scholarly journals A New Method for Optimal Regularization Parameter Determination in the Inverse Problem of Load Identification

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Gao ◽  
Kaiping Yu ◽  
Ying Wu
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Cross Validation ◽  
Regularization Parameter ◽  
New Method ◽  
Parameter Determination ◽  
Load Identification ◽  
Noise Levels ◽  
Regularization Parameters ◽  
Curve Method

According to the regularization method in the inverse problem of load identification, a new method for determining the optimal regularization parameter is proposed. Firstly, quotient function (QF) is defined by utilizing the regularization parameter as a variable based on the least squares solution of the minimization problem. Secondly, the quotient function method (QFM) is proposed to select the optimal regularization parameter based on the quadratic programming theory. For employing the QFM, the characteristics of the values of QF with respect to the different regularization parameters are taken into consideration. Finally, numerical and experimental examples are utilized to validate the performance of the QFM. Furthermore, the Generalized Cross-Validation (GCV) method and theL-curve method are taken as the comparison methods. The results indicate that the proposed QFM is adaptive to different measuring points, noise levels, and types of dynamic load.

An Adaptive Parameter Choosing Approach for Regularization Model

2018 ◽  
Vol 32 (08) ◽  
pp. 1859013 ◽  
Author(s):  
Xiaowei Xu ◽  
Ting Bu
Keyword(s):  
Regularization Method ◽  
Numerical Experiments ◽  
Cross Validation ◽  
Regularization Parameter ◽  
Estimation Algorithm ◽  
Tikhonov Regularization Method ◽  
Practical Applications ◽  
Adaptive Parameter ◽  
Regularization Parameters ◽  
Parameter Estimation Algorithm

The choice of regularization parameters is a troublesome issue for most regularization methods, e.g. Tikhonov regularization method, total variation (TV) method, etc. An appropriate parameter for a certain regularization approach can obtain fascinating results. However, general methods of choosing parameters, e.g. Generalized Cross Validation (GCV), cannot get more precise results in practical applications. In this paper, we consider exploiting the more appropriate regularization parameter within a possible range, and apply the estimated parameter to Tikhonov model. In the meanwhile, we obtain the optimal regularization parameter by the designed criterions and evaluate the recovered solution. Moreover, referred parameter intervals and designed criterions of this method are also presented in the paper. Numerical experiments demonstrate that our method outperforms GCV method evidently for image deblurring application. Especially, the parameter estimation algorithm can also be applied to many regularization models related to pattern recognition, artificial intelligence, computer vision, etc.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

A-optimal design method to determine the regularization parameter of coseismic slip distribution inversion

2020 ◽  
Vol 221 (1) ◽  
pp. 440-450
Author(s):  
Leyang Wang ◽  
Wangwang Gu
Keyword(s):  
Optimal Design ◽  
Regularization Parameter ◽  
Design Method ◽  
Slip Distribution ◽  
Coseismic Slip ◽  
Seismic Slip ◽  
Regularization Parameters ◽  
Curve Method ◽  
Optimal Design Method

ABSTRACT The key to the inversion of a coseismic slip distribution is to determine the regularization parameters. In view of the determination of regularization parameters in seismic slip distribution inversion, the A-optimal design method is proposed in this paper. The L-curve method and A-optimal design method are used to design simulation experiments, and the inversion results show that the A-optimal design method is superior to the L-curve method in determining the regularization parameters. These two methods are also used to determine the regularization parameters of the L'Aquila and Lushan earthquake slip distribution inversions, and the results are consistent with those of other research conducted at home and abroad. Compared with the L-curve method, the A-optimal design method has the advantages of a high accuracy that does not rely on the data fitting accuracy.

