一种求解非线性鞍点问题的交替投影方法

本文针对一类连续非线性Max-Min优化所对应的鞍点问题,提出了一种交替投影算法,证明了算法的收敛性.初步的数值实验表明本文所提出的算法比已有的同类算法具有更高的计算效率.     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

A COMPACT UPWIND SECOND ORDER SCHEME FOR THE EIKONAL EQUATION

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.     

VARIABLE MESH FINITE DIFFERENCE METHOD FOR SELF-ADJOINT SINGULARLY PERTURBED TWO-POINT BOUNDARY VALUE PROBLEMS

A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.     

CHEBYSHEV METHODS WITH DISCRETE NOISE： THE τ-ROCK METHODS

Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345（10）, 2007 and Commun. Math. Sci, 6（4）, 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.     

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.     

基于简单二次函数模型的带线搜索的信赖域算法

基于简单二次函数模型, 结合非精确大步长Armijo线搜索技术, 建立了一个新的求解无约束最优化问题的组合信赖域与线搜索算法, 证明了算法的全局收敛性. 数值例子表明算法是有效的, 适合求解大规模问题.       

在光学实验中开设计算全息实验的探讨

计算全息能很好地实现传统光学全息干涉衍射过程的数字化模拟,与传统实验相比,实验参数调整方便,现象直观。将计算全息引入到光学实验教学中,可以突破传统实验的诸多限制,拓宽学生的专业知识面,促进学生的综合实验技能培养。     

微纳光纤布拉格光栅折射率传感特性研究

利用光纤布拉格光栅方程和光纤基模有效折射率随纤芯半径和环境折射率的函数关系, 建立了微纳光纤布拉格光栅(MNFBG)反射波长随环境折射率变化的数学模型, 给出了波长灵敏度函数, 并指出MNFBG反射波长的变化规律决定于有效折射率随纤芯半径和环境折射率变化的关系. 详细探究了有效折射率及其灵敏度的变化规律, 结果表明: 有效折射率随纤芯半径和环境折射率的减小而非线性减小, 其对环境折射率变化的灵敏度随环境折射率的增大而非线性增加, 而且随纤芯半径减小, 有效折射率的灵敏度、线性度以及线性响应范围均呈递增规律. 通过对纤芯半径为0.5 μm的MNFBG在1.20–1.30和1.33–1.43 环境折射率范围内的波长响应关系拟合, 分别获得了477.33 nm/RIU和856.30 nm/RIU的波长灵敏度以及99.58 %和99.7%的高线性度, 论证了分析结论以及折射率区间划分测量方案的正确性, 为MNFBG折射率传感器的设计、优化以及应用提供了参考依据. 关键词： 微纳光纤 光纤布拉格光栅 折射率传感 数值模拟     

反射率的数值反演

通过分析热流边界条件下的一维热传导过程,提出并实现了由后表面温度变化历史曲线反演前表面反射率变化历史曲线的数值方法。通过一维热传导的正向和反演计算过程,验证了反演程序的可靠性。对数值反演计算和实验测量两种方式得到的反射率曲线进行了对比,结果显示,两条反射率曲线在变化趋势上符合较好,初步验证了一维温度场中数值反演方法的可行性。