Nonlocal Constraints Research Articles

Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal Constraint

Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal Constraint

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

Studies over the existence of a certain impulse-based fuzzy integrodifferential equations of the Sobolev type

The present investigation employs impulses and a non-local constraint to prove the existence are some various types of abstract differential and integrodifferential equations related to the Sobolev type. Semigroup theory, specifically variants of constant formula, is utilized to get the analytical results for those equations. Furthermore, findings using the Banach fixed point approach were examined using fuzzy numbers with values spanning the En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {E}_n$$\\end{document} range, which includes the normal, convex, upper semi-continuous, and compactly supported interval. A description is given for each situation to illustrate the principle.

Integro-differential equations implicated with Caputo-Hadamard derivatives under nonlocal boundary constraints

The goal of this work is to derive a new type of fractional system that arises from the combination of the Caputo-Hadamard derivative with the integro-differential equation. Also, the existence and uniqueness of solutions to this problem have been studied under nonlocal boundary conditions. Moreover, Hyer-Ulam stability has been studied for the considered problem. Finally, to reinforce the theoretical results and provide applications for our paper, two supporting examples have been emphasized.

A Multidimensional Tensor Low Rank Method for Magnetic Resonance Image Denoising.

In this paper, we present the Magnetic Resonance Image (MRI) denoising method via nonlocal multidimensional low rank tensor transformation constraint (NLRT). We first design a nonlocal MRI denoising method by non-local low rank tensor recovery framework. Furthermore, a multidimensional low rank tensor constraint is used to obtain low-rank prior information combined with 3-dimensional structure feature of MRI image cubes. Our NLRT can achieve denoising by retaining more image detail information. The optimization and updating process of the model is solved via the alternating direction method of multipliers (ADMM) algorithm. Several state-of-the-art denoising methods are selected for comparative experiments. In order to reflect the performance of the denoising method, Rician noise with different levels is added to the experiment to analyze the results. The experimental results prove that our NLTR has more outstanding denoising ability and can obtain better MRI images.

Double-Factor Tensor Cascaded-Rank Decomposition for Hyperspectral Image Denoising

Hyperspectral image (HSIs) denoising is a preprocessing step that plays a crucial role in many applications used in Earth observation missions. Low-rank tensor representation can be utilized to restore mixed-noise HSIs, such as those affected by mixed Gaussian, impulse, stripe, and deadline noises. Although there is a considerable body of research on spatial and spectral prior knowledge concerning subspace, the correlation between the spectral continuity and the nonlocal sparsity of the spectral and spatial factors is not yet fully understood. To address this deficiency, in the present study, we determined the correlation between these factors using a cascaded technique, and we describe in this paper the double-factor tensor cascaded-rank (DFTCR) minimization method that was used. The information existing in the nonlocal sparsity property of the spatial factor was employed to promote a geometrical feature representation, and a tensor cascaded-rank minimization approach was introduced as a nonlocal self-similarity to promote restoration quality. The continuity between the difference and nonlocal gradient sparsity constraints of the spectral factor was also introduced to learn the basis. Furthermore, to estimate the solutions of the proposed model, we developed an algorithm based on the alternating direction method of multipliers (ADMM). The performance of the DFTCR method was tested by a comparison with eleven established denoising methods for HSIs. The results showed that the proposed DFTCR method exhibited superior performance in the removal of mixed noise from HSIs.

Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints

Abstract This article focuses on the Cauchy problem of the Benjamin–Bona–Mahony–Burgers (BBMB) equation with nonlocal control constraints in ℝ 3 {\mathbb{R}^{3}} . We employ the Phragmén–Lindelöf decomposition method and Green’s function to investigate global classical solutions and their long-term behaviors, including optimal estimates for finite density initial perturbation. Additionally, optimal solutions near the minimal mass ground state are analyzed.

On a Non Local Initial Boundary Value Problem for a Semi-Linear Pseudo-Hyperbolic Equation in the Theory of Vibration

This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at hand. With respect to the associated linear problem, the uniqueness of its solution is ascertained through an energy inequality, which provides an a priori bound for the solution. Moreover, the solvability of this linear problem is verified by proving that the operator range engendered by the problem is indeed dense. Extending the analysis to the nonlinear problem, an iterative methodology is utilized. This approach is predicated on the insights gained from the linear problem and facilitates the demonstration of both the existence and uniqueness of a solution for the nonlinear problem under study. Consequently, the paper contributes a robust mathematical framework for solving both linear and nonlinear variants of complex initial boundary value problems with non-local constraints.

