Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization

Deconvolution of optically collected axisymmetric flame data is equivalent to solving an ill-posed problem subject to severe error amplification. Tikhonov regularization has recently been shown to be well suited for stabilizing this deconvolution, although the success of this method hinges on choosing a suitable regularization parameter. Incorporating a parameter selection scheme transforms this technique into a reliable automatic algorithm that outperforms unregularized deconvolution of a smoothed data set, which is currently the most popular way to analyze axisymmetric data. We review the discrepancy principle, L-curve curvature, and generalized cross-validation parameter selection schemes and conclude that the L-curve curvature algorithm is best suited to this problem.     

Spectral multivariate calibration with wavelength selection using variants of Tikhonov regularization

Tikhonov regularization (TR) is a general method that can be used to form a multivariate calibration model and numerous variants of it exist, including ridge regression (RR). This paper reports on the unique flexibility of TR to form a model using full wavelengths (RR), individually selected wavelengths, or multiple bands of selected wavelengths. Of these three TR variants, the one based on selection of wavelength bands is found to produce lower prediction errors. As with most wavelength selection algorithms, the model vector magnitude indicates that this error reduction comes with a potential increase in prediction uncertainty. Results are presented for near-infrared, ultraviolet-visible, and synthetic spectral data sets. While the focus of this paper is wavelength selection, the TR methods are generic and applicable to other variable-selection situations.     

Numerical differentiation of the radical artery sphygmogram by Tikhonov regularization

A method is proposed and substantiated for determining the characteristic points of the pulse wave (sphygmogram) of the human radial artery, by numerical differentiation with the aid of the Tikhonov regularization algorithm. The results are given of experiments made to test the stability, to evaluate the accuracy of recovering the first and second derivatives, and to establish the sensitivity of the method to random distortions of the original data. Based on this method, we have solved one of the chief technical problems in the objectivization and automatization of the pulse diagnostics of Tibetan medicine, by the development and construction of an information-computing Tibetan medicine diagnostic complex, for processing medicobiological information in real time.Translated from Izmeritel'naya Tekhnika, No. 11, pp. 60–62, November, 1994.     

Determination of an optimal regularization factor in system identification with Tikhonov regularization for linear elastic continua

This paper presents a geometric mean scheme (GMS) to determine an optimal regularization factor for Tikhonov regularization technique in the system identification problems of linear elastic continua. The characteristics of non‐linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses. It is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution. Based on this observation, the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses. The validity of the GMS is demonstrated through two numerical examples with measurement errors and modelling errors. Copyright © 2001 John Wiley & Sons, Ltd.     

Note on using radial basis functions and Tikhonov regularization method to solve an inverse heat conduction problem

In this note, a meshless numerical scheme for solving an inverse heat conduction problem (IHCP) is considered. The fundamental solution of heat equation is used as a radial basis function. Applying this radial basis function results in a badly ill-condition system of equations. The Tikhonov regularization method is employed for solving this system of equations. Determination of regularization parameter is based on L-curve technique. The numerical results demonstrate the accuracy and ability of this method.     

Deconvolution using signal segmentation

Extraction of peak areas and mass spectral information from chromatography mass spectral data such as obtained in metabolomics measurements requires much effort and the quality is often subjective to the operator that handles the data at hand. In multiple file deconvolution, all samples are processed simultaneously and alignment issues are part of the modeling strategy. However, processing the total data set as a whole is an impossible task and therefore the data processing task requires segmentation. Two intertwined divide and conquer strategies are proposed. The first strategy divides the retention time axis into equal parts and the second strategy divides the total data set into a model and a prediction data set. Dividing the data into smaller segments allows us to conquer the total problem. Post processing of the resulting matrices with peak areas and mass spectra ensures that a matrix with peak areas ready for statistics and a matrix with mass spectral information ready for peak annotation is obtained. The proposed methodology is implemented within a package called TNO-DECO but can easily be implemented in other data pre-processing approaches.     

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.     

Solution for the general integral‐type nonlocal plasticity model with Tikhonov regularization

A new solution approach, based on Tikhonov regularization on the Fredholm integral equations of the first kind, is proposed to find the approximate solutions of the strain softening problems. In this approach, the consistency condition is regularized with the Tikhonov stabilizers along with a regularization parameter, and the internal variable increments are solved from the resulting Euler's equations. It is shown that, as the regularization parameter is increased, the solutions converge to a unique one. A nonlocal yield condition and a nonlocal return mapping algorithm are proposed to carry out the integration of constitutive equations in the time and spatial domains. A global plastic dissipation principle is proposed to relax the classical local plastic dissipation postulate. Numerical examples show that the proposed approach leads to objective, mesh‐independent solutions of the softening‐induced localization problems. A comparison of the results from the proposed approach with those from the gradient‐dependent plasticity model shows that the two models give close solutions.     