Optimal Regularization Methods for Inverse Heat Transfer Problems

Volume 4 ◽  
2004 ◽  
Author(s):  
Kei Okamoto ◽  
Ben Q. Li
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Regularization Method ◽  
Cross Validation ◽  
Tikhonov Regularization Method ◽  
Inverse Algorithm ◽  
Marquardt Method ◽  
Inverse Heat Transfer ◽  
Regularization Parameters ◽  
Levenberg Marquardt

The Tikhonov regularization method has been used to find the unknown heat flux distribution along the boundary when the temperature measurements are known in the interior of a sample. Mathematically, the inverse problem is ill-posed, though physically correct, and prone to instability. This paper discusses the fundamental issues concerning the selection of optimal regularization parameters for inverse heat transfer calculations. Towards this end, a finite-element-based inverse algorithm is developed. Five different methods, that is, the maximum likelihood (ML), the ordinary cross-validation (OCV), the generalized cross-validation (GCV), the L-curve method, and the discrepancy principle, are evaluated for the purpose of determining optimal regularization parameters. An assessment of these methods is made using 1-D and 2-D inverse steady heat conduction problems where analytical solutions are available. The optimal regularization method is also compared with the Levenberg-Marquardt method for inverse heat transfer calculations. Results show that in general the Tikhonov regularization method is superior over the Levenberg-Marquardt method when the input data errors are noisy. With the appropriately determined regularization parameter, the inverse algorithm is applied to estimate the heat flux of spray cooling of a 3-D microelectronic component with an embedded heating source.

scholarly journals NEW REGULARIZATION METHOD FOR CALIBRATED POD REDUCED-ORDER MODELS

2016 ◽  
Vol 21 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Badr Abou El Majd ◽  
Laurent Cordier
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
Weighted Average ◽  
Calibration Method ◽  
Wake Flow ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Normalized Error ◽  
Curve Method

Reduced-order models based on Proper orthogonal decomposition are known to suffer from a lack of accuracy due to the truncation effect introduced by keeping only the most energetic modes. In this paper, we propose a new regularized calibration method aiming at minimizing a weighted average of normalized error, and a term measuring the change of the coefficients from their value obtained by Galerkin projection. We also determine the optimal value of the regularization parameter by analogy of the L-curve method. This paper is a sequel of [8] in which we compared various methods of calibration and introduced a Tikhonov-based regularization method. The proposed approach is assessed for a two dimensional wake flow around a cylinder, characteristic of the configurations of interest.

scholarly journals SOLUTION OF THE PROBLEM OF COMPUTATIONAL STABILITY AND CONSISTENCY OF SAMPLE ESTIMATES OF THE CORRELATION MATRIX OF OBSERVATIONS BY THE METHOD OF DYNAMIC REGULARIZATION

2019 ◽  
Vol 26 ◽  
pp. 47-60
Author(s):  
V. SKACHKOV ◽  
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Correlation Matrix ◽  
Regularization Parameter ◽  
A Priori ◽  
Computational Cost ◽  
Spectral Composition ◽  
Adaptive Antenna ◽  
Computational Stability ◽  
A Priori Uncertainty

The problem of forming sample estimates of the correlation matrix of observations that satisfy the criterion "computational stability – consistency" is considered. The variants in which the direct and inverse asymptotic forms of the correlation matrix of observations are approximated by various types of estimates formed from a sample of a fixed volume are investigated. The consistency of computationally stable estimates of the correlation matrix for their static regularization was analyzed. The contradiction inherent in the problem of regularization of the estimates with a fixed parameter is revealed. The dynamic regularization method as an alternative approach is proposed, which is based on the uniqueness theorem for solving the inverse problem with perturbed initial data. An optimal mean-square approximation algorithm has been developed for dynamic regularization of sample estimates of the correlation matrix of observations, using the law of monotonic decrease in the regularizing parameter with increasing sample size. An optimal dynamic regularization function was obtained for sample estimates of the correlation matrix under conditions of a priori uncertainty with respect to their spectral composition. The preference of this approach to the regularization of sample estimates of the correlation matrix under conditions of a priori uncertainty is proved, which allows to exclude the domain of computational instability from solving the inverse problem and obtain its solution in real time without involving prediction data and additional computational cost for finding the optimal value of the regularization parameter. The application of the dynamic regularization method is shown for solving the problem of detecting a signal at the output of an adaptive antenna array in a nondeterministic clutter and jamming environment. The results of a computational experiment that confirm the main conclusions are presented.

