Truncated Singular Value Decomposition Regularization Research Articles

TSVD Regularization-Parameter Selection Method Based on Wilson-θ and Its Application to Vertical Wheel-Rail Force Identification of Rail Vehicles

A parameter-selection method is proposed to improve the accuracy of the truncated singular value decomposition (TSVD) method, which is based on the Wilson-θ method and the principle of minimum response error, for dynamic load identification. First, using the Green kernel-function matrix, the dynamic load-identification model of the multi-degree-of-freedom system is established. Second, the response corresponding to the dynamic load is identified using the Wilson-θ method, and the minimum error between the response and the input response is obtained. Then, the best regularization parameters are obtained, and the dynamic load is identified using the TSVD regularization method. Finally, the SIMPACK dynamic model of a rail vehicle is established. Taking the German high-interference spectrum as the input, the axle-box displacement and the vertical wheel-rail force of each wheelset at speeds of 100, 160, and 200 km/h are simulated. Taking the simulated axle-box displacement response with 0%, 5%, and 10% noise as the input, the proposed load-identification model and regularization-parameter selection method are used to identify the vertical wheel-rail force of a rail vehicle. The effects of different track spectra on the identification results are considered. The results indicate that this method has a high identification accuracy for the wheel-rail vertical dynamic load. With an increase in the vehicle speed, the correlation coefficient for identifying the dynamic load decreases, but the correlation remains strong. At the speed of 200 km/h, when the input response noise level is 0%, the dynamic load identification correlation coefficient is 0.9556, which corresponds to extremely strong correlation. When the input response contains 5% noise, this method has stronger robustness than L-curve method, and the dynamic load identification correlation coefficient is 0.6354, which corresponds to strong correlation. The proposed load-identification model and regularization-parameter selection method have important theoretical and engineering application value for wheel-rail force monitoring and safety assessment of running trains.

An Adaptive Regularized Solution to Inverse Ill-Posed Models

The ill-posed models are widely encountered in various inversions of geodesy and remote sensing. The regularization approaches can significantly stabilize the solution to ill-posed models since the high-frequency noise is effectively suppressed. Although the famous Tikhonov regularization and truncated singular value decomposition (TSVD) regularization have been widely applied in various geodetic applications, there still remain theoretical drawbacks for either single regularization. For Tikhonov regularization, given a regularization parameter, the low-frequency terms are over regularized, and high-frequency terms are under regularized. For TSVD regularization, some medium-frequency terms will be mistaken for high-frequency terms to be truncated and the hidden signals will be lost. For this reason, we propose an adaptive regularized solution in spectral form, which adaptively divides the terms of different frequencies into three kinds: (i) the low-frequency terms are not regularized; (ii) the medium-frequency terms are regularized by the Tikhonov method; (iii) the high-frequency terms are regularized by TSVD method. The analytical conditions for determining the term sets are derived based on the criteria that the introduced biases should be smaller than the reduced errors, in other words, the mean squared error (MSE) should be reduced. The two examples are presented to demonstrate the performance of our adaptive regularization. The first numerical example is solving the Fredholm integral equation of the first kind, which is widely encountered in remote sensing inversions. The simulations clearly demonstrate that the adaptive regularized solution can improve the MSE of ordinary Tikhonov and TSVD regularized functions by 25.00% and 9.09%, respectively; In the second example, we apply the new method to investigate the mass variation of the Yangtze River Basin based on the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field model. The Tongji-Grace 2018 monthly gravity field solutions from April 2002 to December 2016 are used to construct the mascon observation equation. The results show that our method also outperforms the ordinary Tikhonov and TSVD regularized solutions, with mean MSE reductions of about 13.40% and 11.69%, respectively. Furthermore, the spatial resolution of secular trend derived by our method are improved and the signal-to-noise ratio (SNR) of mass variation series is higher than the other two regularizations.

A Novel Localization Method of Partial Discharge Sources in Substations Based on UHF Antenna and TSVD Regularization

Ultra-high frequency (UHF) technology has been widely used in partial discharge (PD) source localization based on the time difference of arrival (TDOA). However, the TDOA errors will then lead to the ill-conditioned problem of matrix A. A larger condition number will then lead to a more ill severe condition of linear equations <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AX</i> =b, and resulting in a decline in the localization accuracy. In this paper, a novel method for measuring the PD source has been developed based on the UHF antenna and the truncated singular value decomposition (TSVD) regularization method. First, the PD signals are obtained through the cooperation of five UHF antennas. Second, a spherical model linear transformation method is proposed for non-linear measurement equations. Then, to address the ill-conditioned problem of the measurement equations caused by the UHF antenna installation position, the linear measurement equations are improved with the pretreatment method. Finally, the TSVD regularization algorithm is applied to solve the UHF measurement equations with the ill-conditioned problems. The regularization parameter of the TSVD is then optimized by generalized cross-validation (GCV). In addition, the algorithm is verified by theoretical verification and field tests. The experimental results reveal that the proposed localization method can accurately locate the PD source.

