An adaptive greedy technique for inverse boundary determination problem

In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels.     

Singular meshless method using double layer potentials for exterior acoustics

Time-harmonic exterior acoustic problems are solved by using a singular meshless method in this paper. It is well known that the source points cannot be located on the real boundary, when the method of fundamental solutions (MFS) is used due to the singularity of the adopted kernel functions. Hence, if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived. Herein we present an approach to obtain the diagonal terms of the influence matrices of the MFS for the numerical treatment of exterior acoustics. By using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. We also maintain the prominent features of the MFS, that it is free from mesh, singularity, and numerical integration. The normal derivative of the fundamental solution of the Helmholtz equation is composed of a two-point function, which is one of the radial basis functions. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary based on the potential theory. The solutions of three selected examples are used to compare with the results of the exact solution, conventional MFS, boundary element method, and Dirichlet-to-Neumann finite element method. Good numerical performance is demonstrated by close agreement with other solutions.     

用于电介质中空间电荷分布测量的Tikhonov反卷积算法

研究了使用压力波法测量平板电介质试样的空间电荷分布的数值解法,使用基于Tikhonov正则化方法的反卷积算法得到了真实的空间电荷分布.在反卷积算法中使用了相关的技术处理,如小波包过滤高频噪音,Tikhonov正则化方法处理积分方程等.利用数值实验研究了噪声对反卷积算法的影响,结果表明,在无噪或者低噪环境下,反卷积算法能够非常好地计算出电介质中的空间电荷分布;在处理有噪数据时,反卷积的结果受到明显的影响,但仍然有较高的计算精度.正则化参数α对空间电荷分布的数值解起着明显的光滑作用,但是对于解的积分值却影响不大.对实际测量数据进行处理的结果表明,反卷积算法成功地计算出了固体电介质中的空间电荷分布和电场分布.     

Compensation for discarded singular values in vibro-acoustic inverse methods

The paper addresses the inverse problem where source strengths are back-calculated from a sound pressure field sampled at several points. Regularization techniques, such as singular value discarding or Tikhonov regularization, are commonly used to improve estimates of source strength in such situations. However, over-regularization can result in even worse errors. A simple procedure is proposed here to compensate for errors of over-regularization. The basis is to constrain the solution such that the spatial mean of the measured and reconstructed sound pressure are equal. In other words, to set the overall sound power of the equivalent (calculated) sources equal to that of the real source. It is argued that the overall sound power is the most stable and reliable quantity on which to base source strength estimates. Examples of both singular value discarding and Tikhonov regularization are given.     

Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions

In this article, a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed, which allows the definition of a Tikhonov functional on a space of level-set functions. We provide convergence analysis for the Tikhonov approach, including stability and convergence results. Moreover, a numerical investigation of the proposed Tikhonov regularization method is presented. Newton-type methods are used for the solution of the optimality systems, which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance. The whole approach is focused on three dimensional models, better suited for real life applications.     

A meshless method for solving an inverse spacewise-dependent heat source problem

In this paper an effective meshless and integration-free numerical scheme for solving an inverse spacewise-dependent heat source problem is proposed. Due to the use of the fundamental solution as basis functions, the method leads to a global approximation scheme in both spatial and time domains. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by several numerical examples.     

基于 L-曲线参数优化的均匀声场重建算法

为了得到最小二乘法声场重建问题的稳定解,通常需要引入Tikhonov正则化方法。然而正则化程度取决于正则化参数的选择。针对这一问题,提出了一种基于L-曲线法参数选择的均匀声场重建算法。该算法根据重建误差与扬声器功率计算得到L-曲线,该曲线上曲率最大的点所对应的参数值作为Tikhonov正则化参数的选值。确定正则化参数后可进一步得到扬声器权系数以及重建均匀声场。针对不同正则化参数取值方法,对控制区域进行均匀声场重建以及重建性能仿真。仿真结果及实验表明,L-曲线法实现了重建误差与扬声器驱动信号功率之间的平衡。     

