不适定热传导反问题的两种求解方法及其比较

本文以不适定热传导反问题为对象，采用两种方法进行了求解。一种方法基于对具有测量误差的边界条件进行适当的微扰，使之化为适定问题；另一种方法基于Ｔｉｋｈｏｎｏｖ的正则平滑思想，对反问题中的输入数据进行平滑处理，以便使函数及其一阶导数均实现一致逼近。通过计算与求解表明，两种方法均能得到具有一定精度与稳定性的结果，其中以正则化法更为理想     

一类双层正则化GMRES方法

近年来,GMRES方法作为一种求解大规模线性不适定方程组的正则化技术越来越受到人们的关注.然而,单独直接应用GMRES求解正则化效果较弱.将GMRES和不同的正则化参数选取准则相结合一外层应用已知误差水平的后验选取、内层应用未知误差水平准则,提出一类双层正则化GMRES方法.数值试验表明,要使新方法得到较好的正则化效果,重开始策略及双层正则化都是必须的.     

有关非线性图像配准的正则化(英文)

近年来,非线性图像配准各种方法的数值分析已经十分成熟;但是大多数工作都没有考虑理论上的收敛性分析.本文对非线性图像配准极小问题进行了理论上的分析证明.我们得到了极小问题正则解的存在性、稳定性及收敛性,证明了在正则化算子和正则参数分别满足一定的条件下,极小问题的正则解收敛到我们所关心的问题的解.这一理论上的证明过程及结果将对非线性图像配准的正则化算子的选取提供十分重要的依据.     

基于自适应步长双参数正则化算法的超声波过程层析成像图像重建

提出了一种新的自适应步长双参数正则化算法,对超声波层析成像系统检测浆体浓度分布进行图像重建。该算法利用转换矩阵将超定解作为先验信息,嵌入到正则化泛函中,避免重建图像被过度平滑,不仅成像速度较快且重建图像具有较高分辨率。仿真实验结果表明,相比于Tikhonov正则化算法以及Landweber算法,自适应步长双参数正则化算法重建图像的相关系数有明显提高并且边界信息更加可靠。     

用正则化Lanczos迭代法进行模型修正

用灵敏度方法进行有限元模型修正,常遇到病态和亏秩的线性方程组,当模态数据含有测量误差时,最小范数解常常没有物理意义.解决这类问题的有效方法是正则化方法.讨论用正则化Lanczos方法进行有限元修正,其中正则化参数用L曲线确定.数值模拟结果表明,该方法能够较好地进行模型修正.     

求解病态问题的一种新的正则化子与正则化算法

根据紧算子的奇异系统理论,提出了一种新的正则化子,进而建立了一类新的求解病态问题的正则化方法。证明了正则解的收敛性并得到了其最优的渐近收敛阶,数值算例说明文中建立的正则化算法是可行而有效的。     

Tikhonov正则化在Zernike多项式拟合中的应用

Zernike多项式系数的求解问题是一个典型的离散不适定问题,最小二乘法,格拉姆-斯密特正交化法和Householder变换法均无法求得稳定的数值解.本文对导致该问题解的不稳定性的原因进行了分析,并采用Tikhonov正则化法对Zernike多项式系数进行求解,利用L曲线准则确定了正则参数.数值仿真结果表明,Tikhonov正则化法有效的保证了解的稳定性,利用该方法得到的拟合面形很好的反映了面形的真实情况.     

用于近场声全息正则化的共轭梯度法

提出采用共轭梯度正则化方法稳定基于分布源边界点法的近场声全息重建过程,控制测量误差对重建结果的影响.共轭梯度法中的最佳迭代次数通过在全息面与源面之间布置一个小型辅助面,并通过最小化测量值与重建值之间的相对误差来选取.与Landweber迭代正则化方法相比,该方法同样具有较高的重建精度,但迭代时间却大大减少.对实际声源的实验研究验证了采用共轭梯度正则化方法控制近场声全息重建误差影响的有效性及其优越性.     

