Truncated Singular Value Decomposition Research Articles

A New CCERT System With Shielding for Gas-Liquid Two-Phase Flow

This work presents a new capacitively coupled electrical resistance tomography (CCERT) system with shielding for gas-liquid two-phase flow. The new system is developed from the aspects of hardware, image reconstruction algorithm and experiments. The hardware includes a 12-electrode CCERT sensor with shielding, a measurement circuit and an image reconstruction computer. The sensor with shielding is designed and the measurement circuit is developed to obtain the resistance information (real part of the impedance). Based on truncated singular value decomposition (TSVD) and density peaks clustering (DPC), a new image reconstruction algorithm is proposed. TSVD is introduced to obtain the initial image. DPC is improved and used to obtain the final image. Image reconstruction experiments were carried out with the new 12-electrode CCERT system with shielding. Experimental results show that the new CCERT system has good imaging performance. The design of the hardware is successful, and the new proposed image reconstruction algorithm is effective. Compared with the conventional image reconstruction algorithms, the proposed algorithm has higher imaging quality. The maximum relative image errors of the new system are 0.2887 under bubble flow and 0.4177 under stratified flow. Research results also indicate that the shielding design contributes to better imaging stability in interference environment.

A dynamic light scattering inversion method based on regularization matrix reconstruction for flowing aerosol measurement.

Tikhonov regularization, or truncated singular value decomposition (TSVD), is usually used for dynamic light scattering (DLS) inversion of particles in suspension. The Tikhonov regularization method uses a regularization matrix to modify all singular values in the kernel matrix. The modification of large singular values cannot effectively reduce the variance of the estimated values but may introduce bias in the solution, resulting in poor disturbance resistance in the inversion results. The TSVD method, on the other hand, truncates all small singular values, which leads to the loss of particle size information during the inversion process. The shortcomings of the two methods mentioned above do not have a significant impact on the inversion of high signal-to-noise ratio data. However, compared to the classical DLS particle size inversion for non-flowing suspended particles, the DLS inversion of flowing aerosols is more significantly affected by noise, and the extraction of particle size information is more difficult due to the effect of flow velocity, resulting in worse inversion results with increasing aerosol flow velocity for both methods. To improve the accuracy of the particle size distribution (PSD) of flowing aerosols, we introduced a kernel matrix into the regularization matrix, and based on the principles of the two methods, the spectral information of the kernel matrix was utilized to make the modification of small singular values by the regularization matrix. Correspondingly, weak or no modification was made according to the values of large singular values to reduce the introduction of bias. The inversion results of simulated and measured data indicate that the reconstruction of the regularization matrix improves the anti-disturbance performance and avoids the loss of particle size information during the regularization inversion process, thereby significantly improving the PSD accuracy, which is affected by the dual effects of flow velocity and noise in the DLS measurement of flowing particles. The peak error and distribution error of the inversion results by reconstructing the regularization matrix are lower than those of Tikhonov regularization.

Robust state of charge estimation for lithium-ion batteries in variable temperatures using truncated singular value decomposition Cubature Kalman Filter algorithm

Advanced battery management systems play a significant role in the safe and efficient use of battery energy storage systems. While the accurate and reliable estimation of state of charge (SOC) is essential for improving the utilization efficiency of batteries and extending their service life. However, the working environment of battery energy storage systems is highly complex, and temperature changes can often affect the accuracy of modeling and estimation. The traditional methods of SOC estimation often ignore the changes in temperature of the whole system, thus resulting in the cumulative errors in SOC estimation. To address this problem, an improved method is proposed in this paper to estimate the SOC of lithium-ion batteries. It requires the use of Cubature Kalman Filter (CKF) algorithm that is based on Truncated Singular Value Decomposition (TSVD). Additionally, the Forgetting Factor Recursive Least Squares (FFRLS) method is used to determine the parameters of the Thevenin equivalent model for lithium batteries. This algorithm is then validated through simulation analysis and comparison with the experimental data obtained under cyclic operating conditions at a temperature of 5 ℃, 25 ℃, and 40 ℃, respectively. The results demonstrate that the proposed TSVD-CKF algorithm is accurate in reflecting the impact of ambient temperature on the model parameters and is applicable to SOC estimation in a wide range of temperatures, showing a strong robustness.