Spatial resolution limits for the reconstruction of acoustic source strength by inverse methods

One method for deducing the strength of an acoustic source distribution from measurement of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution. In practice, the frequency response function matrix to be inverted may very often be ill-conditioned. This ill-conditioning will also often result in an ill-posed problem and thus regularization algorithms are used to produce reasonable solutions. For this purpose, Tikhonov regularization has been applied, and generalized cross-validation (GCV) has been introduced as an effective method for determining the proper amount of regularization without prior knowledge of either the source distribution or the contaminating errors. In the present work, the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted. However, the use of Tikhonov regularization often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution. Thus, the process of regularization produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution. Furthermore, in the field of Fourier acoustics, the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation. This paper expresses the relationship between estimation accuracy, spatial resolution, noise-level and source/sensor geometry, when a range of inverse sound radiation problems are regularised using Tikhonov regularization with GCV. The results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution that is finer than the intrinsic half-wavelength limit.     

The application of regularization technique based on partial optimization in the nearfield acoustic holography

The regularization technique for stabilizing the reconstruction based on the nearfield acoustic holography(NAH) was investigated on the basis of the equivalent source method.In order to obtain higher regularization effect,a regularization method based on the idea of partial optimization was proposed,which inherits the advantages of the Tikhonov and another regularization method—truncated singular value decomposition(TSVD).Through the numerical simulation,it is proved that the proposed method is stabler than the Tikhonov,and more precise than the TSVD.Finally the validity and the feasibility of the proposed method are demonstrated by an experiment carried out in a semi-anechoic room with two speakers.     

Acoustical inverse problems regularization: Direct definition of filter factors using Signal-to-Noise Ratio

Acoustic imaging aims at localization and characterization of sound sources using microphone arrays. In this paper a new regularization method for acoustic imaging by inverse approach is proposed. The method first relies on the singular value decomposition of the plant matrix and on the projection of the measured data on the corresponding singular vectors. In place of regularization using classical methods such as truncated singular value decomposition and Tikhonov regularization, the proposed method involves the direct definition of the filter factors on the basis of a thresholding operation, defined from the estimated measurement noise. The thresholding operation is achieved using modified filter functions. The originality of the approach is to propose the definition of a filter factor which provides more damping to the singular components dominated by noise than that given by the Tikhonov filter. This has the advantage of potentially simplifying the selection of the best regularization amount in inverse problems. Theoretical results show that this method is comparatively more accurate than Tikhonov regularization and truncated singular value decomposition.     

Spectral semi-blind deconvolution with hybrid regularization

A spectral semi-blind deconvolution with hybrid regularization (SBD-HR) is proposed to recover the spectrum and to estimate the parameter of the point spread function (PSF) adaptively. Firstly, a weighted Tikhonov regularization term about the spectrum is presented to preserve the details of spectrum. Then the regularization term about the PSF is modeled as L1-norm instead of L2-norm to enhance the stability of kernel estimation. The numerical solution processes for recovering the spectrum and for estimating the parameter of the PSF are deduced. Simulation results of infrared spectrum deconvolution demonstrate that the proposed method can recover the spectrum better from the degraded spectrum and estimate the parameter of the PSF accurately.     

A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing

The regularization parameter plays an important role in applying the Tikhonov regularization method to recover the particle size distribution from dynamic light scattering experiments. The so-called V-curve, which is a plot of the product of the residual norm and the norm of the recovered distribution versus all valid regularization parameters, can be used to estimate the result of inversion. Numerical simulation demonstrated that the resultant V-curve can be applied to optimize the regularization parameter. The regularization parameter is optimized corresponding to the minimum value of the V-curve. Simulation and experimental results show that stable distributions can be retrieved using the Tikhonov regularization with optimum parameter for unimodal particle size distributions.     

