确定地下水污染强度的反问题方法

探讨了山东省淄博市张店区沣水南部区域地下水的硫酸污染问题 .根据观测井点的浓度数据 ,将硫酸污染强度的识别问题作为一个已知终值数据的源项反问题 .应用积分恒等式方法分析了未知源函数与已知数据间的数据相容性 ,进而基于最优化策略进行了数据反演 ,计算结果与实际估计值基本吻合     

一维波动方程联合反演的分步正则化法

本文只用一个纵波信息,对一维波动方程的速度和震源函数进行联合反演.并考虑到波动方程的反问题是一不适定问题,对震源函数和波速分别用正则化法分步迭代求解,大大减少了反问题的计算工作量,改善了该反问题的计算稳定性.为计算实际一维地震数据提供了一种方法.文中给出了只用一个反问题补充条件同时进行多参数反演的详细公式,并对相应的数值算例进行了分析和比较.     

源项反演问题的条件稳定性

对于一维扩散方程的源项反演问题,探讨了反问题数据的相容性并应用积分恒等式方法建立了非线性源项反演的一种稳定性.     

空间-时间分数阶变系数对流扩散方程微分阶数的数值反演

考虑终值数据条件下一维空间-时间分数阶变系数对流扩散方程中同时确定空间微分阶数与时间微分阶数的反问题.基于对空间-时间分数阶导数的离散,给出求解正问题的一个隐式差分格式,通过对系数矩阵谱半径的估计,证明差分格式的无条件稳定性和收敛性.联合最佳摄动量算法和同伦方法引入同伦正则化算法,应用一种单调下降的Sigmoid型传输函数作为同伦参数,对所提微分阶数反问题进行精确数据与扰动数据情形下的数值反演.结果表明同伦正则化算法对于空间-时问分数阶反常扩散的参数反演问题是有效的.     

利用远场模式的不完全数据反演声波阻尼系数

1.引言 对声波反散射理论的研究,已经有大量的研究[1.5].[7]利用散射波的远场模式反演边界条件中的阻尼系数.但是在实际问题中,要在物体的一周测量到远场模式的值是不现实的.因此,利用远场模式的不完全数据来进行反演有明显的物理和实际意义.一些文献将此类问题称为声波反散射理论的“limited aperture problem”.本文利用远场模式的不完全数据,反演边界条件中的声波阻尼系数.     

混凝土重力坝多参数弹性位移反演分析不唯一性理论探讨

反演分析是现场监测⁃反演分析⁃工程实践检验⁃正演分析及预测的闭环系统的重要环节,而参数反分析是工程实践中研究最多的反分析问题.针对混凝土重力坝多参数反演分析是否具有唯一性,基于均质地基上重力坝在水压力作用下的位移解析解建立目标函数,进而以目标函数和非空凸集构建一个凸规划问题,然后通过分析目标函数的Hesse矩阵是否是正定矩阵,验证目标函数是否是严格凸函数,从而辨识构建的凸规划问题是否具有唯一全局极小点.对坝体和岩基弹性参数的不同组合方案分析表明,当采用理论值与实测值的差值的l1范数作为目标函数时,目标函数的Hesse矩阵均不能保证为正定矩阵,即混凝土重力坝多参数弹性位移反演分析凸规划问题不具有唯一全局极小点,反演分析不唯一.     

一个分数阶生长-抑制系统的参数反问题

本文研究一个分数阶生长-抑制线性系统模型及其参数反问题.首先利用Laplace逆变换得到正问题解的唯一存在性.其次,考虑一个利用内点观测数据确定微分阶数与衰减率的反问题,应用极值原理在Laplace像空间中证明反演的唯一性.最后,基于正问题的有限差分解,应用同伦正则化算法进行数值反演.计算结果表明算法的收敛性及反问题的数值稳定性.     

一类耦合Korteweg-de Vries方程组输运系数反演问题的Lipschitz稳定性

该文研究了一类耦合Korteweg-de Vries(KdV)方程组中两个仅依赖空间变量的输运系数的反演问题.为证明在单个内部测量数据下反问题的稳定性,该文先证明了该耦合KdV方程组的一个仅含单个局部积分项的卡勒曼估计,然后进一步得到了在先验信息下的反问题的Lipschitz稳定性.     

