Singular Value Decomposition Regularization Research Articles

An improved composite ship magnetic field model with ellipsoid and magnetic dipole arrays.

In order to simultaneously maintain the ship magnetic field modeling accuracy, reduce the number of coefficient matrix conditions and the model computational complexity, an improved composite model is designed by introducing the magnetic dipole array model with a single-axis magnetic moment on the basis of the hybrid ellipsoid and magnetic dipole array model. First, the improved composite model of the ship's magnetic field is established based on the magnetic dipole array model with 3-axis magnetic moment, the magnetic dipole array model with only x-axis magnetic moment, and the ellipsoid model. Secondly, the set of equations for calculating the magnetic moments of the composite model is established, and for the problem of solving the pathological set of equations, the least-squares estimation, stepwise regression method, Tikhonov, and truncated singular value decomposition regularization methods are introduced in terms of the magnetic field, and generalized cross-validation is used to solve the optimal regularization parameters. Finally, a ship model test is designed to compare and analyze the effectiveness of the composite and hybrid models in four aspects: the number of coefficient matrix conditions of the model equation set, the relative error of magnetic field fitting, the relative error of magnetic field extrapolation, and the computational time complexity. The modeling results based on the ship model test data show that the composite model can be used for modeling the magnetic field of ships, and compared with the hybrid model, it reduces the number of coefficient matrix conditions and improves the computational efficiency on the basis of retaining a higher modeling accuracy, and it can be effectively applied in related scientific research and engineering.

Contactless Microwave Tomography via MIMO GPR

This paper presents an imaging approach for Multiple Input Multiple Output Ground Penetrating Radar (MIMO GPR) systems working in down-looking contactless mode. The approach exploits a linear approximation of the scattering phenomenon and is based on a ray-based propagation model, which takes into account the presence of the air-soil interface. Accordingly, the Interface Reflection Point concept is extended to the case of MIMO GPR. The proposed approach performs the imaging in the 2D scalar case and applies the Truncated Singular Value Decomposition regularization scheme to perform the inversion. The effectiveness of the approach is assessed by processing synthetic and real data. The real data are referred to the lunar soil and have been collected by means of the Lunar Regolith Penetrating Radar, installed on the Chang’E-5 lander.

Error Modeling and Anomaly Detection of Smart Electricity Meter Using TSVD+L Method

Using remote data analysis to estimate smart electricity meters (SMs) and detect SMs&#x2019; anomaly, has aroused considerable discussion in power industry, because its lower cost and higher efficiency compared to traditional field calibration. However, the trouble of lacking topology and parameter of the distribution grid and the ill-posed problem in the SMs&#x2019; error estimation and anomaly detection (AD) are not well resolved in most energy conservation theorem-based SMs AD methods. This paper presents a sorted Top-N anomaly detection mechanism to generate a list of suspicious anomalous SMs. The error estimation model (EEM) only using SMs electricity consumption data is investigated. The truncated singular value decomposition regularization with L-curve optimization (TSVD+L) method is proposed to address the model&#x2019;s ill-posedness. Three data processing modes, namely one-pot mode, segmentation mode and sliding window technique (SWT), are suggested to obtain multiple calculation results for SMs error comprehensive evaluation. The top <i>N</i>% SMs in error sequence is proposed for onsite calibration instead of full inspection. The effectiveness and practicality of the proposed method are verified through both simulation case and practical distribution network application. The results show that the proposed method has higher accuracy in SMs anomaly detection, compared with the ordinary least squares (OLS) method, recursive least squares (RLS) method, and Tikhonov regularization (Tik) method.