scholarly journals A Comparative Study of Regularization Method in Structure Load Identification

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Bingrong Miao ◽  
Feng Zhou ◽  
Chuanying Jiang ◽  
Xiangyu Chen ◽  
Shuwang Yang
Keyword(s):  
Structural Parameters ◽  
Railway Track ◽  
Load Identification ◽  
Noise Levels ◽  
Beam Structure ◽  
Identification Procedure ◽  
Vibration Data ◽  
Vibration Responses ◽  
Curve Method ◽  
Quantitative Indicators

This study aims to correlate the vibration data with quantitative indicators of structural health by comparing and validating the feasibility of identifying unknown excitation forces using output vibration responses. First, numerical analysis was performed to investigate the accuracy, convergence, and robustness of the load identification results for different noise levels, sensors numbers, and initial estimates of structural parameters. Then, the laboratory beam structure experiments were conducted. The results show that using the two identification methods Tikhonov (L-curve) and TSVD (GCV-curve) can successfully and accurately identify the different excitation forces of the external hammer. The TSVD based on GCV method has more advantages than the Tikhonov based on L-curve method. The proposed two kinds of load identification procedure based on vibration response can be applied to the safety performance evaluation of the railway track structure in future inverse problems research.

scholarly journals Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Jinlong Dong ◽  
Guogang Zhang ◽  
Zhiqiang Zhang ◽  
Yingsan Geng ◽  
Jianhua Wang
Keyword(s):  
Inverse Problem ◽  
Density Distribution ◽  
Current Density ◽  
Cross Validation ◽  
Regularization Parameter ◽  
Current Density Distribution ◽  
Optimality Criteria ◽  
Parameter Selection ◽  
Selection Strategies ◽  
Regularization Parameter Selection

Current density distribution in electric arcs inside low voltage circuit breakers is a crucial parameter for us to understand the complex physical behavior during the arcing process. In this paper, we investigate the inverse problem of reconstructing the current density distribution in arcs by inverting the magnetic fields. A simplified 2D arc chamber is considered. The aim of this paper is the computational side of the regularization method, regularization parameter selection strategies, and the estimation of systematic error. To address the ill-posedness of the inverse problem, Tikhonov regularization is analyzed, with the regularization parameter chosen by Morozov’s discrepancy principle, the L-curve, the generalized cross-validation, and the quasi-optimality criteria. The provided range of regularization parameter selection strategies is much wider than in the previous works. Effects of several features on the performance of these criteria have been investigated, including the signal-to-noise ratio, dimension of measurement space, and the measurement distance. The numerical simulations show that the generalized cross-validation and quasi-optimality criteria provide a more satisfactory performance on the robustness and accuracy. Moreover, an optimal measurement distance can be expected when using a planner sensor array to perform magnetic measurements.

Identifying an Unknown Source in the Poisson Equation with a Super Order Regularization Method

2019 ◽  
Vol 17 (07) ◽  
pp. 1950030 ◽  
Author(s):  
Zhi Li ◽  
Zhenyu Zhao ◽  
Zehong Meng ◽  
Baoqin Chen ◽  
Duan Mei
Keyword(s):  
Poisson Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
New Method ◽  
Tikhonov Regularization Method ◽  
Optimal Error ◽  
Penalty Term ◽  
The Poisson Equation ◽  
Unknown Source

This paper develops a new method to deal with the problem of identifying the unknown source in the Poisson equation. We obtain the regularization solution by the Tikhonov regularization method with a super-order penalty term. The order optimal error bounds can be obtained for various smooth conditions when we choose the regularization parameter by a discrepancy principle and the solution process of the new method is uniform. Numerical examples show that the proposed method is effective and stable.

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