《高等数值分析》教学案例的建设与思考

《高等数值分析》是一门与实际联系紧密的数学公共课程,它理论深刻,应用广泛.本文结合实际应用,为《高等数值分析》中常微分方程数值解部分设计了一个教学案例,通过理论分析和数值实验向学生展示了刚性问题的概念和相关数值方法,并对《高等数值分析》课程教学案例的设计进行了思考.     

随机椭圆最优控制问题的随机配置方法

本文讨论求解随机系数泊松方程约束最优控制问题的有效数值方法.通过应用有限元方法和随机配置法,将原最优控制问题离散转化为最优化问题,再利用交替方向乘子法求解最优化问题.之后,对所提出的算法进行了收敛性分析,并通过数值实验验证了算法的有效性.     

油、水两相渗流驱动问题的离散算子法及其数值分析

本文应用离散算子法解决油、水两相渗流驱动问题的数值模拟并进行了理论分析.最后我们根据油田的实际地质参数进行了模型试算,并对结果进行分析.对于单相渗流数值模拟的理论分析我们已有工作.对于不考虑毛细管力的弹性驱动问题的有限元方法我们已有工作.     

结合数学建模 改革《数值分析》教学

数值分析是一门与实际紧密结合的课程.在教学过程中,既要注重介绍课程的思想和方法,更要通过实际问题阐述方法的应用.探讨了将数学建模引入数值分析教学中的必要性和可行性,并通过实例说明方法的有效性.     

基于气体运动论的差分算法对管道流动数值研究

从分析研究求解Boltzmann模型方程的气体运动论数值计算方法特点出发,设计了几种求解离散速度分布函数不同精度的差分显式与隐式气体运动论数值格式.通过对不同Knudsen数下一维非定常激波管内流动、二维槽道流问题计算研究与应用测试,分析了不同差分格式数值离散效应对计算结果的影响,研究讨论了提高气体运动论数值算法计算效率的途径和差分离散处理所适用的计算准则等问题.     

数值分析双语教学的探讨

首先指出了数值分析双语教学的目的和意义.其次,分析了数值分析双语教学中存在的问题.最后,给出了数值分析双语教学的一些措施.     

炸药爆轰物质点法三维数值模拟

炸药爆轰以及多点起爆所产生的爆轰波汇聚问题很难应用有限元法进行模拟分析,尤其当网格发生畸变时,导致有限元法计算效率和数值精度严重下降,甚至无法得到正确结果.为此,该文应用显式积分算法的物质点法对炸药两点起爆和按时间序列的多点起爆的爆轰过程进行数值模拟,与炸药爆轰的理论计算结果相吻合.物质点法不但可以有效地避免网格畸变问题,而且为炸药爆轰的数值模拟提供了新的思路.     

求解大型最小二乘问题的混合贪婪随机坐标下降法

线性最小二乘问题是科学计算与工程领域普遍存在的问题,有着广泛的应用背景.本文提出了两个新的贪婪随机坐标下降算法来求解大规模的线性最小二乘问题.理论上分析了算法的收敛性.数值实验结果进一步表明了算法的可行性和有效性.     

半导体器件探测器计算流体力学的数值方法和分析

对三维光导型半导体探测器瞬态问题的计算流体力学数值模拟,提出了一类修正迎风分数步差分格式.应用变分形式、能量方法、归纳法假定、微分方程的先验估计理论和技巧,得到最佳阶误差估计.该文数值方法已成功地应用到光导型半导体探测器瞬态问题的数值模拟中.     

具有时滞的基因表达模型的稳定性与分支分析

应用频域法研究了一类具有三个时滞的基因表达模型的Hopf分支问题.基于Nyquist稳定性准则和Hopf分支定理,选取三个时滞的和τ作为分支参数,发现当τ超过某个临界值时,系统产生了Hopf分支.最后,对系统进行了数值仿真,数值仿真的结果验证了理论分析的正确性.     

The stability analysis of dynamic SPECT systems

Summary. SPECT (Single Photon Emission Computed Tomography) techniques have been applied to a wide range of medical studies. The stability of a SPECT model depends strongly upon the data collected. We show that a SPECT model is full rank and well-conditioned (stable) if the projection data are large enough. Condition number estimates for a linear model are given. Numerical results for a class of linear models confirm our theoretical analysis. Received February 1, 1996 / Revised version received August 2, 1996     

Regularized Kalman filtering in dynamic SPECT

This paper presents an algorithm for reconstructing a dynamic SPECT image. Projections of a time-dependent activity are acquired in discrete time frames. This leads to a highly underdetermined set of equations for the images corresponding to each time frame. We reconstruct these images using a Kalman filter algorithm which helps is processing the time-varying information, To obtain meaningful results a positivity constraint must be enforced. Due to the ill-posedness of the problem, regularization is required. We compared Tikhonov and total bounded variation regularization schemes, and found the latter to be more effective, producing superior results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints

In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted ℓ _p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem.     

THE IMPROVED RECONSTRUCTION METHOD FOR NONUNIFORM ATTENUATED SPECT DATA

In this article,we study reconstruction of nonuniform attenuated SPECT data and present analytic reconstruction formulae which are similar to Novikov's inversion formula.Furthermore,we extend Natterer's results.     

RECONSTRUCTION OF THE ATTENUATED RADON TRANSFORM IN π-SCHEME SHORT-SCAN SPECT

In this work, the image reconstruction in π-scheme short-scan single-photon emission computed tomography （SPECT） with nonuniform attenuation is derived in its most general form when π-scheme short-scan SPECT entails data acquisition over disjoint angular intervals without conjugate views totaling to π radians. The reconstruction results are based on decomposition of Novikov＇s inversion operator into three parts bounded in the L2 sense. The first part involves the measured partial data; the second part is a skew-symmetric operator; the third part is a symmetric and compact contribution. It is showed firstly that the operators involved belong to L（L^2（B）. Furthermore numerical simulations are conducted to demonstrate the effectiveness of the developed method.     

Comparison of four natural pixel bases for SPECT imaging

In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function f(P)$f(P)$ is defined on a plane circular domain Ω$\Omega$, and has to be restored from data collected by a SPECT machine. Through an approach known as natural pixel discretization, the solution is expressed as a linear combination of functions belonging to a suitable basis. We consider here four different bases, all of them giving a highly structured coefficient matrix. The linear system obtained in this way can be solved efficiently by means of the fast Fourier transform. The computational cost and the performance of the bases are compared. When the data are contaminated by Poissonian noise, the numerical experimentation shows that all the bases are almost equivalent from the point of view of the restoration efficiency. Hence the choice of a basis should rely on other considerations, as for instance the computational cost.     

Radon变换和衰减Radon变换的分析研究

衰减Radon变换出现在单光子放射型计算机层析成像中。本文首先回顾和研究了Radon变换和衰减Radon变换及其反演的有关结论，进而提出了Tretiak-Metz结果的一种新证明方法，对于一般对象，本文用变换方法非滤子背投影法导出了衰减Radon变换的反演公式。     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.