Norm Estimates Research Articles

The impact of channel density, inverse solutions, connectivity metrics and calibration errors on OPM-MEG connectivity analysis: A simulation study.

Magnetoencephalography (MEG) systems based on optically pumped magnetometers (OPMs) have rapidly developed in the fields of brain function, health, and disease. Functional connectivity analysis related to the resting-state has gained popularity as a field of research in recent years. Several studies have attempted to use OPM-based MEG (OPM-MEG) for brain network estimation research; however, the choice of source connectivity analysis pipeline may lead to outcome variability. Several methods and related parameters must be selected carefully at each step of the analysis. Therefore, this study assessed the effect of such analytical variability on the OPM-MEG connectivity analysis by conducting simulations. Synthetic MEG data corresponding to two default mode networks (DMN) with six or ten DMN regions were generated using the Gaussian Graphical Spectral (GGS) model. Six intersensor spacings were constructed, and six inverse algorithms and six functional connectivity measures were selected to assess their impact on the network reconstruction accuracy. Three potential sources of error-errors in the sensor gain, crosstalk, and angular errors of the sensitive axis of the OPM-were also assessed. Analytical variability with regard to the tested intersensor spacings, inverse solutions, and functional connectivity measures led to high result variability. Crosstalk exerted a significant impact on the accuracy, which may lead to network reconstruction failure. The accuracy improvement caused by an increase in the sensor density may be reduced by gain and angular errors. The minimum norm estimate (MNE) and weighted minimum norm estimate (wMNE) exhibited low robustness to sensor noise and calibration errors. Hence, a calibration workflow for accurate sensor parameters, such as the gain and direction of the sensitive axis, before commencing OPM-MEG measurement and a careful choice of different method combinations play crucial roles in ensuring that OPMs yield optimal results for functional connectivity analysis. A thorough framework for analyzing brain connectivity networks was provided herein.

Optimized PID controller and model order reduction of reheated turbine for load frequency control using teaching learning-based optimization

Load frequency control (LFC) systems in power grids face challenges in maintaining stability while managing computational complexity. This research presents an optimized approach combining model order reduction techniques with Teaching Learning-Based Optimization (TLBO) for PID controller tuning in single-area LFC systems. Three reduction methods—Routh Approximation, Balanced Truncation, and Hankel Norm Approximation—were implemented to reduce system order from 4th to 2nd order, achieving a 47.3% reduction in computational time. The TLBO-optimized PID controller was compared with conventional tuning methods (Ziegler-Nichols, AMIGO, S-IMC, and CHR), demonstrating superior performance with a 38.2% decrease in settling time and 42.7% reduction in peak overshoot. The Routh Approximation method exhibited optimal performance with minimum settling time (2.8s) and peak overshoot (8.4%). Sensitivity analysis revealed stable system behavior with phase margin maintained at 84.25 degrees across parameter variations. The proposed approach achieved a 56.8% reduction in Integral Square Error compared to conventional methods, establishing its effectiveness for modern power grid applications. This research provides a robust framework for implementing efficient load frequency control in power systems while maintaining system stability and performance.

A Method for Estimating Fluorescence Emission Spectra from the Image Data of Plant Grain and Leaves Without a Spectrometer

This study proposes a method for estimating the spectral images of fluorescence spectral distributions emitted from plant grains and leaves without using a spectrometer. We construct two types of multiband imaging systems with six channels, using ordinary off-the-shelf cameras and a UV light. A mobile phone camera is used to detect the fluorescence emission in the blue wavelength region of rice grains. For plant leaves, a small monochrome camera is used with additional optical filters to detect chlorophyll fluorescence in the red-to-far-red wavelength region. A ridge regression approach is used to obtain a reliable estimate of the spectral distribution of the fluorescence emission at each pixel point from the acquired image data. The spectral distributions can be estimated by optimally selecting the ridge parameter without statistically analyzing the fluorescence spectra. An algorithm for optimal parameter selection is developed using a cross-validation technique. In experiments using real rice grains and green leaves, the estimated fluorescence emission spectral distributions by the proposed method are compared to the direct measurements obtained with a spectroradiometer and the estimates obtained using the minimum norm estimation method. The estimated images of fluorescence emissions are presented for rice grains and green leaves. The reliability of the proposed estimation method is demonstrated.

LittleFaceNet: A Small-Sized Face Recognition Method Based on RetinaFace and AdaFace.