Random unitaries, Robustness, and Complexity of Entanglement

It is widely accepted that the dynamic of entanglement in presence of a generic circuit can be predicted by the knowledge of the statistical properties of the entanglement spectrum. We tested this assumption by applying a Metropolis-like entanglement cooling algorithm generated by different sets of local gates, on states sharing the same statistic. We employ the ground states of a unique model, namely the one-dimensional Ising chain with a transverse field, but belonging to different macroscopic phases such as the paramagnetic, the magnetically ordered, and the topological frustrated ones. Quite surprisingly, we observe that the entanglement dynamics are strongly dependent not just on the different sets of gates but also on the phase, indicating that different phases can possess different types of entanglement (which we characterize as purely local, GHZ-like, and W-state-like) with different degree of resilience against the cooling process. Our work highlights the fact that the knowledge of the entanglement spectrum alone is not sufficient to determine its dynamics, thereby demonstrating its incompleteness as a characterization tool. Moreover, it shows a subtle interplay between locality and non-local constraints.

Boosting quantification accuracy of chemical exchange saturation transfer MRI with a spatial-spectral redundancy-based denoising method.

Chemical exchange saturation transfer (CEST) is a versatile technique that enables noninvasive detections of endogenous metabolites present in low concentrations in living tissue. However, CEST imaging suffers from an inherently low signal-to-noise ratio (SNR) due to the decreased water signal caused by the transfer of saturated spins. This limitation challenges the accuracy and reliability of quantification in CEST imaging. In this study, a novel spatial-spectral denoising method, called BOOST (suBspace denoising with nOnlocal lOw-rank constraint and Spectral local-smooThness regularization), was proposed to enhance the SNR of CEST images and boost quantification accuracy. More precisely, our method initially decomposes the noisy CEST images into a low-dimensional subspace by leveraging the global spectral low-rank prior. Subsequently, a spatial nonlocal self-similarity prior is applied to the subspace-based images. Simultaneously, the spectral local-smoothness property of Z-spectra is incorporated by imposing a weighted spectral total variation constraint. The efficiency and robustness of BOOST were validated in various scenarios, including numerical simulations and preclinical and clinical conditions, spanning magnetic field strengths from 3.0 to 11.7 T. The results demonstrated that BOOST outperforms state-of-the-art algorithms in terms of noise elimination. As a cost-effective and widely available post-processing method, BOOST can be easily integrated into existing CEST protocols, consequently promoting accuracy and reliability in detecting subtle CEST effects.

An Adaptive Fuzzy C-Means Noise Image Segmentation Algorithm Combining Local and Regional Information

Noisy image segmentation using fuzzy c-mean clustering (FCM) algorithm is an important direction in the field of image segmentation, among which FCM using local information or non-local information constraints has made some progress. However, as the noise density increases, the local information cannot restore the real image. The non-local information-based segmentation method overly retains the noise and thus produces false edges resulting in the degradation of image segmentation accuracy. To address these problems, this paper use the regional and local information to develop an adaptive FCM image segmentation algorithm(RLFCM). First, the regional information and local information are obtained by the regional information filtering method and the local information filtering method. Then the weighted image is constructed in a certain way. Secondly, to make the objective function adaptively adjust the constraint ratio of the original image and the weighted image pixel by pixel, the degree of constraint is adaptively changed by using the difference between them as the coefficients of both through certain transformation. Finally, to speed up the convergence of the objective function, we add the number of clustering samples to the denominator of the objective function by certain transformations to reduce the number of iterations of the objective function. The proposed algorithm achieves more than 90% in image segmentation accuracy (SA), and mean Intersection-over-Union (mIoU) through synthetic and real-world image tests and acquires good performance in other indexes.

Few-shot IoT attack detection based on SSDSAE and adaptive loss weighted meta residual network

The Internet of Things (IoT) is an open and comprehensive network of smart objects. Unfortunately, it is also becoming increasingly vulnerable to security attacks during the increasing popularity of the IoT. It can lead traditional antivirus software to be less likely to prevent this threat. Therefore, it is necessary to design a model for the IoT attack detection. Current detection models are trained using massive big data samples. However, the distribution of traffic samples is few in specific scenarios. Also, existing models are also susceptible to noise interference in IoT environments, lowering detection efficiency and accuracy. In this work, we propose a few-shot IoT attack detection approach using a semi-supervised deep sparse autoencoder (SSDSAE) and an adaptive loss weighted meta residual network (ALWM-ResNet). First, an SSDSAE feature extraction model for local graph embedding is designed using local and non-local graph embedding constraints. Then, we design an ALWM-ResNet model to achieve IoT attack detection with few-shot samples under noise labels. A weighted function map is established using a weighted network and a meta-model, and weights are adaptively learned from the noise labels. Finally, we validate our approach using four IoT datasets. Several experimental results demonstrate the superior performance of our approach in IoT attack detection under few-shot samples.