Robust adaptive beamforming algorithm based on damped singular value decomposition regularization

When adaptive beamformers are applied to actual problems, various model mismatches are present in the sample covariance matrix and signal steering vector, which severely deteriorates the performance of the beamformer. To address this issue, we develop a novel adaptive beamforming algorithm in which the weight vector is a linear combination of the presumed steering vector and basis vectors of the orthogonal supplement subspace of the steering vector. Then, we convert the common constrained optimization problem into an unconstrained one to establish the new beamformer. Moreover, we use damped singular value decomposition regularization, which employs the L-curve method to adaptively determine the regularized factor (loading level), for suppressing the effects of model mismatches. In addition, we perform simulations of the proposed algorithm and other existing beamforming algorithms by considering several commonly encountered mismatches (e.g., the direction-of-arrival mismatch, sensor gain, phase error and location perturbation, coherent local scattering, and mutual coupling) and demonstrate the superior performance of the new beamforming algorithm.

Observation and Inversion of Aerosol Particle Size Distribution over Yinchuan Area

Particle size distribution is one of the important microphysical parameters to characterize the aerosol properties. The aerosol optical depth is used as the function of wavelength to study the particle size distribution of whole atmospheric column. However, the inversion equation of the particle size distribution from the aerosol optical depth belongs to the Fredholm integral equation of the first kind, which is usually ill-conditioned. To overcome this drawback, the integral equation is first discretized directly by using the complex trapezoid formula. Then, the corresponding parameters are selected by the L curve method. Finally the truncated singular value decomposition regularization method is employed to regularize the discrete equation and retrieve the particle size distribution. To verify the feasibility of the algorithm, the aerosol optical depths taken by a sun photometer CE318 over Yinchuan area in four seasons, as well as hazy, sunny, floating dusty and blowing dusty days, were used to retrieve the particle size distribution. In order to verify the effect of truncated singular value decomposition algorithm, the Tikhonov regularization algorithm was also adopted to retrieve the aerosol PSD. By comparing the errors of the two regularizations, the truncated singular value decomposition regularization algorithm has a better retrieval effect. Moreover, to understand intuitively the sources of aerosol particles, the backward trajectory was used to track the source. The experiment results show that the truncated singular value decomposition regularization method is an effective method to retrieve the particle size distribution from aerosol optical depth.

Method of fundamental solutions and a high order continuation for bifurcation analysis within Föppl-von Karman plate theory

An efficient numerical model is developed in this work for the bifurcation analysis within Föppl-von Karman plate theory. This procedure is based on the use of the method of fundamental solutions in a high order continuation method that relies on Taylor series expansion with respect to a path parameter. This numerical model has an adaptive step length, which is effective especially for solving nonlinear problems and detecting bifurcation points. Despite of the huge application field of nonlinear elasticity, the method of fundamental solution was very rarely applied in this field and never to a nonlinear elastic plate model. The governing equations are strongly formulated in terms of the two unknowns: the deflection and the stress function. The singular value decomposition regularization is used to overcome the difficulty of the resulting system ill-conditioning. The accuracy and efficiency of the numerical model are illustrated on buckling numerical examples.

Theory of adaptive SVD regularization for deep neural networks

Deep networks can learn complex problems, however, they suffer from overfitting. To solve this problem, regularization methods have been proposed that are not adaptable to the dynamic changes in the training process. With a different approach, this paper presents a regularization method based on the Singular Value Decomposition (SVD) that adjusts the learning model adaptively. To this end, the overfitting can be evaluated by condition numbers of the synaptic matrices. When the overfitting is high, the matrices are substituted with their SVD approximations. Some theoretical results are derived to show the performance of this regularization method. It is proved that SVD approximation cannot solve overfitting after several iterations. Thus, a new Tikhonov term is added to the loss function to converge the synaptic weights to the SVD approximation of the best-found results. Following this approach, an Adaptive SVD Regularization (ASR) is proposed to adjust the learning model with respect to the dynamic training characteristics. ASR results are visualized to show how ASR overcomes overfitting. The different configurations of Convolutional Neural Networks (CNN) are implemented with different augmentation schemes to compare ASR with state-of-the-art regularization methods. The results show that on MNIST, F-MNIST, SVHN, CIFAR-10 and CIFAR-100, the accuracies of ASR are 99.4%, 95.7%, 97.1%, 93.2% and 55.6%, respectively. Although ASR improves the overfitting and validation loss, its elapsed time is not significantly greater than the learning without regularization.

