Decomposition Of Operator Research Articles

Wavelet transform-based mode decomposition for EEG signals under general anesthesia.

Mode decomposition methods are used to extract the characteristic intrinsic mode function (IMF) from various multidimensional time series signals. We analyzed an electroencephalogram (EEG) dataset for sevoflurane anesthesia using two wavelet transform-based mode decomposition methods, comprising the empirical wavelet transform (EWT) and wavelet mode decomposition (WMD) methods, and compared the results with those from the previously reported variational mode decomposition (VMD) method. To acquire the EEG data, we used the software application EEG Analyzer, which enabled the recording of raw EEG signals via the serial interface of a bispectral index (BIS) monitor. We also created EEG mode decomposition software to perform empirical mode decomposition (EMD), VMD, EWT, and WMD operations. When decomposed into six IMFs, the EWT enables narrow band separation of the low-frequency bands IMF-1 to IMF-3, in which all central frequencies are less than 10 Hz. However, in the upper IMF of the high-frequency band, which has a center frequency of ≥ 10 Hz, the dispersion within the frequency band covered was widespread among the individual patients. In WMD, a narrow band of clinical interest is specified using a bandpass filter in a Meyer wavelet filter bank within a specific mode-decomposition discipline. When compared with the VMD and EWT methods, the IMF that was decomposed via WMD was accommodated in a narrow band with only a small variance for each patient. Multiple linear regression analyses demonstrated that the frequency characteristics of the IMFs obtained from WMD best tracked the changes in the BIS upon emergence from general anesthesia. The WMD can be used to extract subtle frequency characteristics of EEGs that have been affected by general anesthesia, thus potentially providing better parameters for use in assessing the depth of general anesthesia.

Decomposition of nonlinear collision operator in quantum Lattice Boltzmann algorithm

We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.

Identification of fractional order time delay system with measurement noise using variable period integration operational matrix

This paper proposes a parameter identification method for fractional-order time-delay systems with measurement noise, based on the Legendre wavelet variable-period integration operational matrix. A variable-period integration operational matrix and a variable-period time-delay integration operational matrix are constructed by variable-period sampling principle. For the impulse noise, global Gaussian noise and local Gaussian noise present in the identification data, the variable-period sampling, operational matrix global coefficient decomposition and matrix local coefficient decomposition methods are used to reconstruct the variable-period integration operational matrix to separate the noise from the data; the fractional-order time-delay system is expressed as an integral equation by using the reconstructed variable-period operational matrix, this reduces the influence of measurement noise on system identification accuracy. To further improve the parameter identification accuracy, the integral equation is solved by the multi-innovation least squares algorithm. The proposed method’s effectiveness is verified through several simulations and an experimental study of a constant-temperature system.

Frozen condition of quantum coherence

Quantum coherence as an important physical resource plays the key role in implementing various quantum tasks, whereas quantum coherence is generally nonincreasing under incoherent operations. In this paper, we analyse under which dynamical conditions the l 1-norm or the relative entropy of coherence can remain unchanged under strictly incoherent operations (freezing coherence). We provide a detailed analysis of their structure together with exact geometric conditions of freezing coherence. It reveals a remarkable feature: any strictly incoherent operation freezing coherence can be decomposed as a convex combination of unitary operations. This partially answers an open question named unitary decomposition of doubly-stochastic quantum operations [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)]. Based on this analysis, we also give a complete classification of coherent states from operational coherence theory. This builds the counterpart of entanglement classification under LOCC.

Research on Indirect Influence-Line Identification Methods in the Dynamic Response of Vehicles Crossing Bridges

The bridge influence line can effectively reflect its overall structural stiffness, and it has been used in the studies of safety assessment, model updating, and the dynamic weighing of bridges. To accurately obtain the influence line of a bridge, an Empirical and Variational Mixed Modal Decomposition (E-VMD) method is used to remove the dynamic component from the vehicle-induced deflection response of a bridge, which requires the preset fundamental frequency of the structure to be used as the cutoff frequency for the intrinsic modal decomposition operation. However, the true fundamental frequency is often obtained from the picker, and the testing process requires the interruption of traffic to carry out the mode decomposition. To realize the rapid testing of the influence lines of bridges, a new method of indirectly identifying the operational modal frequency and deflection influence lines of bridge structures from the axle dynamic response is proposed as an example of cable-stayed bridge structures. Based on the energy method, an analytical solution of the first-order frequency of vertical bending is obtained for a short-tower cable-stayed bridge, which can be used as the initial base frequency to roughly measure the deflection influence line of the cable-stayed bridge. The residual difference between the deflection response and the roughly measured influence line under the excitation of the vehicle is operated by Fast Fourier Transform, from which the operational fundamental frequency identification of the bridge is realized. Using the operational fundamental frequency as the cutoff frequency and comparing the influence-line identification equations, the empirical variational mixed modal decomposition, and the Tikhonov regularization to establish a more accurate identification of the deflection influence line, the deflection influence line is finally identified. The accuracy and practicality of the proposed method are verified by real cable-stayed bridge engineering cases. The results show that the relative error between the recognized bridge fundamental frequency and the measured fundamental frequency is 0.32%, and the relative error of the recognized deflection influence line is 0.83%. The identification value of the deflection influence line has a certain precision.