For surveillance video management in university laboratories, issues such as occlusion and low-resolution face capture often arise. Traditional face recognition algorithms are typically static and rely heavily on clear images, resulting in inaccurate recognition for low-resolution, small-sized faces. To address the challenges of occlusion and low-resolution person identification, this paper proposes a new face recognition framework by reconstructing Retinaface-Resnet and combining it with Quality-Adaptive Margin (adaface). Currently, although there are many target detection algorithms, they all require a large amount of data for training. However, datasets for low-resolution face detection are scarce, leading to poor detection performance of the models. This paper aims to solve Retinaface's weak face recognition capability in low-resolution scenarios and its potential inaccuracies in face bounding box localization when faces are at extreme angles or partially occluded. To this end, Spatial Depth-wise Separable Convolutions are introduced. Retinaface-Resnet is designed for face detection and localization, while adaface is employed to address low-resolution face recognition by using feature norm approximation to estimate image quality and applying an adaptive margin function. Additionally, a multi-object tracking algorithm is used to solve the problem of moving occlusion. Experimental results demonstrate significant improvements, achieving an accuracy of 96.12% on the WiderFace dataset and a recognition accuracy of 84.36% in practical laboratory applications.

Study of the Relationship Between Two Types of Mixed-Order Evolution Equations and the Existence of Their Solutions

The purpose of this article is to study the relationship between two types of mixed-order evolution equations and the existence of their solutions. Initially, the mild solutions of the two system are obtained by Laplace transform. Then, we use semigroup theorem and sector operator theorem to get the norm estimation results of under- standing operator through special paths in the complex plane. Further, the sufficient conditions for existence and uniqueness of mild solution of the proposed system are verified by applying fixed point theorems. Finally, examples are provided to illustrate the main results.

Discontinuous Galerkin discretization of coupled poroelasticity–elasticity problems

Abstract This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.

Local error estimates and post processing for the Galerkin BEM on polygons

In this paper we give local error estimates in Sobolev norms for the Galerkin method applied to strongly elliptic pseudodifferential equations on a polygon. By using the K-operator, an operator which averages the values of the Galerkin solution, we construct improved approximations.

Essential Norm of Sum of Generalized Weighted Composition Operators between Weighted Spaces of Analytic Functions

Let D be the open unit disc in the complex plane C , H ( D ) be the space of analytic functions on D and S ( D ) be the set of analytic self-maps on D . Let Ψ = ( ψ j ) j = 0 k be such that ψ j ∈ H ( D ) and φ ∈ S ( D ) . We obtain estimates of the essential norms of the following sum of generalized weighted composition operators T Ψ , φ k f = ∑ j = 0 k ψ j · f ( j ) ° φ = ∑ j = 0 k D ψ j , φ j f , f ∈ H ( D ) , between weighted Banach spaces of analytic functions H v ∞ ( H v 0 ) and H w ∞ ( H w 0 ) , which unifies the study of products of composition operators, multiplication operators and differentiation operators. As an application, we give the estimates of the essential norm of the operators D ψ , φ n : B v → B w , B v → H w ∞ and T μ 1 , μ 2 , φ n : B v → H w ∞ . Also, we give the new characterizations of the estimates of the essential norms of the operators T Ψ , φ k and D ψ j , φ j .

Invariant region property of weak Galerkin method for semilinear parabolic equations

In this paper, we establish invariant region properties (IRPs) for the time-continuous and full-discrete weak Galerkin (WG) schemes of the semilinear parabolic equations. The scheme employs the semi-implicit scheme in the time direction and P1-P0 WG method in the space direction, respectively. The full-discrete scheme is proved to preserve the IRP unconditionally on triangular meshes, and the optimal convergence order estimates in both L2 and H1 norms are obtained for semi-discrete and full-discrete schemes. Some numerical results are presented to validate the theory of IRP and error estimates.