Truncated Singular Value Decomposition Regularization for Estimating Terrestrial Water Storage Changes Using GPS: A Case Study over Taiwan

It is a typical ill-conditioned problem to invert GPS-measured loading deformations for terrestrial water storage (TWS) changes. While previous studies commonly applied the 2nd-order Tikhonov regularization, we demonstrate the truncated singular value decomposition (TSVD) regularization can also be applied to solve the inversion problem. Given the fact that a regularized estimate is always biased, it is valuable to obtain estimates with different methods for better assessing the uncertainty in the solution. We also show the general cross validation (GCV) can be applied to select the truncation term for the TSVD regularization, producing a solution that minimizes predictive mean-square errors. Analyzing decade-long GPS position time series over Taiwan, we apply the TSVD regularization to estimate mean annual TWS variations for Taiwan. Our results show that the TSVD estimates can sufficiently fit the GPS-measured annual displacements, resulting in randomly distributed displacement residuals with a zero mean and small standard deviation (around 0.1 cm). On the island-wide scale, the GPS-inferred annual TWS variation is consistent with the general seasonal cycle of precipitations. However, on smaller spatial scales, we observe significant differences between the TWS changes estimated by GPS and simulated by GLDAS land surface models in terms of spatiotemporal pattern and magnitude. Based on the results, we discuss some challenges in the characterization of TWS variations using GPS observations over Taiwan.

A Method for the Localization of Partial Discharge Sources in Transformers Using TDOA and Truncated Singular Value Decomposition

Ultrasonic sensor detection has been widely used in partial discharge (PD) location. However, this PD localization method based on an ultrasonic sensor is computationally expensive, inaccurate, and sensitive to the time differences. A localization method for PD based on time difference of arrival (TDOA) and truncated singular value decomposition (TSVD) regularization is proposed in this paper, which can effectively ensure the accuracy of PD location in a transformer. First, the PD signal is received by the ultrasonic sensor, and an oscilloscope is used to receive the ultrasonic signal from the operational amplifier. Second, a spherical linear transformation method is used to transform the general nonlinear localization equations. Third, a TSVD regularization method is used to solve the linear equations. Then, to eliminate the influence of coordinate selection on the matrix ${b}$ , the matrix ${b}$ is centred. By dealing with the balance of matrix ${A}$ , the ill-conditioned degree of linear equations is weakened. Finally, the optimal solution serves as a good approximation of the original problem. The proposed method is applied to the PD location in a transformer, and the localization results are compared with direct Newton iteration and direct TSVD methods. A comparison of the experimental results shows that the method in this paper can reduce the ill-conditioned degree of a linear equation, and effectively weaken the influence of time difference error on the solution. The localization results of this method are better than the other two methods.

Reconstruction of Current Density Distribution in Low-Voltage Switching Devices Through TSVD and TGSVD Regularization

This article examined two regularization methods for reconstructing the arc current density distribution of a switching device in an arc-quenching chamber. The finite-element method (FEM) was used to discretize the current-flow region of the arc-quenching chamber. Both truncated generalized singular value decomposition (TGSVD) and truncated singular value decomposition (TSVD) regularization were used to reconstruct the current density distribution. TGSVD addressed the poor reconstruction accuracy caused by the ill-posedness of the Biot–Savart operator. Simulation results showed that the reconstruction of current density distribution was more accurate with higher signal-to-noise ratios (SNRs) and lower mean-square errors (MSEs). Both regularization methods were accurate with low noise levels, although the reconstruction accuracy of TGSVD was more sensitive to noise than that of TSVD regularization.

A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau

Limited by the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) measurement principle and sensors, the spatial resolution of mass flux solutions is about 2–3° in mid-latitudes at monthly intervals. To retrieve a mass flux solution in the Tibetan Plateau (TP) with better visual spatial resolution, we combined truncated singular value decomposition (TSVD) and Tikhonov regularization to solve for a mascon modeling. The monthly mass flux parameters resolved at 1° are smoothed to about 2° by truncating the eigen-spectrum of the normal equation (i.e., using the TSVD approach), and then Tikhonov regularization is applied to the truncated normal equation. As a result, the terms beyond the native resolution of GRACE/GRACE-FO data are truncated, and the errors in higher degree and order components are dampened by Tikhonov regularization. In terms of root mean squared errors, the improvements are 27.2% and 12.7% for the combined method over TSVD and Tikhonov regularization, respectively. We confirm a decreasing secular trend with −5.6 ± 4.2 Gt/year for the entire TP and provide maps with 1° resolution from April 2002 to April 2019, generated with the combined TSVD and Tikhonov regularization method.