Use of non-negative constraint in Tikhonov regularization for particle sizing based on forward light scattering

Abstract

The set of linear equations in the inversion of particle size distribution (PSD) based on forward light scattering is an ill-posed problem. In order to solve the inverse problem of this kind, a number of inversion algorithms have been proposed. The regularization algorithm can reconstruct the PSD, but in usual case, the solution may contain negative values and is strongly oscillating. Owing to the natural reason, the solution should be non-negative and smooth. In this paper, a simple non-negative constraint (NNC) is used with a combination of the Tikhonov regularization. Simulations and experiments show that the regularization with NNC can achieve more reasonable results.     

Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems

In this article, we study the balancing principle for Tikhonov regularization in Hilbert scales for deterministic and statistical nonlinear inverse problems. While the rates of convergence in deterministic setting is order optimal, they prove to be order optimal up to a logarithmic term in the stochastic framework. The two-step approach allows us to consider a data-driven algorithm in a general error model for which an exponential behaviour of the tail of the estimator chosen in the first step is valid. Finally, we compute the overall rate of convergence for a Hammerstein operator equation and for a parameter identification problem. Moreover, we illustrate these rates for the last application after we study some large sample properties of the local polynomial estimator in a general stochastic framework.     

Temperature measurement of axisymmetric flame under the influence of magnetic field using lensless Fourier transform digital holography

In this paper the effect of uniform magnetic field B on the temperature and temperature profile of the diffusion flame is investigated using lensless Fourier transform digital holographic interferometry. The evaluation of temperature profile reveals that the width of flame as well as the maximum value of temperature inside the flame is increased.     

基于Tikhonov正则化及奇异值分解的载荷识别方法

针对矩阵求逆法应用中存在的病态逆问题,用Tikhonov正则化及奇异值分解法解决。通过对平板模型仿真分析,利用频响函数法矩阵条件数评价系统的病态性,系统病态性不同时用奇异值分解法与基于不同正则化参数选择的Tikhonov方法对载荷进行识别。研究表明,条件数大于1000时,Tikhonov正则化方法识别误差较小;反之,奇异值分解法较优。提出综合使用Tikhonov正则化与奇异值分解的载荷识别方法,给出方法流程。仿真与实验结果表明该方法可提高结构载荷识别精度,具有一定工程应用价值。     

Computer simulation of fluids using axisymmetric molecules

Molecular dynamics simulations have been performed using the Gaussian overlap potential in order to develop and test thermodynamic property predictive methods and to investigate the possibility of modeling real fluids as well. A first-order anisotropic reference-based perturbation theory and a temperaturedependent sphericalized potential have been tested and these methods are shown to work well, qualitatively for the original Gaussian overlap potential and qualitatively as well as quantitatively for the modified Gaussian overlap potential. It is also shown that the modified Gaussian overlap potential with the quadrupole moment works well for modeling benzene. Future possibilities are also discussed.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Wavelength selection for multivariate calibration using tikhonov regularization

Prediction of sample properties using spectroscopic data with multivariate calibration is often enhanced by wavelength selection. This paper reports on a built-in wavelength selection method in which the estimated regression vector contains zero to near-zero coefficients for undesirable wavelengths. The method is based on Tikhonov regularization with the model 1-norm (TR1) and is applied to simulated and near-infrared (NIR) spectral data. Models are also formed from wavelength subsets determined by the standard method of stepwise regression (SWR). Harmonious (bias/variance tradeoff) and parsimonious considerations are compared with and without wavelength selection for principal component regression (PCR), ridge regression (RR), partial least squares (PLS), and multiple linear regression (MLR). Results show that TR1 models generally contain large baseline regions of near-zero coefficients, thereby essentially achieving built-in wavelength selection. For example, wavelengths with spectral interferences and/or poor signal-to-noise ratios obtain near zero regression coefficients. Results often improve with TR1 models, compared to full wavelength PCR, RR, and PLS models. The SWR subset results are similar to those for the TR1 models using the NIR data and worse with the simulated spectral situations. In general, wavelength selection improves prediction accuracy at a sacrifice to a potential increase in variance and the parsimony remains nearly equivalent compared to full wavelength models. New insights gained from the reported studies provide useful guidelines on when to use full wavelengths or use wavelength selection methods. Specifically, when a small number of large wavelength effects (good sensitivity and selectivity) exist, subset selection by SWR (with caution) and TR1 do well. With a small to moderate number of large to moderate sized wavelength effects, TR1 is better. Lastly, when a large number of small effects are present, full wavelengths with the methods of PCR, RR, or PLS are best.     

