实验观测数据的最优正则平滑方法

为了滤除测量噪声 ,提出了一种对实验观测数据进行最优化正则平滑的数据处理方法 .文中阐述了方法的基本原理 ,并就稳定泛函和正则参数的选择等关键问题作了分析和论述 .通过一个数学模拟实例对正则化平滑方法的效果进行了验证 ,这种正则化平滑方法在数学物理反问题求解等领域具有独特的优点     

基于二阶平滑先验的图像保边平滑快速算法

研究了计算机图形处理与计算视觉处理中的图像保边平滑（保持图像边缘平滑）处理。考虑到基于优化方法的保边平滑算法多使用一阶平滑先验作为能量函数的正则项,但它会使平滑结果产生阶梯状的平滑效果,提出了一种基于二阶平滑先验的保边平滑算法,该算法能够避免一阶平滑先验存在的阶梯状平滑偏差,同时锋利地保持图像中显著的边缘。针对该算法的连续变量与01变量的混合优化问题,使用了一种快速的求解方法,该方法在使用图形处理器（GPU）并行加速的情况下能够快速获取平滑结果。通过实验验证了该算法在深度图保边平滑处理、JPEG卡通图像压缩瑕疵恢复以及边缘提取问题中的应用效果。     

动态载荷时域识别的联合去噪修正和正则化预优迭代方法

系统响应可表示为单位脉冲响应函数与激励载荷的卷积,将其离散化一组线性方程组,则载荷识别问题即转化为求解线性方程组的反问题。针对响应中带有噪音时载荷识别的困难,提出了联合奇异熵去噪修正和正则化预优的共轭梯度迭代识别方法。一方面对含噪信号进行基于奇异熵的去噪处理,提高反问题求解中输入数据的精度。另一方面利用正则化方法对共轭梯度迭代算法进行预优,改善反问题的非适定性。由于从输入的响应数据去噪和正则化算法两方面同时改善动态载荷识别反问题的求解,因此可以有效地抑制噪声,提高识别精度。通过数值算例分析,表明在不同的噪声水平干扰下,其识别精度均优于常规的正则化方法,能够实现有效稳定地识别动态载荷。最后通过实验研究进一步验证了该方法的正确性和有效性。     

非线性热传导反问题参数辨识

基于一种时域正演精细算法,引入Bregman距离加权函数作为正则项,应用Tikhonov正则化方法,对非线性热传导反问题进行求解。所建正/反演数值模型在便于敏度分析的同时,能够对非线性内热源强度、导温系数和边界条件等多个热学参数进行有效组合识别。该文给出了相关的数值算例,并对信息误差以及不同正则项的计算效率作了探讨,得到满意的计算结果。数值结果表明所提的求解策略在求解非线性热传导反问题时,不仅能够对相关的热学参数进行有效的组合识别,而且具有较高的计算精度、较好的稳定性和一定的抗噪性,采用加权的Bregman距离函数作正则项可以提高计算效率。     

动态载荷辨识问题的正则化求解新方法

Tikhonov正则化方法是求解载荷辨识问题的有效方法,正则化参数的确定是影响求解准确性的关键问题。提出了一种新方法——虚拟载荷法,以虚拟载荷向量的范数最小作为正则化参数确定准则,将载荷辨识问题转化为单参数优化问题,用一维搜索法求得最优解。通过将法方程的系数矩阵预先进行三对角化,建立了高效的求解算法。通过两个数值算例对新方法进行了检验,结果表明新的参数确定准则是有效的,基于该准则得到的载荷辨识结果具有满意的精度。     

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.     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

基于L1范数正则化和最小二乘优化的冲击载荷识别研究#br#

为了改善冲击载荷识别问题的病态特性，最大限度提高识别精度，在时域内提出一种基于L1范数正则化和最小二乘优化的改进冲击载荷识别方法。采用L1范数正则化方法构建冲击载荷稀疏反卷积模型，使用截断牛顿内点法求解L1范数的最小二乘优化问题，同时根据预条件共轭梯度法确定最优搜索路径和计算方向。最后，考虑不同冲击工况、不同响应位置对识别结果的影响。通过对铝合金板进行冲击载荷识别试验进行验证，发现在铝板受单次冲击和多次冲击工况下所识别载荷与施加的实际载荷吻合良好。结果还表明，与Tikhonov 正则化方法相比，该方法能够提高冲击载荷识别的准确性和稳定性。     

线性不适定问题中选取Tikhonov正则化参数的线性模型函数方法

如何选取正则化参数是不适定问题Tikhonov正则化的一个重要问题。基于吸收的Morozov偏差原理,研究了正则化参数选取的线性模型函数方法。在从Hermite插值角度导出线性模型函数后,讨论了选取正则化参数的两种线性模型函数算法(基本算法与改进算法)及其收敛性。为克服基本算法的局部收敛性,提出了一种新的线性模型函数松弛算法。并且,提出了两种具有全局收敛性的组合算法,即线性与线性模型函数算法、双曲型与线性模型函数算法。数值实验说明了所提算法的有效性。     

基于Tikhonov正则化迭代求解的结构损伤识别方法

求解以结构物理与模态信息所构成的线性方程组,而获得结构的损伤位置和损伤程度,是进行结构损伤检测的一种常用做法。然而,在噪声影响下,其求解往往会出现振荡发散的情况,导致损伤检测结果不准确。Tikhonov正则化方法广泛应用于噪声条件下的线性系统求解,该方法执行的关键是选择合理的正则化矩阵及正则化参数。提出了一种迭代化的Tikhonov正则化方法,通过迭代的方式重构正则化矩阵,在充分抑制噪声的同时,保留了真实的损伤信息。同时,提出了奇异值二分法,自适应地调整正则化参数,避免了传统"L-曲线"方法选取正则化参数时需要进行大量试算等诸多问题。选取一海洋平台结构对提出方法的有效性进行验证,并与传统Tikhonov正则化方法进行对比,结果表明:提出的迭代型Tikhonov正则化方法具有更好的损伤识别结果。     

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.     

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.      

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

Strong convergence of subgradient extragradient method with regularization for solving variational inequalities

The paper concerns with the two numerical methods for approximating solutions of a monotone and Lipschitz variational inequality problem in a Hilbert space. We here describe how to incorporate regularization terms in the projection method, and then establish the strong convergence of the resulting methods under certain conditions imposed on regularization parameters. The new methods work in both cases of with or without knowing previously the Lipschitz constant of cost operator. Using the regularization aims mainly to obtain the strong convergence of the methods which is different to the known hybrid projection or viscosity-type methods. The effectiveness of the new methods over existing ones is also illustrated by several numerical experiments.

     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

超声逆散射成像问题中正则化参数研究

研究了二维理想情况下,基于精确场描述的超声逆散射成像问题,先用矩量法将波动方程化为离散形式,分别用BI和DBI算法进行迭代重建。影响整个算法的一个关键因素是散射场方程的正则化求解,具有明显的不适定性。文章基于L曲线法,提出以解的范数和残差变化量的加权形式作为确定正则化参数的依据,在迭代过程根据问题不适定性程度,自适应地调整搜索范围。仿真结果表明,该算法可快速地找到最优正则化参数。     

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.     

Speckle reduction by I-divergence regularization in optical coherence tomography

For optical coherence tomography (OCT), ultrasound, synthetic-aperture radar, and other coherent ranging methods, speckle can cause spurious detail that detracts from the utility of the image. It is a problem inherent to imaging densely scattering objects with limited bandwidth. Using a method of regularization by minimizing Csiszar's I-divergence measure, we derive a method of speckle minimization that produces an image that both is consistent with the known data and extrapolates additional detail based on constraints on the magnitude of the image. This method is demonstrated on a test image and on an OCT image of a Xenopus laevis tadpole.     

Inverse problems for estimating bottom boundary conditions of natural waters

Unknown boundary conditions for natural waters are estimated using an inverse problem methodology. In the formulation of the inverse problem, expressed as a non‐linear constrained optimization problem, its objective function is given by a square difference term, between experimental and computed data, added to a regularization operator. The computed data are obtained by solving the radiative transfer direct problem using the LTS_N method. A key aspect to get a good reconstruction is played by observed data quality. The reconstruction strategy is examined for in situ radiance and irradiance data for many arrangements of the experimental grid of the measurement devices, in order to plan good designs for experimental works. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimally Generalized Regularization Methods for Solving Linear Inverse Problems

In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be equilibrated or diagonally dominated to derive the optimal scales in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved that the iterative sequences are monotonically convergent to the true solution. The presently proposed optimally generalized regularization methods are able to overcome the ill-posedness of linear inverse problems, and provide rather accurate numerical solution.