Application of Regularization Methods in the Sky Map Reconstruction of the Tianlai Cylinder Pathfinder Array

The Tianlai cylinder pathfinder is a radio interferometer array to test 21 cm intensity mapping techniques in the post-reionization era. It works in passive drift scan mode to survey the sky visible in the northern hemisphere. To deal with the large instantaneous field of view and the spherical sky, we decompose the drift scan data into m-modes, which are linearly related to the sky intensity. The sky map is reconstructed by solving the linear interferometer equations. Due to incomplete uv coverage of the interferometer baselines, this inverse problem is usually ill-posed, and regularization method is needed for its solution. In this paper, we use simulation to investigate two frequently used regularization methods, the Truncated Singular Value Decomposition (TSVD), and the Tikhonov regularization techniques. Choosing the regularization parameter is very important for its application. We employ the generalized cross validation method and the L-curve method to determine the optimal value. We compare the resulting maps obtained with the different regularization methods, and for the different parameters derived using the different criteria. While both methods can yield good maps for a range of regularization parameters, in the Tikhonov method the suppression of noisy modes are more gradually applied, produce more smooth maps which avoids some visual artefacts in the maps generated with the TSVD method.

Developing analytical models for the post‐detection SINR of TSVD‐based techniques with imperfect CSI

SummaryIn massive multiple input multiple output (m‐MIMO) uplink, the performance gap between zero forcing (ZF) or direct pseudo‐inverse and minimum mean square error (MMSE) detection schemes increases with an increase in the load factor. To report this issue, truncated singular value decomposition (TSVD) is used in both ZF and direct pseudo‐inverse detection techniques using dynamic threshold. Moreover, the analytical models for the post‐detection signal‐to‐interference‐plus‐noise ratio (PDSINR) of TSVD‐based detection schemes are derived with imperfect channel state information (CSI), and subsequently, bit error rate (BER) is evaluated. Later, an optimum hard threshold for the TSVD‐based detection schemes is deduced from the empirical analysis. Further, the tightness of the analysis is verified through Monte Carlo trails. The simulation result shows that at lower values of average signal‐to‐noise ratio (SNR), TSVD‐based detection schemes with imperfect CSI offer comparable performance and low complexity when compared with the MMSE technique. However, the BER of TSVD‐based detection schemes decreases and gets saturated with an increase in the average SNR, whereas the BER of the MMSE detection scheme increases and approaches the BER of ZF or direct pseudo‐inverse detection schemes.

A method to calculate inverse solutions for steady open channel free-surface flow

The inverse problem of steady two-dimensional open channel free-surface flow is considered, with the focus on determining two types of disturbances: a surface pressure distribution and solid channel bottom topography. A closed-form expression for the inverse surface pressure is derived, and a linear Fredholm equation of the first kind is shown to describe the inverse topography problem, which then needs to be descretised and solved numerically. However, the equation for the channel bottom is prone to instability, so the truncated singular value decomposition (TSVD) method is proposed as a way to stabilise the associated discrete solution. The effectiveness of the TSVD method is demonstrated through several numerical examples, and its performance in the presence of error-contaminated input data is also examined. The results show that the TSVD method can recover the topography accurately from the forward free-surface problem, and provide good approximations even with noisy input data.

Interval strategy-based regularization approach for force reconstruction with multi-source uncertainties

A force reconstruction method for uncertain structures instead of direct measurement is proposed in this paper. Considering the multi-source uncertainties as unknown-but-bounded (UBB) parameters including unknown structural dynamics and measurement errors, this study proposed a novel force reconstruction method using an interval strategy-based regularization approach with truncated singular value decomposition (TSVD). To overcome the drawbacks of incompleted models and limited information, the overall modelling and inversion of the interval force are reconstructed based on the non-probabilistic uncertainty theory. First, The interval uncertainty-based kernel function for force inversion is deduced using the second-order interval perturbation method, in which the uncertainties of the reconstructed force originate from the uncertain singular values and measurement errors. Based on the regularization of the TSVD, the interval bounds of the singular values can be accurately obtained. Combining the two typical regularization approaches, a novel interval strategy-based regularization approach is proposed to optimize the best regularization parameter based on the uncertain interval distribution of singular values. This novel strategy can not only balance the two typical regularization approaches but also reduce the burden of force inversion. Two numerical examples including a truss and space capsule are applied to access the proposed method, and the estimated accurate force can prove the effectiveness.