Source strength identification problem for the three-dimensional inverse heat conduction equations

ABSTRACTIn this paper, we consider the source strength identification problem for the three-dimensional inverse heat conduction equations. The problem is to determine an unknown heat source strength from the measurement data for a specified location. In this process, the direct problem is solved by applying the Green’s function method. Then, this problem can be converted into a Volterra integral equation of the first kind. Further, the Tikhonov and truncated singular value decomposition regularization methods are developed to identify the unknown source strength based on the discrepancy principle for choosing the regularization parameter. Finally, numerical examples are presented to show the feasibility and efficiency of the proposed method.

The inverse methods based on S-FEMs with an adaptive SVD regularization technique for solving Cauchy inverse heat transfer problems

This paper presents three inverse models based on S-FEMs with adaptive singular value decomposition (SVD) regularization technique for solving inverse heat transfer problems of Cauchy type with noisy input data “measured” on the domain boundary. The smoothed finite element methods (S-FEMs) are used as forward solvers, including edge-based S-FEM (ES-FEM), node-based S-FEM (NS-FEM) and cell-based S-FEM (CS-FEM that is the same as the standard FEM using T3 elements). First, the full-size S-FEM system equation is created based on the S-FEM theory. A set of local nodal temperature equations is then extracted from the full-size equations through a matrix partitioning operation, considering the nodes related to the heat flux on the Cauchy boundary. The Fourier heat convection theory is used to establish a relationship between the local nodal temperatures and the known heat fluxes on the Cauchy boundary. This procedure effectively converts a Cauchy type inverse problem to a forward-like problem with a set of ill-posed equations for inverse analyses. In order to mitigate the ill-posedness and to achieve a reliable and accurate inverse solution, we propose an adaptive SVD technique to regularize the solution, through minimizing the error between the approximate heat fluxes of the inverse model and the given (“measured”) flux values on the Neumann boundaries. The proposed adaptive procedure effectively determines the number of small singular values to be deleted, and hence the noises in the input data cannot be magnified. The present techniques are tested using a number of Cauchy type heat transfer problems, and it is concluded that our inverse procedure is much more effective compared to the widely used Tikhonov regularization technique, and it is a systematic procedure for accurate, reliable and stable inverse solutions with noisy input data.

The blackbody radiation inversion problem: A numerical perspective utilizing Bernstein polynomials

The blackbody radiation inversion (BRI) problem can be formulated as a Fredholm integral equation of the first kind for which area as a function of temperature must be solved. This paper utilizes the Bernstein polynomials as a basis to approximate the solution of the BRI problem. By discretization, the BRI problem can be reduced to an ill-posed algebraic equation whose right-hand side is the measured power spectrum. The error in the right-hand side commonly causes a large variation in the solution of BRI. In order to eliminate the numerical instability, the truncated singular value decomposition regularization method is employed for solving the resulting linear system and an error analysis is made to estimate the upper bound between the recovered and the exact area temperature distribution. Numerical examples also shed light on the efficiency and accuracy of this method.

噪声动态光散射数据Tikhonov与截断奇异值正则化反演

噪声动态光散射数据Tikhonov与截断奇异值正则化反演

Static Magnetic Field Shimming Calculation Using TSVD Regularization With Constraints of Iron Piece Placements

A new shimming calculation method for a static magnetic field in cylindrical magnets has been developed and the feasibility was confirmed by a test shimming. The calculation is based on truncated singular value decomposition regularization to solve a linear system of equations with a response matrix from iron piece volumes in shim-pockets to the magnetic field distribution. Singular value decomposition obtains eigenmodes which are numbered in a descending sequence of the singular. Low-ordered (large singular value) eigenmodes are chosen and summed to get the iron piece placements, considering the number of the main coil blocks in the magnet and tuning their eigenmodes strengths. High-ordered (small singular value) eigenmodes, which have a small magnetic field, are utilized to make the iron volumes within constraints in each pocket using iterative calculations. During the test shimming, iron pieces were placed according to the calculation and good homogeneities were obtained. The shimmed magnetic field agreed well with the field predicted by the calculation. These results have shown that the developed shimming calculation method works well to get homogeneous magnetic field.