An efficient inversion algorithm for atmospheric remote sensing with application to UV limb observations

In this paper we present a retrieval algorithm for atmospheric remote sensing. The algorithm combines Tikhonov regularization and the iteratively regularized Gauss-Newton method and is devoted to the solution of multi-parameter inverse problems with simple bounds on the variables. The basic features of the algorithm: the solution of the bound-constrained minimization problem, the selection of the optimal regularization parameter, the derivation of the global regularization matrix and the characterization of the solution (error analysis) are discussed in detailed. The inversion algorithm is applied to ozone retrieval from SCIAMACHY limb scatter measurements in the ultraviolet spectral range.     

炉膛三维温度场重建中Tikhonov正则化和截断奇异值分解算法比较

在煤粉锅炉诊断中火焰辐射能图像扮演着越来越重要的角色, 通过电荷耦合器件(CCD)获得的辐射能图像可以重建出炉内火焰三维温度场, CCD 用于获取视场角内的辐射能图像. 温度场重建的矩阵方程是一个严重病态的方程, 本文使用两种算法(Tikhonov正则化算法和截断奇异值分解(TSVD)算法)来重建温度场. 应用广义交叉检验算法来选取正确的正则化参数. 数值模拟的环境为一个10 m×10 m×10 m的三维炉膛, 系统被划分为10×10×10的1000个网格, 每个网格单元都是边长为1 m的立方体. 在正问题求解所得到的CCD接受信号基础上加上不同随机误差以模拟测量时的CCD接受信号. 研究两种算法重建后的温度重建误差、两者的重建时间, 以及最高温度的重建效果. 初步的研究结果显示, 一般情况下基于Tikhonov算法重建的温度场比基于TSVD算法重建的温度场误差要小, 计算所需时间短, 最高温度重建更准确.     

Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation

The mixed boundary value problem of the Laplace equation is considered. The method of fundamental solutions (MFS) approximates the exact solution to the Laplace equation by a linear combination of independent fundamental solutions with different source points. The accuracy of the numerical solution depends on the distribution of source points. In this paper, a weighted greedy QR decomposition (GQRD) is proposed to choose significant source points by introducing a weighting parameter. An index called an average degree of approximation is defined to show the efficiency of the proposed method. From numerical experiments, it is concluded that the numerical solution tends to be more accurate when the average degree of approximation is larger, and that the proposed method can yield more accurate solutions with a less number of source points than the conventional GQRD.     

A threshold for the use of Tikhonov regularization in inverse force determination

In the analysis of structure-borne sound from installed machinery, it is important to be able to estimate the operational forces. Assuming that their location is known, indirect approaches based on matrix inversion can be used to reconstruct the operational forces from a set of measured operational responses and corresponding matrix of frequency response functions. In common with many such inverse problems, matrix ill-conditioning can affect the reliability of the results. Methods such as pseudo-inversion of over-determined matrices, singular value rejection, and Tikhonov regularization have been used previously to overcome this and it has been found that Tikhonov regularization generally performs well in reducing the errors in the reconstructed forces. However, full-rank pseudo-inversion (unregularized solution) gives better results than Tikhonov regularization in some cases, particularly with low condition numbers. Since the need for regularization is greatest when the matrix is ill-conditioned, this suggests the introduction of a threshold above which Tikhonov regularization is used and below which pseudo-inversion is used. In this study, the extent to which response errors are amplified in the force estimates is considered and plotted against the matrix condition number. This allows a threshold condition number to be identified above which Tikhonov regularization gives improved results. It is found that the threshold is related not only to the condition number but also to the matrix dimensions including the extent of over-determination. A simple empirical formula is obtained for this threshold that is usable for matrices in a wide range of matrix dimensions.     