删失数据回归误差项的非参数密度估计的一致相合性(英文)

回归误差项是不可观测的. 由于回归误差项的密度函数在实际中有许多应用, 故使用非参数方法对其进行估计就成为回归分析中的一个基本问题. 针对完全观测数据回归模型, 曾有作者对此问题进行了研究. 然而在实际应用中, 经常会有数据被删失的情况发生, 在此情况下, 可以利用删失回归残差, 并使用核估计的方法对回归误差项的密度函数进行估计. 本文研究了该估计的大样本性质, 并证明了估计量的一致相合性.     

基于数据驱动与遗传计算的海域组合单元水质模型多参数分步耦合优化反演方法研究

考虑水质模型参数时域和地域差异性,建立了模型参数在各单元与时段内独立赋值的海域组合单元水质模型.结合多种反演方式的效率问题,通过数据驱动模型、遗传算法和海域组合单元水质模型的分步耦合,提出了水质模型多参数优化反演的新方法:将数据驱动模型同海域组合单元水质模型有机结合进行初步反演,获得多参数匹配关系;以误差函数为适应度,将海域组合单元水质模型嵌入遗传算法模型中,以多参数匹配关系初值为约束条件,进行多参数精确反演.最后以渤海湾海域组合单元水质模型多参数反演的"孪生"试验验证方法的有效性,数值结果表明分步耦合反演新方法具有较高的精度和效率;组合单元式的参数赋值与反演方式有利于提高模型验证精度.     

Numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm

This paper deals with an inverse problem of identifying a nonlinear source term g=g(u) in the heat equation u_t-u_xx=a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M>0, with which an optimal perturbation algorithm is applied to determine the source function g(u) on u∈[0,M]. Numerical inversions are carried out for g(u) with functional forms of polynomial, trigonometric and index functions. The inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here. By the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources. The inversion algorithm seems to be very sharp if the solution’s maximum M of the forward problem is relatively small; otherwise, the deviations in the source solutions become large especially near the endpoint of u=M.     

基于部分基变量的LP问题矩阵算法

基于部分基变量提出了LP问题的矩阵算法. 该算法以最优基矩阵的一个充分必要条件为基础,首先将一个初始矩阵转化为右端项和检验数均满足要求的矩阵,再转为检验数满足要求的基矩阵,最后转化为最优基矩阵.该算法具有使用范围广、计算规模小、计算过程简化、计算机易于实现的优势.矩阵算法的核心运算是求逆矩阵的运算,提出了矩阵算法的求逆问题,讨论并给出了求逆快速算法,该算法充分利用了矩阵算法迭代过程中提供的原来的逆矩阵的信息经过简单的变换得到新的逆矩阵,该算法比直接求逆法计算效率更高.     

One-dimensional equilibrium model and source parameter determination for soil-column experiment

One-dimensional equilibrium soil-column experiment models with source (sink) reaction terms are discussed in this paper. In the case of occurring high-order chemical reactions, the zero production term in traditional models should be modified to a nonlinear term related with time (or space) and solute concentration, and then a mathematical model with nonlinear terms is put forward. Furthermore, an actual soil-column experiment in Zhangdian, Zibo is investigated. By applying an optimal perturbation algorithm, the source coefficient in the model is determined both in the cases of accurate data and inaccurate data. The inversion results show that for such inverse source coefficient problems with limited additional data, some optimal methods could be more efficient than regularization strategies, and for some real equilibrium soil-column experiments, the process of source (sink) reactions could be a key factor in the solute transportation.     

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

We consider finite difference approximations of solutions of inverse Sturm‐Liouville problems in bounded intervals. Using three‐point finite difference schemes, we discretize the equations on so‐called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm‐Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand‐Levitan formulation. © 2005 Wiley Periodicals, Inc.     

DETERMINATION OF PERMANENT OPTIMAL DATA POINTS AND AN EFFICIENT ALGORITHM FOR LAD PROBLEM

This paper gives a definition of permanent optimal data point of Least Absolute Deviation(LAD)problem.Some theoretical results on non-degenerate LAD problem are obtained.For computing LAD problem,an efficient,algorithm is given according to the idea of permanent optimal data point.Numerical experience shows that our algorithm is better than many of others,including the famous B R algorithm.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

This paper is concerned with the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. We extend the Kirsch-Kress method originally developed for inverse obstacle scattering problems to the above inverse transmission problem with unbounded interfaces. To this end, we reformulate our inverse problem as a nonlinear optimization problem with a Tikhonov regularization term. We prove the convergence of the optimization problem when the regularization parameter tends to zero. Finally, numerical experiments are carried out to show the validity of the inversion algorithm.