Optimization methods for regularization-based ill-posed problems: a survey and a multi-objective framework

Ill-posed problems are widely existed in signal processing. In this paper, we review popular regularization models such as truncated singular value decomposition regularization, iterative regularization, variational regularization. Meanwhile, we also retrospect popular optimization approaches and regularization parameter choice methods. In fact, the regularization problem is inherently a multi-objective problem. The traditional methods usually combine the fidelity term and the regularization term into a single-objective with regularization parameters, which are difficult to tune. Therefore, we propose a multi-objective framework for ill-posed problems, which can handle complex features of problem such as non-convexity, discontinuity. In this framework, the fidelity term and regularization term are optimized simultaneously to gain more insights into the ill-posed problems. A case study on signal recovery shows the effectiveness of the multi-objective framework for ill-posed problems.

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.

Retrieval of the Ocean Skin Temperature Profiles From Measurements of Infrared Hyperspectral Radiometers—Part I: Derivation of an Algorithm

The retrieval of vertical profiles of the sea surface temperature through the electromagnetic and thermal skin layers is demonstrated. The inversion equation applied to the high-resolution radiance spectra measurements obtained from hyperspectral infrared spectrometers is introduced and is known to be nonlinear and ill-conditioned. To constrain the solution and reduce the errors in this ill-conditioned retrieval problem, the truncated singular value decomposition regularization technique is adopted. This technique's feasibility is demonstrated on synthetic data with and without the addition of noise. Results sometimes include an unphysical temperature inversion in the solution which was avoided by iteratively adjusting the lower boundary condition. The presence of noise required an additional step of averaging the radiance spectrum into 11 wavenumber intervals to solve the issue of an overconstrained problem and reduce the errors involved.

Retrieval of the Ocean Skin Temperature Profiles From Measurements of Infrared Hyperspectral Radiometers—Part II: Field Data Analysis

To obtain the vertical temperature profile of the thermal skin layer on the aqueous side of the ocean–atmosphere interface from radiance measurements of hyperspectral infrared radiometers, regularization methods are used to address the nonlinearity and ill-conditioning of the inverse problem associated with the retrieval. This paper demonstrates the truncated singular value decomposition regularization technique on field data sets obtained from the Marine-Atmospheric Emitted Radiance Interferometer (M-AERI). The M-AERI takes highly accurate and spectrally resolved radiance measurements in the infrared, which allows for the sensing of temperature values within thermal skin layer depths of less than 1 mm below the ocean surface. Results showed temperature inversions found in the solution, but these are deemed unphysical, based on previous tests on synthetic data and by a scaling analysis using the Rayleigh number, which are presented here. The importance of sea surface emissivity for performing the atmospheric correction to M-AERI measurements to derive surface emissivity spectra is also discussed. The retrieved profiles exhibit a smooth exponential-like structure that agrees well with theory.

Structural finite element model updating using incomplete ambient vibration modal data

This paper presents an effective approach for updating finite element dynamic model from incomplete modal data identified from ambient vibration measurements. The proposed method is based on the relationship between the perturbation of structural parameters such as stiffness and mass changes and the modal data measurements of the tested structure such as measured mode shape readings. Structural updating parameters including both stiffness and mass parameters are employed to represent the differences in structural parameters between the finite element model and the associated tested structure. These updating parameters are then evaluated by an iterative solution procedure, giving optimised solutions in the least squares sense without requiring an optimisation technique. In order to reduce the influence of modal measurement uncertainty, the truncated singular value decomposition regularization method incorporating the quasi-optimality criterion is employed to produce reliable solutions for the structural updating parameters. Finally, the numerical investigations of a space frame structure and the practical applications to the Canton Tower benchmark problem demonstrate that the proposed method can correctly update the given finite element model using the incomplete modal data identified from the recorded ambient vibration measurements.