Minimization of the Compliance under a Nonlocal p-Laplacian Constraint

This work is an extension of the paper by Cea and Malanowski to the nonlocal and nonlinear framework. The addressed topic is the study of an optimal control problem driven by a nonlocal p-Laplacian equation that includes a coefficient playing the role of control in the optimization problem. The cost functional is the compliance, and the constraint on the states are of the Dirichlet homogeneous type. The goal of the present work is a numerical scheme for the nonlocal optimal control problem and its use to approximate solutions in the local setting. The main contributions of the paper are a maximum principle and a uniqueness result. These findings and the monotonicity properties of the p-Laplacian operator have been crucial to building an effective numerical scheme, which, at the same time, has provided the existence of optimal designs. Several numerical simulations complete the work.

A knee point driven Kriging-assisted multi-objective robust fuzzy clustering algorithm for image segmentation

To enhance the segmentation performance and optimization efficiency of multi-objective evolutionary clustering algorithms for noisy images, this study presents a knee point driven Kriging-assisted multi-objective robust fuzzy clustering algorithm for image segmentation (K2 MORFC). First, pixel and region-level fitness functions are developed to comprehensively consider the spatial constraints and region consistency in an image. In the pixel-level fitness function, a Kullback–Leibler (KL) divergence-based spatial constraint term is employed and combined with intra-class compactness to preserve the membership similarity between an arbitrary image pixel and its non-local neighbors. In the region-level fitness function, image edge information is utilized to evaluate the regional consistency of the partition. Moreover, an adaptive determination mechanism of the weight factor for the spatial constraint term is designed in the pixel-level fitness function to control the influence of the spatial constraint for each pixel. Furthermore, considering that the preference toward knee points can effectively improve the convergence performance, a knee point driven Kriging-assisted evolutionary algorithm (K2EA) is proposed to efficiently optimize these two fitness functions, including a dynamic subspace knee point searching strategy, a knee point guided environmental selection strategy, and a Kriging model management mechanism. Finally, we construct a fuzzy clustering validity index with a KL divergence-based non-local spatial constraint term to select the optimal solution from the nondominated solution set. Competitive experimental results on the DTLZ benchmark verify the effectiveness of K2EA. Segmentation results on color images from the Berkeley dataset and magnetic resonance images from the BrainWeb and IBSR datasets confirm the performance of K2MORFC.

Adaptive differentiable grids for cryo-electron tomography reconstruction and denoising.

Tilt-series cryo-electron tomography is a powerful tool widely used in structural biology to study 3D structures of micro-organisms, macromolecular complexes, etc. Still, the reconstruction process remains an arduous task due to several challenges: The missing-wedge acquisition, sample misalignment and motion, the need to process large data, and, especially, a low signal-to-noise ratio. Inspired by the recently introduced neural representations, we propose an adaptive learning-based representation of the density field of the captured sample. This representation consists of an octree structure, where each node represents a 3D density grid optimized from the captured projections during the training process. This optimization is performed using a loss that combines a differentiable image formation model with different regularization terms: total variation, boundary consistency, and a cross-nodes non-local constraint. The final reconstruction is obtained by interpolating the learned density grid at the desired voxel positions. The evaluation of our approach using captured data of viruses and cells shows that our proposed representation is well adapted to handle missing wedges, and improves the signal-to-noise ratio of the reconstructed tomogram. The reconstruction quality is highly improved in comparison to the state-of-the-art methods, while using the lowest computing time footprint. The code is available on Github at https://github.com/yuanhaowang1213/adaptivediffgrid_ex.

A Novel Semi-Supervised Graph-Guided Approach for Intelligent Health State Diagnosis of a 3-PRR Planar Parallel Manipulator

As one of the most important operation and maintenance approaches, health state diagnosis technology plays a crucial role in ensuring the safety and reliability of mechanical equipment. The planar parallel manipulator, as a typical actuator, is widely employed in the field of precision manufacturing due to its advantages of high stiffness, large load support capability, and high precision. However, compared with common key functional components (such as bearings and gearboxes), planar parallel manipulators have more complicated operating mechanisms and failure behaviors. To satisfy the health state diagnosis demands of planar parallel manipulators in the scenario of insufficient label information, a novel intelligent health state diagnosis approach, termed semi-supervised graph-guided network with perception attention (SGN-PA), is developed for a 3-PRR (P and R represent prismatic and revolute pairs, respectively) planar parallel manipulator. Specifically, an improved multiorder graph perception module is constructed to extract multiscale feature information, and achieve feature fusion by combining perceptual attention mechanism, which enables the proposed SGN-PA model to have adaptation adjustment capabilities. Following that, local and nonlocal feature constraint strategies are employed with pseudo-label technology to reduce intraclass differences and maximize interclass differences, and then to fit the demands of health state diagnosis tasks. Eventually, based on the simulation and experimental scenarios of a 3-PRR planar parallel manipulator, the effectiveness and feasibility of the proposed SGN-PA model is extensively confirmed, and the diagnosis results show that it can significantly relax the constraints of label information while maintaining superior performances.