AN INVERSE AEROACOUSTIC PROBLEM ON ROTOR WAKE/STATOR INTERACTION

An inverse aeroacoustic model on rotor wake/stator interaction is proposed based on the linearized Euler equations. The sound field is related to the pressure distribution on the stator surface in the form of a Fredholm integral equation of the first kind which is a well-known ill-posed problem. Since the sound field is fully specified, numerical inversion allows the reconstruction of the pressure distribution on the stator surface. For solving the discrete ill-posed problem, the singular-value decomposition technique coupled with the Tikhonov regularization method is applied to stabilize the solution. The optimal regularization parameter is chosen by the generalized cross-validation criterion and the discrete Picard condition is employed to analyze the ill-posedness of the inverse problem. Numerical results show that the reconstruction is fairly good when the signal-to-noise ratio is not very low. The results become inaccurate when the noise dominates over the observer signal. In addition, numerical results also indicate the importance of the reduced frequency. The higher the reduced frequency, the better the reconstruction results.     

RRT: the regularized resolvent transform for high-resolution spectral estimation

A new numerical expression, called the regularized resolvent transform (RRT), is presented. RRT is a direct transformation of the truncated time-domain data into a frequency-domain spectrum and is suitable for high-resolution spectral estimation of multidimensional time signals. One of its forms, under the condition that the signal consists only of a finite sum of damped sinusoids, turns out to be equivalent to the exact infinite time discrete Fourier transformation. RRT naturally emerges from the filter diagonalization method, although no diagonalization is required. In RRT the spectrum at each frequency s is expressed in terms of the resolvent R(s)(-1) of a small data matrix R(s), that is constructed from the time signal. Generally, R is singular, which requires certain regularization. In particular, the Tikhonov regularization, R(-1) approximately [R(dagger)R + q(2)](-1)R(dagger) with regularization parameter q, appears to be computationally both efficient and very stable. Numerical implementation of RRT is very inexpensive because even for extremely large data sets the matrices involved are small. RRT is demonstrated using model 1D and experimental 2D NMR signals. Copyright 2000 Academic Press.     

基于迭代Tikhonov正规化的三刺激值重建光谱方法研究

光谱图像中的反射率光谱数据维数高,且与光源、设备均无关,能够比较全面、真实、客观地描述图像中物体的颜色信息。针对三色相机的光谱图像获取系统中三维色度数据重建多维光谱数据产生的光谱信息丢失、以及伴随而生的颜色信息丢失问题,提出了迭代Tikhonov正规化的光谱重建方法。首先依据色度学理论中色度值与反射率光谱之间的关系,构建反射率光谱重建方程建立起相机所获三维色度数据与高维反射率光谱数据的映射关系;然后,通过反射率光谱重建方程的病态分析,在Moore-Penrose伪逆矩阵求解思想的基础上构建迭代Tikhonov正规化方法求解反射率光谱,并利用训练样本数据通过L-曲线方法训练获取迭代Tikhonov正规化的最优正规化参数,以有效控制并改善反射率光谱重建方程求解的病态、减少重建光谱的光谱信息丢失。实验通过选取样本数据对光谱重建方法进行验证。验证实验的结果表明所提出的光谱重建方法改善了三色相机的光谱图像获取系统中重建光谱的光谱信息丢失程度,使得重建光谱的光谱误差和色度误差较其他光谱重建方法均有明显降低。     

The quantification of structure-borne transmission paths by inverse methods. Part 2: Use of regularization techniques

The inversion of an ill-conditioned matrix of measured data lies at the heart of procedures for the quantification of structure-borne sources and transmission paths. In an earlier paper the use of over-determination, singular value decomposition and the rejection of small singular values was discussed. In the present paper alternative techniques for regularizing the matrix inversion are considered. Such techniques have been used in the field of digital image processing and more recently in relation to nearfield acoustic holography. The application to structure-borne sound transmission involves matrices, which vary much more with frequency and from one element to another. In this study Tikhonov regularization is used with the ordinary cross-validation method for selecting the regularization parameter. An iterative inversion technique is also studied. Here a form of cross-validation is developed allowing an optimum value of the iteration parameter to be selected. Simulations are carried out using a rectangular plate structure to assess the relative merits of these techniques. Experiments are also performed to validate the results. Both techniques are found to give considerably improved results compared to singular value rejection.