Image Enhancement Using Bidimensional Empirical Mode Decomposition and Morphological Operations for Brain Tumor Detection and Classification.

The three steps of brain image processing – preprocessing, segmentation, and classification are becoming increasingly important in patient care. The aim of this article is to present a proposed method in the mentioned three-steps, with emphasis on the preprocessing step, which includes noise removal and contrast enhancement. The fast and adaptive bidimensional empirical mode decomposition and the anisotropic diffusion equation as well as the modified combination of top-hat and bottom-hat transforms are used for noise reduction and contrast enhancement. Fast C-means clustering with enhanced image is used to detect tumors and the tumor cluster corresponds to the maximum centroid. Finally, Ensemble learning is used for classification. The Figshare brain tumor dataset contains magnetic resonance images used for data selection. The optimal parameters for both noise reduction and contrast enhancement are investigated using a tumor contaminated with Gaussian noise. The results are evaluated against state-of-the-art results and qualitative performance metrics to demonstrate the dominance of the proposed approach. The fast C-means algorithm is applied to detect tumors using twelve enhanced images. The detected tumors were compared to the ground truth and showed an accuracy and specificity of 99% each, and a sensitivity and precision of 90% each. Six statistical features are retrieved from 150 enhanced images using wavelet packet coefficients at level 4 of the Daubechies 4 wavelet function. These features are used to develop the classifier model using ensemble learning to create a model with training and testing accuracy of 96.7% and 76.7%, respectively. When this model is applied to classify twelve detected tumor images, the accuracy is 75%; there are three misclassified images, all of which belong to the pituitary disease group. Based on the research, it appears that the proposed approach could lead to the development of computer-aided diagnosis (CADx) software that physicians can use as a reference for the treatment of rain tumor.

Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method.

A Study on Fault Localization Method of Three-Terminal Multi-Section Overhead Line–Cable Hybrid Line Using MEEMD Combined with Teager Energy Operator Algorithm

An improved fault localization method combining total aggregate empirical modal decomposition (MEEMD) and Teager energy operator (TEO) is proposed to address the fault localization issue of three-terminal multi-segment overhead line–cable hybrid transmission lines. This method solves the fault localization problem caused by wave impedance discontinuity in hybrid lines. First, the MEEMD algorithm, which improves modal aliasing, and the Teager energy operator, which reflects transient energy changes, are combined for the accurate detection of faulty traveling wave heads. The fault line section determination condition within the fault branch is used to determine the overhead line section or cable section where the fault is located after determining the faulty branch line. This condition is based on the time difference between the initial traveling wave of the fault arriving at each end measurement point and the T-node. Ultimately, the fault distance is determined using the double-ended traveling wave technique. By combining PSCAD and MATLAB for simulation verification, the accuracy of the ranging method is compared and analyzed, and the results show that the method is able to ensure the accuracy of ranging at different fault locations and transition resistances. Additionally, the localization error is basically kept within 30 m, which significantly improves the accuracy of fault localization.

Computational decomposition and composition technique for approximate solution of nonstationary problems

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit–implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition–composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Boundary controllability for a 1D degenerate parabolic equation with drift, a singular potential, and a Neumann boundary condition

We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability using a representation theorem for analytic functions of exponential type.

LIASM-NRID: Constructing an atmospheric scattering model for low-light conditions and dehazing nighttime road images