自适应正则化算法的粒径测量数值研究

针对光散射颗粒测量技术中需要求解病态线性方程组的问题,通过研究正则化与矢量相似度预测算法的反演精度和抗噪性能以及这2种算法的缺陷,将这2种算法相结合,提出一种可以有效求解病态问题的自适应正则化算法。结果表明:该算法可以针对不同形态的粒径分布结果做出不同的处理,粒径分布较窄时去掉多余的小峰,粒径分布较宽时进行平滑处理,改善了反演的精度;标准颗粒测试实验证实了算法的有效性。     

一种基于正则化和Watson视觉模型的图像指纹算法

大量多媒体应用的发展使得数字图像很容易地被非法操作和篡改,提出一种基于图像正则化和视觉特性的图像指纹算法,可以有效地实现图像的认证和识别.首先对图像进行正则化预处理,消除几何形变对图像的影响,然后对图像进行分块DCT变换,利用Watson视觉模型对DCT系数进行处理,增大人眼敏感的频域系数在计算图像特征时的权重,经过量化形成最终的指纹序列.在图像指纹序列生成过程中,加入密钥控制,提高了指纹的安全性.实验结果表明,该方法的冲突概率在10-7数量级,对JPEG压缩、旋转、缩放等操作具有较好的稳健性.     

Implementation of edge-preserving regularization for frequency-domain diffuse optical tomography

In this study, we first propose the use of edge-preserving regularization in optimizing an ill-conditioned problem in the reconstruction procedure for diffuse optical tomography to prevent unwanted edge smoothing, which usually degrades the attributes of images for distinguishing tumors from background tissues when using Tikhonov regularization. In the edge-preserving regularization method presented here, a potential function with edge-preserving properties is introduced as a regularized term in an objective function. With the minimization of this proposed objective function, an iterative method to solve this optimization problem is presented in which half-quadratic regularization is introduced to simplify the minimization task. Both numerical and experimental data are employed to justify the proposed technique. The reconstruction results indicate that edge-preserving regularization provides a superior performance over Tikhonov regularization.     

A new spatial regularization scheme for the identification of the geometric shape of an inclusion in a finite body

This paper presents a system identification scheme to determine the geometric shape of an inclusion in a finite body. The proposed algorithm is based on the minimization of the least‐squared errors between the measured displacement field and calculated displacement field by the finite element model. The domain parameterization technique is adopted to manipulate the shape variation of an inclusion. To stabilize the optimization process, a new regularization function defined by the length of the boundary curve of an inclusion is added to the error function. A variable regularization factor scheme is proposed for a consistent regularization effect. The modified Newton method with the active set method is adopted for optimization. Copyright © 1999 John Wiley & Sons, Ltd.     

L∞范数拟合正则化方法在飞行器动态载荷识别中的应用

基于变限积分理论,构造了响应函数最小二乘意义下的加权变限积分。通过适当次数的积分滑动平均,有效过滤测量噪声中的高频噪声。针对测量响应中残留的低频噪声,使用L_∞范数拟合正则化方法识别载荷,提出了一种选取L_∞范数拟合正则化方法最优正则化参数的单调性检验方法。数值仿真及试验验证说明单调性检验方法可以有效确定L_∞范数拟合正则化方法的最优正则化参数,得到比传统L_2范数正则化方法拟合性质更好精度更高的识别载荷;针对遥测数据中噪声特点,使用模拟遥测数据利用L_∞范数拟合正则化方法对冲击载荷进行了有效识别。     

1‐Norm‐based regularization scheme for system identification of structures with discontinuous system parameters

This paper presents a new class of regularization functions and the associated regularization scheme for structural system identification. In particular, 1‐norm regularization functions are investigated to overcome the smearing effect of 2‐norm regularization functions for the identification of discontinuous system parameters of structures. The truncated singular value decomposition is employed to filter out noise‐polluted solution components and to impose the 1‐norm regularization function on SI. The bilinear fitting method is proposed for selecting an optimal truncation number of the truncated singular value decomposition. The validity of the proposed method is demonstrated through the identification of an inclusion in a square plate and damaged members in a two‐span truss. Copyright © 2006 John Wiley & Sons, Ltd.     

Reconstructing dynamic events from time‐limited spatially distributed data

This paper presents a method of reconstructing dynamic events from limited photographically recorded responses. It is based on a wavelet representation of the unknown applied loading in conjunction with a general finite element program. The incorporation of regularization terms provides a robust solution to these usually ill‐conditioned problems. The method allows for the data collected to be of dissimilar type (e.g. strain and displacement), as well as for the data to be collected remotely. As a demonstration of the method, multiple force reconstructions for an impacted plate with a hole are determined using synthetically generated Moiré displacement data. Copyright © 2002 John Wiley & Sons, Ltd.     