Active control of electromagnetic fields in layered media

This article presents a numerical strategy for actively manipulating electromagnetic (EM) fields in layered media. In particular, we develop a scheme to characterize an EM source that will generate some predetermined field patterns in prescribed disjoint exterior regions in layered media. The proposed question of specifying such an EM source is not an inverse source problem (ISP) since the existence of a solution is not guaranteed. Moreover, our problem allows for the possibility of prescribing different EM fields in mutually disjoint exterior regions. This question involves a linear inverse problem that requires solving a severely ill-posed optimization problem (i.e. suffering from possible non-existence or non-uniqueness of a solution). The forward operator is defined by expressing the EM fields as a function of the current at the source using the layered media Green's function (LMGF), accounting for the physical parameters of the layered media. This results to integral equations that are then discretized using the method of moments (MoM), yielding an ill-posed system of linear equations. Unlike in ISPs, stability with respect to data is not an issue here since no data is measured. Rather, stability with respect to input current approximation is important. To get such stable solutions, we applied two regularization methods, namely, the truncated singular value decomposition (TSVD) method and the Tikhonov regularization method with the Morozov Discrepancy Principle. We performed several numerical simulations to support the theoretical framework and analyzes, and to demonstrate the accuracy and feasibility of the proposed numerical algorithms.

Analysis of underwater acoustic propagation induced by structural vibration in arctic ocean environment based on hybrid FEM-WSM solver

This paper investigates the underwater acoustic propagation characteristics induced by structural vibrations in the arctic ocean environment. For this purpose, a hybrid solver based on the finite element method (FEM) and the wave superposition method (WSM) is proposed. Leveraging the unique strengths of each method, the FEM is employed to perform structural-acoustic coupling analysis, while the WSM is used to analyze underwater acoustic propagation in the arctic ocean environment. In addition, the truncated singular value decomposition (TSVD) technique is introduced to mitigate the ill-conditioned effects of the Green's function matrix provided by the WSM. Numerical results demonstrate the accuracy and efficiency of the proposed solver in comparison to COMSOL Multiphysics finite element analysis software. Furthermore, the influence of sound speed stratification and ice layer thickness on the underwater acoustic propagation characteristics is numerically investigated in detail. These findings may provide valuable insights for the application of underwater acoustic communication and ocean exploration in the Arctic region.

Color ghost imaging based on optimized random speckles and truncated singular value decomposition

The ambient noise always harms the image quality in ghost imaging. In this letter, we propose a scheme of color ghost imaging to reduce the noise impact. In addition to the method of truncated singular value decomposition (TSVD), the measurement matrix is optimized with low-pass filters. In experiment, four (Gaussian, averaging, circular averaging, and median) filters are applied to the patterns of random speckles, which reforms the measurement matrix. By using the TSVD method, the pseudo-inverse matrix of the optimized measurement matrix is obtained and used to reconstruct the images. With proper truncation rates, the mean square errors of the reconstructed images are greatly decreased. Therefore, the image quality in our scheme is greatly improved. Further simulation analysis shows that the point spread functions are greatly optimized after filtering the random speckles. Our technique strengthens the anti-noise ability of ghost imaging, especially in complex environment.

Evaluation of the Resolution in Inverse Scattering of Dielectric Cylinders for Medical Applications.