Computation of stresses in axisymmetric elastostatical problems using BEM

The application of the conjugate residual method to the indefinite system resulting from the finite element approximation of the incompressible Navier-Stokes equations is studied. The possibility to use an element-by-element preconditioner for the acceleration of the convergence is also discussed. With this scheme, one can achieve significant savings in storage without ever forming the large total matrix. Furthermore, the hybrid finite element mesh, a mixture of the structured and unstructured mesh subdivisions, is employed to minimize the storage requirement. The numerical simulations of the Karman vortex street, a driven cavity flow and the flow in a safety valve are presented to demonstrate the applicability of the proposed strategies.     

Small animal PET image super-resolution using Tikhonov and modified total variation regularisation

Positron emission tomography (PET) is an imaging procedure used mainly in the diagnosis and treatment of diseases. PET is also used in the preclinical research studies of small animals. However, researchers may have difficulty interpreting the particularly low-resolution images obtained via this procedure. This paper presents a new method of increasing the resolution of PET images through the use of super-resolution techniques. Aside from being resistant to the noise and other degradations that plague PET images, our proposed algorithm is also capable of preserving important structures (e.g. lesions). To this end, the proposed objective function includes a term based on the modified total variation model which allows the user to preserve texture and to deal with noise without incurring the artefacts that typically arise when the total variation norm is used. The present study shows the effectiveness of the method in recovering structures and details and indicates that, in most cases, it outperforms other state-of-the-art methods.     

Intrinsic properties of the optical coupling between axisymmetric Gaussian beams

On the basis of the overlap integral method, an approximate analytical model is derived to estimate the coupled optical power between axisymmetric Gaussian beams when transverse, axial, and angular misalignments simultaneously exist in three dimensions. Seven optical properties are derived from a detailed analysis of the model. Because the model is an approximate analytical solution to the overlap integral method, the existence of each property is also investigated by a numerical solution. Results show that all seven properties are intrinsic to the optical coupling phenomenon between Gaussian beams. Because numerous single-mode device-to-fiber coupling systems can be well described by use of Gaussian beams, the seven properties provide a solid basis to develop model-based algorithms for single-mode device-to-fiber alignment automation.     

Estimating thermal contact resistance using sensitivity analysis and regularization

Characterization of the thermal contact resistance is important in modeling of multi-component thermal systems which feature mechanically mated surfaces. Thermal resistance is phenomenologically quite complex and depends on many parameters including surface characteristics of the interfacial region and contact pressure. Although most studies seek a single value as a function of these parameters, in general, the contact resistance is non-uniform over the interface. In this paper, a technique is developed for extracting non-uniform contact resistance values from experiments in two-dimensional configurations. To begin, a two-dimensional model problem is formulated for a known contact resistance between two mated surfaces. An inverse problem is devised to estimate the variation of the contact resistance by using the BEM to determine sensitivity coefficients for specific temperature measurement points in the geometry. Temperature measured at these discrete locations can be processed to yield the contact resistance between the two mating surfaces using a simple matrix inversion technique. The inversion process is sensitive to noise and requires using a regularization technique to obtain physically possible results. The regularization technique is then extended to a genetic algorithm for performing the inverse analysis. Numerical simulations are carried out to demonstrate the approach. Random noise is used to simulate the effect of input uncertainties in measured temperatures at the sensors.     

Detection of flame radicals using light-emitting diodes

Lasers are common tools in the field of combustion diagnostics. In some respects, however, they have disadvantages. Therefore, there is a need for new light sources delivering radiation in the required wavelength regions with high stability and reliability at low cost. Light-emitting diodes (LED) in the near- and mid-infrared spectral region have proven their potential for spectroscopic applications in the past. In the present work we demonstrate the feasibility of using ultraviolet LEDs for flame diagnostics. For this purpose, OH and CH radicals are detected in premixed methane/air flames. The LED emission is found to be stable after thermal equilibrium is reached. This was the case after a warming-up period in the order of minutes. The spectral characteristics were stable during a 24-h test.