Dehazing of nighttime road images holds significant practical relevance for autonomous driving and intelligent transportation systems operating in nocturnal conditions. To address the limitations of traditional atmospheric scattering models in accurately describing the imaging process in low illumination environments such as nighttime, this study constructs a low illumination environment atmospheric scattering model. Based on this newly constructed model, we propose an algorithm specifically designed for dehazing road images at nighttime. The proposed model incorporates an incident light attenuation factor, an additive noise factor, and a global compensation factor related to the scene's depth. The proposed nighttime road image dehazing algorithm firstly suppresses the interference of additive noise in the original image using wavelet decomposition and soft thresholding operation. Then, a transmittance-solving method containing the primary structural information of the image is designed based on the color attenuation linear model, and the solved transmittance is refined based on the interval gradient image texture structure method. Consequently, the global light map estimation process is formulated as an optimization problem to ensure the accuracy of the atmospheric light value calculation. In the final stages, a novel RGB color equalization method is introduced to address the issue of non-uniform incident light color bias in the dehazed images. Additionally, to eliminate the halo problem in the dehazed images, the adaptive histogram equalization algorithm is modified by incorporating a normalized gamma correction function. The experimental results show that the proposed nighttime road image dehazing algorithm exhibits robust performance and generality across various nighttime scenes. Compared with the representative algorithms for nighttime image dehazing, it can get more excellent dehazing results and has more outstanding performance in all numerical evaluation indexes.

Analytical solution for horizontal vibration of partially embedded pipe pile group in layered saturated soil

AbstractBased on Biot's dynamic wave theory and Novak's plane strain model (PSM), an analytical model for the horizontal vibration of a partially embedded pipe pile group (PEPPG) in layered saturated soil (LSS) that considers the soil‐plugging effect was established: First, the frequency‐domain analytical expressions of the saturated surrounding soil (SSS) and soil plug were obtained by introducing operator decomposition theory and the variable separation method. Second, the analytical solution of the horizontal impedance of a partially embedded pipe pile in the LSS is derived using the coupled condition of the pile–soil interface and transfer matrix method. Finally, an analytical solution for the horizontal impedance of PEPPG in LSS was derived by introducing the horizontal dynamic interaction factor of piles and the superposition method. The derived solution is then reduced and compared with existing theoretical solutions to verify its rationality. In addition, further parametric analysis was conducted to investigate the influences of pile spacing, embedment ratio, properties of layered SSS, and soil‐plugging on the horizontal vibration characteristics (HVCs) of PEPPG.

Detection of Buried Nonlinear Targets Using DORT

Ground-penetrating radars (GPR) based on a variety of techniques have been proposed to improve the performance of buried target (e.g., landmines, threat devices) detection. However, the small radar cross section (RCS) of small electronic devices poses difficulties for target detection, especially when they are buried in lossy and inhomogeneous media. This paper presents a novel buried nonlinear target detection method based on the decomposition of the time-reversal operator (DORT) that uses a multistatic system to overcome the limitations of conventional GPR. Using harmonic radar, which detects the harmonic responses scattered from electronic devices, and DORT processing, which enables focusing/imaging of the detected target, the detection performance is verified by conducting simulation and measurements. The overall results demonstrate that the proposed method achieves accurate detection of buried targets with small RCS.

Dynamic response of bored piles with gradient defects during low strain integrity test

With the proposal of the gradient ring-soil pile model, the possibility of utilizing the low strain integrity test for identifying gradient defects of the bored piles embedded in unsaturated soil is confirmed. The gradient ring-soil pile model realizes the simulation of soil-pile interaction at the soil-pile interfaces with arbitrary shapes in the horizontal projection. Compared to existing theoretical answers, this paper extends the low strain test theory to the identification of gradient defects. With the adoption of the Laplace transform, differential operator decomposition theory, variable separation method, and impedance function transfer method, a semi-analytical solution is derived. Taking advantage of the semi-analytical solution, a parametric study is conducted to analyze the variation of the defect shapes on the test signal of the bored pile. Based on the findings, a series of methods for identifying gradient defects in engineering applications are summarized. The main findings can be summarized as follows: 1. The soil filled in the gradient necking defects would significantly decrease the magnitudes of reflected signals at these defects, making them more challenging to be identified; 2. If simply modeling the gradient defects as sudden defects, the severity of defects can be underestimated, resulting in the overestimation of pile capacity; 3. The shape of the gradient defect, as well as the length of the gradient defect, has a significant influence on the reflected signal.

Sound Source Localization Method Based on Time Reversal Operator Decomposition in Reverberant Environments

Predicting sound sources in reverberant environments is a challenging task because reverberation causes reflection and scattering of sound waves, making it difficult to accurately determine the position of the sound source. Due to the characteristics of overcoming multipath effects and adaptive focusing of the time reversal technology, this paper focuses on the application of the time reversal operator decomposition method for sound source localization in reverberant environments and proposes the image-source time reversal multiple signals classification (ISTR-MUSIC) method. Firstly, the time reversal operator is derived, followed by the proposal of a subspace method to achieve sound source localization. Meanwhile, the use of the image-source method is proposed to calculate and construct the transfer matrix. To validate the effectiveness of the proposed method, simulations and real-data experiments were performed. In the simulation experiments, the performance of the proposed method under different array element numbers, signal-to-noise ratios, reverberation times, frequencies, and numbers of sound sources were studied and analyzed. A comparison was also made with the traditional time reversal method and the MUSIC algorithm. The experiment was conducted in a reverberation chamber. Simulation and experimental results show that the proposed method has good localization performance and robustness in reverberant environments.

