions

This paper focuses on the relationship between the strong solvability of a certain system involving a Z-function, and the strict semimonotonicity of such a function. Our main result shows that, for a system defined by a continuous, superhomogeneous Z-function, the additional condition of strict semimonotonicity is both necessary and sufficient for strong solvability.     

Nonnegativity of solutions of nonlinear fractional differential-algebraic equations

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.     

Numerical methods for fourth order nonlinear degenerate diffusion problems

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.     

Hilbert空间线性二次最优控制问题中的一个算子的可逆性

对于如下出现在Hilbert空间线性二次最优控制问题中的线性算子■其中F_3,F_5是自伴算子．本文得到了它具有有界逆的充分必要条件,并举例验证了结果的有效性．     

Nonnegative surface fitting with Powell-Sabin splines

Algorithms are presented for fitting a nonnegative Powell-Sabin spline to a set of scattered data. Existing necessary and sufficient nonnegativity conditions for a quadratic polynomial on a triangle are used to compose a set of necessary and sufficient nonnegativity constraints for the PS-spline. The PS-spline is expressed as a linear combination of locally supported basis functions, of which the Bernstein-Bézier representation is considered to improve the efficiency. Numerical examples illustrate the profit of nonnegative surface fitting with Powell-Sabin splines.     

On local weak solutions to Nernst–Planck–Poisson system

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well as electrons and holes transport in semiconductors. The considered boundary conditions allow the physical system to be not only closed but also open. Theorems on existence, uniqueness, and nonnegativity of local weak solutions are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.     

On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions

We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting. By use of the entropy estimate, strong convergence results for discrete solutions are obtained. In particular, the limit of discrete fluxes will be identified with the flux in the continuous setting. As a by-product, first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting.