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。     

局部近场声全息的仿真与实验研究

声场的局部测量不能满足基于快速傅里叶变换近场声全息理论推导的前提条件，所以该方法无法实现局部声场的精确重建。统计最优近场声全息在空间域直接实现声场的重建，避免由于使用快速傅里叶变换而产生的各种误差。结合不同的正则化方法，研究了统计最优近场声全息对局部声场的重建效果，分析了重建面边缘区域以及中心区域误差对总误差的贡献。仿真与实验结果表明：统计最优近场声全息可以实现局部声场的精确重建，重建面边缘区域的误差大于中心区域的误差；正则化技术方面，基于Engl误差最小化原则的正则化参数选择法，使得Tikhonov正则化方法更为实用。     

一类积分方程的数值解法

研究了形如∫^t 0H(T,τ)f(τ)dτ=g(t)一类积分方程的数值求解，从讨论病态性质入手，基于吉洪诺夫的正则化思想，构造了正则化算了,而给出了求解这类积分方程稳定的数值方法，并给出了一些数值算例。     

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.     

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     

广义Tikhonov正则化及其正则参数的先验选取

对于算子与右端都有扰动的第一类算子方程建立了一种广义Tikhonov正则化。应用紧算子的奇异系统及正则化子的性质先验选取正则参数,证明了正则解具有最优的渐近阶。     

用RRQR迭代法进行模型修正

在有限元模型修正中,由正交条件导出的线性方程组的系数矩阵通常是病态和亏秩的,当测量模态数据含有误差时,其最小二乘解通常没有物理意义的修正参数。解决这类问题的有效方法是正则化方法。讨论用示秩QR分解(RRQR)方法进行有限元模型修正,正则化参数用L曲线和GCV准则确定。数值模拟结果表明,这些方法能够较好地进行模型修正。     

非均匀损伤介质中利用波传播理论对损伤参数的反演

介绍了波在非均匀损伤介质中的传播的反问题,根据波的响应来反演介质的损伤。首先应用Newton迭代法把非线性问题化为线性问题,得到一组不适定的线性方程式组,采用Tikhonov正则化求解,对于正则参数选取采用L线曲线准则,求出了原问题的稳定的、有效的解,获得了与真实损伤度较为吻合的反演参数。工程实例的模拟分析计算证明了上述算法的有效性。     

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.     

A multi-population genetic algorithm approach for solving ill-posed problems

In this paper we develop a multi-population genetic algorithm (GA) for regularizing nonlinear ill-posed problems. Real coded genetic algorithms are used for calculating the minimizers of the Tikhonov functional. The algorithm is based on evolving separate populations for various values of the regularization parameter. The rate of convergence of the algorithm is substantially increased by exchanging information between neighbouring populations by the process of migration.     

An Alternating Iterative MFS Algorithm for the Cauchy Problem in Two-Dimensional Anisotropic Heat Conduction

In this paper, the alternating iterative algorithm originally proposed by Kozlov, Maz'ya and Fomin (1991) is numerically implemented for the Cauchy problem in anisotropic heat conduction using a meshless method. Every iteration of the numerical procedure consists of two mixed, well-posed and direct problems which are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems related to heat conduction in two-dimensional anisotropic solids to confirm the numerical convergence, stability and accuracy of the method.     

ITERATIVE SOLUTION OF LARGE THREE-DIMENSIONAL BEM ELASTOSTATIC ANALYSES USING THE GMRES TECHNIQUE

The GMRES (Generalized Minimal RESidual) iterative method is receiving increased attention as a solver for the large dense and unstructured matrices generated by boundary element elastostatic analyses. Existing published results are predominantly for two-dimensional problems, of only medium size. When these methods are applied to large three-dimensional problems, which actually do require efficient iterative methods for practical solution, they fail. This failure is exacerbated by the use of the pre-conditioning otherwise desirable in such problems. The cause of the failure is identified as being in the orthogonalization process, and is demonstrated by the divergence of the ‘true’ residual, and the residual calculated during the GMRES algorithm. It is shown that double precision arithmetic is required for only the small fraction of the work comprising the orthogonalization process, and exploitation of this largely removes the penalties associated with the use of double precision. Additionally, it is shown that full re-orthogonalization can be employed to overcome the lack of convergence, extending the applicability of the GMRES to significantly larger problems. The approach is demonstrated by solving three-dimensional problems comprising ∼4000 and ∼5000 equations. © 1997 John Wiley & Sons, Ltd.     

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analysis and a conditional stability result are given. This problem is ill-posed and Landweber iteration regularization method is used to deal with this problem. Convergence estimates are presented under the priori and the posteriori regularization choice rules. For the a priori and the a posteriori regularization parameters choice rules, the convergence error estimates are all order optimal. Numerical examples are given to show that the regularization method is effective and stable for dealing with this ill-posed problem.     

An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation

We investigate the numerical implementation of the alternating iterative algorithm originally proposed by Kozlov et al. (Comput Math Math Phys 31:45–52) for the Cauchy problem associated with the two-dimensional modified Helmholtz equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method.     

Weakly singular stress‐BEM for 2D elastostatics

A weakly singular stress‐BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C⁰ elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress‐BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self‐regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright © 1999 John Wiley & Sons, Ltd.     

A posteriori error estimates for numerical solutions to inverse problems of elastography

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be found is smooth or discontinuous. Tikhonov regularization with appropriate regularizing functionals is applied to solve these problems. In particular, discontinuous Young’s modulus distribution can be found on the class of 2D functions with bounded variation of Hardy–Krause type. It is shown in the paper that a variant of Tikhonov regularization provides for such discontinuous distributions the so-called piecewise uniform convergence of approximate solutions as the error levels of the data vanish. The problem of practical a posteriori estimation of the accuracy for obtained approximate solutions is under consideration as well. A method of such estimation is presented. As illustrations, model inverse problems with smooth and discontinuous solutions are solved along with a posteriori estimations of the accuracy.     

Regularizing active set method for retrieval of the atmospheric aerosol particle size distribution function

The determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind [S. Twomey, J. Comput. Phys.18, 188 (1975);Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007)], since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature of the problem, regularization techniques were developed. However, most of the literature focuses on the application of Phillips-Twomey regularization and its variants, which are unstable in several cases. As is known, the particle size distribution is always nonnegative, and we are often faced with incomplete data. Therefore, we study the active set method and propose a regularizing active set algorithm for ill-posed particle size distribution function retrieval and for enforcing nonnegativity in computation. Our numerical tests are based on synthetic data for theoretical simulations and the field data obtained with a CE 318 Sun photometer for the Po Yang lake region of Jiang Xi Province, China, and are performed to show the efficiency and feasibility of the proposed algorithms.     

Regularized spectral multipole BEM for plane elasticity

A multipole algorithm for plane elasticity based on the direct boundary element method (BEM) is presented. The kernels in the BEM are approximated as truncated Taylor series with expansion points taken from a uniform grid. The algorithm replaces the usual BEM elemental summations with correlation sums on the regular grid in terms of the sampled kernel data and density moments. Far field influences are rapidly computed in the frequency domain using the fast Fourier transform (FFT). The resultant linear system of equations is solved with GMRES. The multipole method is extended to whole-body regularized forms of the standard displacement-BIE and the stress-BIE. Free-term coefficients which arise from regularization in the far field are also rapidly computed as correlation sums with the FFT. The algorithm is shown to be faster than the traditional BEM for models with over 400 quartic elements while maintaining an acceptably high level of accuracy.