The inverse scattering problem has numerous significant applications, including in geophysical explorations, medical imaging, and radar imaging. To achieve better performance of the imaging system, theoretical knowledge of the resolution of the algorithm is required for most of these applications. However, analytical investigations about the resolution presently feel inadequate. In order to estimate the achievable resolution, we address the point spread function (PSF) evaluation of the scattered field for a single frequency and the multi-view case both for the near and the far fields and the scalar case when the angular domain of the incident field and observation ranges is a round angle. Instead of the common free space condition, an inhomogeneous background medium, consisting of a homogeneous dielectric cylinder with a circular cross-section in free space, is assumed. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an analytical approximation of the resolution is also considered. For the sake of its comparison, the truncated singular value decomposition (TSVD) algorithm can be used to implement the exact PSF. We show how the behavior of the singular values and the resolution change by varying the permittivity of the background medium. The usefulness of the theoretical discussion is demonstrated in localizing point-like scatterers within a dielectric cylinder, so mimicking a scenario that may occur in breast cancer imaging. Numerical results are provided to validate the analytical investigations.

Enhancing Kinematic Calibration Accuracy for Parallel Manipulators Based on Truncated Total Least-Square Regularization

Abstract This article proposes a novel kinematic calibration approach to address the ill-conditioned identification matrix problem and enhance the accuracy of parallel manipulators (PMs). The kinematic calibration approach applies the truncated total least-square (T-TLS) regularization method to the inverse kinematic calibration of PMs and does not have the unit inconsistency issue and configuration selection issue. The performance of the T-TLS method in the inverse kinematic calibration of PMs that have the ill-conditioned identification matrix problem is compared to that of classical regularization methods, including the truncated singular value decomposition (TSVD) method and Tikhonov method, in simulations. Then, the kinematic calibration approach is applied to a 3-PSS/S spherical PM prototype in the real world. For the 3-PSS/S spherical PM prototype, the kinematic calibration approach can reduce the absolute angular errors of the end-effector from over 1.0 deg before calibration to less than 0.1 deg after calibration.

System Matrix Reconstruction Algorithm for Thermoacoustic Imaging With Magnetic Nanoparticles Based on Acoustic Reciprocity Theorem.

According to the acoustic reciprocity theorem (ART), we propose a system matrix reconstruction algorithm of thermoacoustic imaging for magnetic nanoparticles (MNPs) by a single-pulse magnetic field. In both cases of inhomogeneous and homogeneous acoustic velocity, we respectively derive the linear equation between the sound pressure detection value and the distribution of MNPs. The image reconstruction problem is converted to an inverse matrix solution by using the truncated singular value decomposition (TSVD) method. In forward problem, the calculated forward results are consistent with the simulated thermoacoustic signal signals. In inverse problem, we build the two-dimensional breast cancer model. The TSVD method based on the ART faithfully reflects the distribution of abnormal tissue labeled by the MNPs. In the experiment, the biological sample injected with the MNPs is used as the imaging target. The reconstructed image well reflects the cross-sectional images of the MNPs area. The TSVD method based on the ART takes into account energy attenuation and inhomogeneous acoustic velocity, and use a non-focused broadband ultrasonic transducer as the receiver to obtain a larger imaging field-of-view (FOV). By comparing the image metrics, we prove that the algorithm is superior to the traditional time reversal method. The TSVD method based on the ART can better suppress noise, which is expected to reduce the cost by reducing the number of detectors. It is of great significance for future clinical applications.

Boundary element method for contactless estimation of spatially varying internal heat transfer coefficient in circular pipes

In the present study it is proposed a solution approach of the Inverse Heat Conduction Problem (IHCP) intended at assessing the local heat transfer coefficient on the inner wall of a tube, under a forced convection problem. The estimation method is established on the Boundary Element Method (BEM) combined with the Truncated Singular Value Decomposition (TSVD) methodology, employed to manage the ill-conditioned nature of the problem. The numerical results of the direct problem, built by the BEM, are firstly validated, and consequently adopted as synthetic data inputs to resolve the IHCP and corroborate the whole assessment procedure. This approach is also tested with experimental data about forced convection problem in coiled pipes. In these geometries, the convective heat transfer coefficient changes considerably along the wall periphery and for this reason, it constitutes a perfect example to test the ability of the presented method to infer a spatially varying internal heat transfer coefficient distribution.