Influence of blood/tissue differences in contrast agent relaxivity on tracer-based MR perfusion measurements.

Perfusion assessment by monitoring the transport of a tracer bolus depends critically on conversion of signal intensity into tracer concentration. Two main assumptions are generally applied for this conversion; (1) contrast agent relaxivity is identical in blood and tissue, (2) change in signal intensity depends only on the primary relaxation effect. The purpose of the study was to assess the validity and influence of these assumptions. Blood and cerebral tissue relaxivities r1, r2, and r2* for gadodiamide were measured in four pigs at 1.5 T. Gadolinium concentration was determined by inductively coupled plasma atomic emission spectroscopy. Influence of the relaxivities, secondary relaxation effects and choice of singular value decomposition (SVD) regularization threshold was studied by simulations. In vivo relaxivities relative to blood concentration [in s(-1) mM(-1) for blood, gray matter (GM), white matter (WM)] were for r1 (2.614 ± 1.061, 0.010 ± 0.001, 0.004 ± 0.002), r2 (5.088 ± 0.952, 0.091 ± 0.008, 0.059 ± 0.014), and r2* (13.292 ± 3.928, 1.696 ± 0.157, 0.910 ± 0.139). Although substantial, by a nonparametric test for paired samples, the differences were not statistically significant. The GM to WM blood volume ratio was estimated to 2.6 ± 0.9 by r1, 1.6 ± 0.3 by r2, and 1.9 ± 0.2 by r2*. Secondary relaxation was found to reduce the tissue blood flow, as did the SVD regularization threshold. Contrast agent relaxivity is not identical in blood and tissue leading to substantial errors. Further errors are introduced by secondary relaxation effects and the SVD regularization.

Study of the influence factors on the accuracy of sound field reconstruction based on wave superposition method

Wave superposition method is a numerical method for sound field reconstruction, which can be applied in sound source identification and fault diagnosis. This study aims to obtain some guidelines to increase the reconstruction accuracy. A number of influence factors are analyzed through theoretical analysis and numerical simulation, including equivalent sources, measurement array, reconstruction distance and regularization methods. Results show that the distribution of the equivalent sources has a great impact on the reconstruction accuracy. The equivalent sources should be collocated interior to the radiator. It's better that the equivalent source surface conforms to the radiator's surface, while the measurement array doesn't need to be conformal to the radiator's surface. Relatively, the reconstruction distance has a little influence on the reconstruction accuracy. At last, four different regularization methods together with two regularization parameter selection approaches are combined with wave superposition method to solve the ill-posed problem. Comparative results indicate that Tikhonov and truncate singular value decomposition regularization methods work well with wave superposition method. In sum, these conclusions are useful to get higher reconstruction accuracy and consequently help wave superposition method become an effective fault diagnosis tool.

A robust algorithm for high-resolution dynamic MRI based on the partially separable functions model

A recently developed partially separable functions (PSF) model can be used to generate high-resolution dynamic magnetic resonance imaging (MRI). However, this method could not robustly reconstruct high-quality MR images because the estimation of the PSF parameters is often interfered by the noise of the sampled MR data. To improve the robustness of MRI reconstruction using the PSF model, we proposed a new algorithm to estimate the PSF parameters by jointly using robust principal component analysis and modified truncated singular value decomposition regularization methods, instead of using the least square fitting method in the original PSF model. The experiment results of in vivo cardiac MRI demonstrated that the proposed algorithm can robustly reconstruct dynamic MR images with higher signal-to-noise ratio and clearer anatomical structures in comparison with the previous PSF model.