On TSVD regularization for a Broyden-type algorithm

Abstract An applied inverse problem in the form of a nonlinear operator equation on a pair of two Hilbert spaces is considered. In the lack of stability, a version of Broyden’s method regularized by truncated singular value decomposition (TSVD) is proposed in order to suppress the potential noise propagation and to reduce the cost associated with storage and inversion of the Fréchet derivative operator. An attractive feature of this study is that it does not rely on any restrictions on the nonlinearity of the operator and/or on the spectrum of the Fréchet derivative. For the numerical analysis of the proposed algorithm, a parameter identification problem in epidemiology is investigated.

Passive Shimming of MRI Static Magnetic Field Using Regularization of Truncated Singular Value Decomposition.

Purpose:To develop a new shimming calculation method, which can calculate iron piece placements rapidly to make the magnetic field homogeneous at intended homogeneity and then to make the shimming working time short.Materials and Methods:The shimming calculation yields magnetic moment (MM) distribution, which is calculated by the truncated singular value decomposition (SVD) from the measured magnetic field. The MM distribution is described by a superposition of eigenmodes obtained by SVD of a response matrix from the moment distributions to magnetic fields at the field of view (FOV). The homogeneity is regulated by a truncation number of the superposed eigenmodes. The magnetic moments are converted into iron volumes with the assumption of saturated magnetization and the iron pieces are placed according to the calculation results. Since the SVD calculation can be done in advance, the computational time at the shimming site is short.Results:Trial applications on a 0.5T magnetic resonance imaging (MRI) magnet were done using the new shimming calculation method, which was proved to work well. However, since the iron piece volumes had tolerances, the work was repeated until enough homogeneity was obtained. As a result, an intended homogeneity of 8.9 ppm (peak-to-peak) on 40 cm diameter spherical surface was successfully obtained from measured homogeneity of 543 ppm with short computational and working time.Conclusion:The test shimming work showed that the developed shimming calculation method with truncated SVD regularization is applicable to the shimming work on the MRI magnets.

Inversion of photon correlation spectroscopy based on truncated singular value decomposition and cascadic multigrid technology

For the low accuracy of single-scale inversion method in photon correlation spectroscopy technology, a cascadic multigrid (CMG)-truncated singular value decomposition (TSVD) inversion method that combines the TSVD regularization with CMG technology is proposed. This method decomposes the original problem into several subproblems in different scale grid space. According to the particle sizes inverted from the coarsest scale to the finest scale, the solution of an original inversion problem can be obtained. For the inversion of each subproblem, TSVD method is used. The simulation and experimental data were respectively inverted by TSVD and CMG-TSVD methods. The inversion results demonstrate that the CMG-TSVD method has higher accuracy, more strong noise immunity and better smoothness than the TSVD method.

A New Method for TSVD Regularization Truncated Parameter Selection

The truncated singular value decomposition (TSVD) regularization applied in ill-posed problem is studied. Through mathematical analysis, a new method for truncated parameter selection which is applied in TSVD regularization is proposed. In the new method, all the local optimal truncated parameters are selected first by taking into account the interval estimation of the observation noises; then the optimal truncated parameter is selected from the local optimal ones. While comparing the new method with the traditional generalized cross-validation (GCV) andLcurve methods, a random ill-posed matrices simulation approach is developed in order to make the comparison as statistically meaningful as possible. Simulation experiments have shown that the solutions applied with the new method have the smallest mean square errors, and the computational cost of the new algorithm is the least.

INVERSE ANALYSIS APPLIED TO AN ILLUMINATION DESIGN

This work investigates the application of the inverse analysis to an illumination design of a three-dimensional rectangular enclosure. The problem consists of finding the luminous fluxes on the light source elements, located on the top of the enclosure, that satisfies a prescribed uniform luminous flux on the design surface, located on the bottom surface. The solution assumes that all the surfaces emit and reflect diffusely, and that the hemispheric spectral emissivities are wavelength independent in the visible region of the spectrum. The inverse analysis is described by a system of linear equations that is expected to be ill-conditioned since it involves the solution of a Fredholm integral equation of the first kind. To tackle the ill-conditioned system of equations, the TSVD (Truncated Singular Value Decomposition) regularization method is applied. In addition to presenting a methodology to solve for the luminous exchanges in a enclosure, starting from a thermal radiation energy balance, this work considers two design cases: one in which the light source elements cover the entire top surface; and one in which a reduced number of light sources are considered, a more practical solution. In both cases, the proposed inverse design is capable of providing a solution that satisfies the prescribed luminous flux on the design surface within average and maximum errors less than 1.0 % and 5.0 %, respectively.