Regularization Methods for Extracting Doppler Broadening From Hall Thruster Plasma

Extracting Doppler broadening from laser-induced fluorescence (LIF) measurements is essential for determining species temperature in the plasma. The fluorescence signal is a convolution of many broadening effects of which Doppler broadening is the most significant, but several others contribute as well: collisional (pressure) broadening (particle-to-particle interactions), natural lifetime broadening, and power broadening. The process that extracts Doppler broadening is an <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ill-posed</b> inverse problem. The regularization approach addresses the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ill-posed</b> problem to ensure a unique and stable solution with the best possible accuracy. This study reported the Xe II temperature by applying truncated singular value decomposition (TSVD) and Tikhonov regularization (zeroth, first and second-order) methods for deconvolution of Doppler broadening. Tikhonov first-order and TSVD employing the generalized cross-validation (GCV) method provide accurate reconstruction within 2.0%–8.0% of the actual measurements.

Three dimensional seismic data reconstruction based on truncated nuclear norm

The rank reduction method is widely used to reconstruct three dimensional (3D) seismic data. The traditional multichannel singular spectrum analysis (MSSA) utilizes truncated singular value decomposition (TSVD) to approximately solve the rank function of the block Hankel structure matrix constructed by frequency slices of seismic data. However, the TSVD algorithm discards all nonzero singular values except a few largest singular values and ignores the useful seismic information in small nonzero singular values. To further utilize the seismic data information contained in these small singular values, this paper proposes a method to recover 3D seismic data with truncated nuclear norm (TNN). The proposed method imposes different constraints on large singular values and small singular values. Essentially, the TNN is closer to the rank function than the TSVD. Finally, the proposed model is addressed by the alternating direction method of multipliers, and closed-form solutions are used to update the optimization variables. The experimental results demonstrate that the proposed method achieves superior reconstruction results than the traditional MSSA method.

New regularization techniques for ill-conditioning problems and their applications: Choices of regularization parameters

This is a continued study of Li et al. (2011) for the regularization of ill-conditioning problems, where the minimal singular value σmin of discrete matrices is infinitesimal. To remove the effects of high frequency caused by the singular vector of σmin better, we combine the Tikhonov regularization (TR) with the truncated singular value decomposition (TSVD) (simply denoted by T-TR). New computational formulas of the traditional condition number (Cond) and the effective condition number (Cond_eff) are derived, and a brief error analysis is made. The regularization parameter λ is involved in the TR, and its better (or optimal) choices are essential in both theory and computation. For ill-conditioning problems, stability is more imperative than accuracy. The most important criterion is to reduce the Cond. Consider the linear algebraic equations Ax = b. The optimal regularization parameter for the TR is derived as λ=σmaxσmin, where σmax and σmin(>0) are the maximal and the minimal singular values of matrix A, respectively. The L-curve techniques emphasize the ill-conditioning of ‖xλ‖ (see Hansen (1998)). Note that the norm ‖xλ‖ is only part of the Cond in stability. In fact, by using regularization, the Cond may be greatly reduced, while the errors do not increase much. The second important criterion is for both stability and solution accuracy. The sensitivity index proposed in Zhang et al. (2021) indicates the severity of ill-conditioning via accuracy for comparing different numerical methods and techniques. The sensitivity index is effective not only to select the source nodes in the method of fundamental solutions (MFS), but also to select the regularization parameter in the TR. The regularization choices of the T-TR are similar to those of the TR because differences of the Cond and the errors between the TR and the T-TR are insignificant. Numerical experiments are reported by the MFS for Laplace’s equation, to support the new regularization techniques proposed. For data fitting, imaging processing, pattern recognition, and machine learning, the ‖xλ‖ is, however, more important than Cond for ill-conditioning. The L-curve techniques are used in wide applications, and discussed in Hansen (1998). The parameter choices are rather complicated if the crossover region is not small. In this paper, we apply the sensitivity index to L-curves, the optimal parameter λL−curve can be found by the minimal sensitivity index. The λL−curve may also be found by the lowest line that passes the origin. The new algorithms are simple and effective.