A distribution network traveling wave localization method based on CEEMD-DEO3S

A traveling wave head calibration way according to Complementary Ensemble Empirical Mode Decomposition (CEEMD) and Three-point Symmetric Differential Energy Operator (DEO3S) is proposed to address the issues of noise interference and difficulty in precise positioning in local area network wave positioning. The way filters out low-frequency bandwidth in the fault data by performing CEEMD decomposition on the signal and then uses the DEO3S algorithm to define the time of arrival of the fault wave head. The emulation results indicate the proposed way can precisely extract wave heads even in the presence of a large amount of noise interference, and has good robustness and accuracy. Compared with traditional traveling wave positioning methods, it has obvious advantages and further improves the accuracy and reliability of traveling wave positioning.

Two-pronged feature reduction in spectral clustering with optimized landmark selection

Spectral clustering is widely employed for clustering data points, particularly for non-linear and non-convex structures in high-dimensional spaces. However, it faces challenges due to the high computational cost of eigen decomposition operations and the performance limitations with high-dimensional data. In this paper, we introduce BVA_LSC, a novel spectral clustering algorithm designed to address these challenges. Firstly, we incorporate an advanced feature reduction stage utilizing Barnes-Hut t-SNE and a deep Variational Autoencoder (VAE) to efficiently reduce the dimensionality of the data, thereby accelerating eigen decomposition. Secondly, we propose an adaptive landmark selection strategy that combines the Grey Wolf Optimizer (GWO) with a novel objective function and K-harmonic means clustering. This strategy dynamically determines an optimal number of landmarks, enhancing the representativeness of the data and reducing the size of the similarity matrix. We assess the performance of our algorithm on various standard datasets, demonstrating its superiority over state-of-the-art methods in terms of accuracy and efficiency.

Low-depth simulations of fermionic systems on square-grid quantum hardware

We present a general strategy for mapping fermionic systems to quantum hardware with square qubit connectivity which yields low-depth quantum circuits, counted in the number of native two-qubit fSIM gates. We achieve this by leveraging novel operator decomposition and circuit compression techniques paired with specifically chosen low-depth fermion-to-qubit mappings and allow for a high degree of gate cancellations and parallelism. Our mappings retain the flexibility to simultaneously optimize for qubit counts or qubit operator weights and can be used to investigate arbitrary fermionic lattice geometries. We showcase our approach by investigating the tight-binding model, the Fermi-Hubbard model as well as the multi-orbital Hubbard-Kanamori model. We report unprecedentedly low circuit depths per single Trotter layer with up to a 70% improvement upon previous state-of-the-art. Our compression technique also results in significant reduction of two-qubit gates. We find the lowest gate-counts when applying the XYZ-formalism to the DK mapping. Additionally, we show that our decomposition and compression formalism produces favourable circuits even when no native parameterized two-qubit gates are available.

Pseudo-Helmholtz decomposition for an elastic VTI wavefield based on wavefront phase direction

P and S waves are coupled when propagating in anisotropic elastic media. The separation of P and S waves helps to study the characteristics of different types of seismic waves as well as mitigating crosstalk artifacts in elastic reverse time migration and elastic full-waveform inversion. At present, the methods of seismic wave mode separation in anisotropic media are mainly built on divergence- and curl-like operations, pseudo-Helmholtz decomposition, and low-rank approximation. We develop a new pseudo-Helmholtz decomposition operator based on eigenform analysis and the wavefront phase direction to decompose vertically transversely isotropic elastic wavefields. The corresponding P-/S-wave decoupling formulas are also derived in detail. Compared with the divergence- and curl-like methods, the new method does not change the phase of P and S waves. Compared with existing pseudo-Helmholtz decomposition methods based on eigenform analysis, our method achieves more accurate wavefield separation than the zero-order pseudo-Helmholtz decomposition operator. Our method requires solving one vector Poisson equation only, resulting in much less computational cost than the existing first-order pseudo-Helmholtz decomposition methods. In addition, the accuracy of our method is analyzed by providing homogeneous media with different parameter settings. Finally, the numerical examples demonstrate that the new pseudo-Helmholtz decomposition method is effective, efficient, and robust against random noise.

Complementation and Lebesgue‐type decompositions of linear operators and relations

AbstractIn this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of . The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.