Depth Map Super-Resolution via Joint Local Gradient and Nonlocal Structural Regularizations

Depth maps have been widely used in many real world applications, such as human-computer interaction and virtual reality. However, due to the limitation of current depth sensing technology, the captured depth maps usually suffer from low resolution and insufficient quality. In this paper, we propose a depth map super-resolution method via joint local gradient and nonlocal structural regularizations. Depth maps contain mainly smooth areas separated by textures which demonstrate distinct geometry direction characteristic. Motivated by this, we classify depth map patches according to their geometrical directions and learn a compact online dictionary in each class. We further introduce two regularization terms into the sparse representation framework. Firstly, a multi-directional total variation model is proposed to characterize the local patterns in the gradient domain. Secondly, a nonlocal autoregressive model is introduced to provide nonlocal constraint to the local structures, which can effectively restore image details and suppress noise. Quantitative and qualitative evaluations compared with state-of-the-art methods demonstrate that the proposed method achieves superior performance for various configurations of magnification factors and datasets.

Soliton solutions for two kinds of fourth-order nonlinear nonlocal Schrödinger equations

In this paper, we will take two symmetric reduction conditions to obtain two kinds of fourth-order nonlinear nonlocal Schrödinger (NLS) equations, namely the fourth-order nonlocal reverse space and reverse space–time NLS equations. For the former, we will construct one- and N-fold Darboux Transformations (DTs) to obtain the symmetry preserving and broken solutions. In addition, we will study the different combinations of collision scenarios of dark and anti-dark soliton solutions. This process will be expressed in the form of quasi-determinant. Finally, we will get the symmetry preserving and broken solutions can exist at the same time. For the latter, the one- and N-fold DTs will be constructed. Taking the seed solutions as zero and continue wave solutions, we will obtain different kinds of exact solutions under nonlocal constraints, such as soliton, breather and rogue wave solutions, which is different from the classical case.

Representation formulas for stationary solutions of a cell polarization model

We are interested in the global bifurcation structure of solutions for a nonlinear boundary value problem with nonlocal constraint (SLP) that appears in a cell polarization model with mass conservation proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. We obtained primitive representation formulas of all solutions and investigated a surface \({\mathcal {S}}\) consisting of all bifurcation diagrams with heights. However, we could not find any parameterization of the surface \({\mathcal {S}}\). In this paper, we show parameterizations of the surface \({\mathcal {S}}\) and concrete representation formulas of all global bifurcation diagrams of (SLP). These results are also beneficial for numerical computations to get all bifurcation diagrams. We propose new methods in view of them.

A reduced-order fast reproducing kernel collocation method for nonlocal models with inhomogeneous volume constraints

This paper is concerned with the implementations of the meshfree-based reduced-order model (ROM) to time-dependent nonlocal models with inhomogeneous volume constraints. Generally, when using ROM for nonlocal models, the projection of nonlocal volume constraints needs to be computed in every time step to handle the nonlocal boundary conditions. Up to now, only finite element methods (FEM) can work well in constructing ROM for nonlocal models, since the interpolation property of the FEM basis functions makes it easy to obtain such a projection. But if one tries to develop ROM based on existing meshfree methods for nonlocal models, the projection in every time step will lead to a full-order discrete system and is highly time-consuming, since the basis functions of these methods do not meet interpolation property. To overcome the above difficulties, we introduce a mixed reproducing kernel (RK) approximation with nodal interpolation property to develop a meshfree collocation method for nonlocal models and use it to construct ROM. Thanks to the nodal interpolation property, the projection of nonlocal boundary conditions can be obtained explicitly. This ROM is developed using numerical results as snapshots by a full-order model in a small time interval [ 0 , t 1 ] . The surrogate model, which is constructed by POD (proper orthogonal decomposition)-Galerkin approach, leads to a discrete system with far fewer degrees of freedom than the original meshfree method. Numerical experiments for nonlocal problems including nonlocal diffusion and peridynamics are presented to show that our method meets almost the same accuracy with a very small computational cost compared with the full-order meshfree approach.