Evaluation of the Number of Degrees of Freedom of the Field Scattered by a 3D Geometry.

The solution to an ill-posed linear inverse problem requires the use of regularization methods to achieve a stable approximation solution. One powerful approach is the truncated singular value decomposition (TSVD), but it requires an appropriate choice of the truncation level. One suitable option is to take into account the number of degrees of freedom (NDF) of the scattered field, which is defined by the step-like behavior of the singular values of the relevant operator. Then, the NDF can be estimated as the number of singular values preceding the knee or the exponential decay. Therefore, an analytical estimation of the NDF is significant for obtaining a stable, regularized solution. This paper addresses the analytical estimation of the NDF of the field scattered by the surface of a cube geometry for a single frequency and the multi-view case in the far-zone. In addition, a method is proposed to find the minimum numbers of plane waves and their directions to achieve the total estimated NDF. The main results are that the NDF is related to the measure of the surface of the cube and can be achieved by only considering a limited number of impinging plane waves. The efficiency of the theoretical discussion is demonstrated through a reconstruction application for microwave tomography of a dielectric object. Numerical examples are provided to confirm the theoretical results.

Prediction and visualization of gene modulated ultralow cadmium accumulation in brown rice grains by hyperspectral imaging

Monitoring (including prediction and visualization) the gene modulated cadmium (Cd) accumulation in rice grains is one of the most important steps for identification of key transporter genes responsible for grain Cd accumulation and breeding low grain-Cd-accumulating rice cultivars. A method to predict and visualize the gene modulated ultralow Cd accumulation in brown rice grains based on the hyperspectral image (HSI) technology is proposed in this study. Firstly, the Vis-NIR HSIs of brown rice grain samples with 48Cd content levels induced by gene modulation (ranging from 0.0637 to 0.1845 mg/kg) are collected using HSI system. Then, Kernel-ridge (KRR) and random forest (RFR) regression models based on full spectral data and the data after feature dimension reduction (FDR) with kernel principal component analysis (KPCA) and truncated singular value decomposition (TSVD) algorithms are established to predict the Cd contents. RFR model shows poor performance due to the over-fitting based on the full spectral data, while the KRR model can obtain a good predict accuracy with Rp2 of 0.9035, RMSEP of 0.0037 and RPD of 3.278. After the FDR of the full spectral data, the RFR model combined with TSVD reaches the optimum prediction accuracy with Rp2 of 0.9056, RMSEP of 0.0074 and RPD of 3.318, and the best prediction precision of KRR model can also be further enhanced by TSVD with Rp2 of 0.9224, RMSEP of 0.0067 and RPD of 3.512. Finally, the visualization of the predicted Cd accumulation in brown rice grains are realized based on the best regression model (KRR + TSVD). The results of this work indicate that Vis-NIR HSI has great potential for detection and visualization gene modulation induced ultralow Cd accumulation and transport in rice crops.

A New Algorithm for Ill-Posed Problem of GNSS-Based Ionospheric Tomography

Ill-posedness of GNSS-based ionospheric tomography affects the stability and the accuracy of the inversion results. Truncated singular value decomposition (TSVD) is a common algorithm of ionospheric tomography reconstruction. However, the TSVD method usually has low inversion accuracy and reconstruction efficiency. To resolve the above problem, a truncated mapping singular value decomposition (TMSVD) algorithm is presented to improve the reconstructed accuracy and computational efficiency. To authenticate the effectiveness and the advantages of the TMSVD algorithm, a numerical test scheme is devised. Finally, ionospheric temporal–spatial variations of the selected reconstructed region are studied using the GNSS observations under different geomagnetic conditions. The reconstructed results of TMSVD can accurately reflect semiannual anomalies, diurnal variations, and geomagnetic storm effects. In contrast with the ionosonde data, it is found that the reconstructed profiles of the TMSVD method are more consistent with than those of the IRI 2016. The study suggests that TMSVD is an efficient algorithm for the tomographic reconstruction of ionospheric electron density (IED).