Effects of a priori parameter selection in minimum relative entropy method on inverse electrocardiography problem

The goal in inverse electrocardiography (ECG) is to reconstruct cardiac electrical sources from body surface measurements and a mathematical model of torso–heart geometry that relates the sources to the measurements. This problem is ill-posed due to attenuation and smoothing that occur inside the thorax, and small errors in the measurements yield large reconstruction errors. To overcome this, ill-posedness, traditional regularization methods such as Tikhonov regularization and truncated singular value decomposition and statistical approaches such as Bayesian Maximum A Posteriori estimation and Kalman filter have been applied. Statistical methods have yielded accurate inverse solutions; however, they require knowledge of a good a priori probability density function, or state transition definition. Minimum relative entropy (MRE) is an approach for inferring probability density function from a set of constraints and prior information, and may be an alternative to those statistical methods since it operates with more simple prior information definitions. However, success of the MRE method also depends on good choice of prior parameters in the form of upper and lower bound values, expected uncertainty in the model and the prior mean. In this paper, we explore the effects of each of these parameters on the solution of inverse ECG problem and discuss the limitations of the method. Our results show that the prior expected value is the most influential of the three MRE parameters.     

Helmholtz-Type Regularization Method for Permittivity Reconstruction Using Experimental Phantom Data of Electrical Capacitance Tomography

Electrical capacitance tomography (ECT) attempts to image the permittivity distribution of an object by measuring the electrical capacitance between sets of electrodes placed around its periphery. Image reconstruction in ECT is a nonlinear ill-posed inverse problem, and regularization methods are needed to stabilize this inverse problem. The reconstruction of complex shapes (sharp edges) and absolute permittivity values is a more difficult task in ECT, and the commonly used regularization methods in Tikhonov minimization are unable to solve these problems. In the standard Tikhonov regularization method, the regularization matrix has a Laplacian-type structure, which encourages smoothing reconstruction. A Helmholtz-type regularization scheme has been implemented to solve the inverse problem with complicated-shape objects and the absolute permittivity values. The Helmholtz-type regularization has a wavelike property and encourages variations of permittivity. The results from experimental data demonstrate the advantage of the Helmholtz-type regularization for recovering sharp edges over the popular Laplacian-type regularization in the framework of Tikhonov minimization. Furthermore, this paper presents examples of the reconstructed absolute value permittivity map in ECT using experimental phantom data.      

Improved inversion procedure for particle size distribution determination by photon correlation spectroscopy

We propose a minimum variation of solution method to determine the optimal regularization parameter for singular value decomposition for obtaining the initial distribution for a Chahine iterative algorithm used to determine the particle size distribution from photon correlation spectroscopy data. We impose a nonnegativity constraint to make the initial distribution more realistic. The minimum variation of solution is a single constraint method and we show that a better regularization parameter may be obtained by increasing the discrimination between adjacent values. We developed the S-R curve method as a means of determining the modest iterative solution from the Chahine algorithm. The S-R curve method requires a smoothing operator. We have used simulated data to verify our new method and applied it to real data. Both simulated and experimental data show that the method works well and that the first derivative smoothing operator in the S-R curve gives the best results.     

去噪正则化模型修正方法在桥梁损伤识别中的应用

以传统基于灵敏度分析的有限元模型修正方法为基础,提出一种结合小波去噪过程的正则化模型修正损伤识别方法.为改进模型修正方法损伤识别效果,一方面利用有损结构模态与模态噪声的波形在时频域内的差异,以结构有限元模型为基准,对实测模态差进行小波去噪处理,并利用修正后的模态构造目标函数;一方面采用正则化方法改善反问题求解的非适定性.由于从输入数据和求解过程两方面同时改善了结构损伤识别反问题的求解,因此可以有效抑制实测模态参数中噪声的影响,正确识别结构损伤.以连续梁桥模型为例的损伤识别数值模拟表明,所提出方法在保持识别算法鲁棒性、抑制噪声的同时,可有效提高桥梁结构损伤的识别精度.     

Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method

In this paper, we consider the backward problem for diffusion equation with space fractional Laplacian, i.e. determining the initial distribution from the final value measurement data. In order to overcome the ill-posedness of the backward problem, we present a so-called negative exponential regularization method to deal with it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate are established under a-priori bound assumption for the exact solution. Finally, several numerical examples are proposed to show that the numerical methods are effective.