Comparison Between Different Source Localization and Connectivity Metrics of Spiky and Oscillatory MEG Activities

Epilepsy is considered the second neurological disease in a coma after stroke. Famous markers of epilepsy are repetitive seizures, their origin is stroma and cortical deformation. A neurologist would be assisted by identifying Epileptogenic Zones EZ when diagnosing epilepsy.. Source localization is utilized to identify regions known as EZ, which are of excessive discharges. It consists of both forward and inverse problems. The forward problem models the head through analytical and numerical methods. The inverse problem can be resolved using several techniques to locate the cerebral abnormal sources, via the electrophysiological recording biomarkers. In our study, we will investigate four distributed inverse problem methods: minimum norm estimation MNE, standardized low-resolution brain electromagnetic tomography sLORETA, maximum entropy on the mean MEM, Dynamic statistical parametric maps dSPM, to define epileptic networks connectivity of spiky and oscillatory events. We will examine the epileptic network connectivity using Phase Locking Value (PLV), Phase Transfert Entropy (PTE) for oscillatory events, cross-correlation (CC), and Granger Causality (GC) for spiky events applied on 5 pharmaco resistant subjects. We suggest rating the effectiveness of these networks in locating EZ through a phase of confrontation within iEEG transitory and oscillatory networks connectivity by exploring concordant nodes, their distance, propagation delays connection strength, and their cooperation in recognition of seizure onset zone. All studied techniques of the inverse problem, connection metrics, for both biomarkers of the 5 patients succeed in detecting at least one part of SOZ. sLORETA provides the highest concordant nodes and the closed one for spiky events using CC and GC. sLORETA also depicts the lowest propagation delay for oscillatory events using PTE. Through the 5 patients, MEM, dSPM, and MNE using CC, CG for spiky events, and PTE, PLV for oscillatory activities provide about 72 % of concordant nodes between MEG and iEEG.

Well-posedness and finite element analysis for the elastic scattering problem with a modified DtN map

As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application, and then the well-posedness of the resulting boundary value problem (BVP) becomes a challenging issue. By introducing a corresponding eigensystem to the bilinear form together with appropriate truncated norm estimates, we prove the well-posedness of the corresponding BVP in a weak sense. In addition, a priori error estimates that incorporate the effects of the finite element discretization and the truncation of infinite series are derived. Finally, numerical tests are implemented to validate the theoretical results.

Whole-course power evolution in childhood absence epilepsy: A multi-frequency magnetoencephalography study

ObjectiveThis study explores the whole-course neuromagnetic activity changes in childhood absence epilepsy (CAE) using multifrequency magnetoencephalogram (MEG) analysis. We aim to uncover the underlying neurophysiological mechanisms and identify functional signal targets with potential clinical applications. MethodsWe recruited 37 drug-naive children with CAE and collected magnetoencephalography (MEG) data from 62 seizures and interictal periods using a CTF-275 channel MEG system. The seizure course was segmented with temporal unification and subjected to dynamic frequency band analysis. Minimum norm estimation combined with Welch's method was employed for spectral power calculation, followed by correlation analysis between power and seizure duration. ResultsWhole-brain magnetic source power changes in 2–60 Hz largely paralleled the progression of spike and wave discharges (SWDs), while power in 60–90 Hz was suppressed during seizures. Alpha band (8–12 Hz) activity showed a prompt loss of occipital dominance at seizure onset, with concurrent elevation in frontal alpha activity. This frontal alpha dominance persisted throughout the ictal period and reverted to occipital dominance at termination. Beta and gamma1 band (15–59 Hz) activity characteristically declined before SWDs cessation. The power of SWDs during ictal period was negatively correlated with seizure duration. ConclusionSpectral power analysis of neuromagnetic signals throughout CAE process identifies specific frequency-dependent characteristic changes, among which, the distribution of alpha band (8–12 Hz) activity is closely related to absence manifestations, beta band (15–29 Hz) power decline induces seizure termination. Additionally, ictal SWDs power can serve as a neuroimaging indicator of epilepsy severity.

New error estimates for the conjugate gradient method

The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix A. The development of techniques for estimating the norm of the error in iterates computed by this method has received considerable attention. Available methods for bracketing the A-norm of the error evaluate pairs of Gauss and Gauss–Radau quadrature rules to determine lower and upper bounds. The latter rule requires a user to allocate a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally demands further computations to estimate the location of the smallest eigenvalue; see, e.g., Golub and Meurant (1997), Golub and Meurant (2010), Golub and Strakoš (1994), Meurant (1997), Meurant (1999). An approach that avoids the need to know a lower bound for the smallest eigenvalue is to replace the Gauss–Radau quadrature rule by an anti-Gauss rule as described by Calvetti et al. (2000). However, this approach may sometimes yield inaccurate error norm estimates. This paper proposes the use of pairs of Gauss and associated optimal averaged Gauss quadrature rules to estimate the A-norm of the error in iterates determined by the conjugate gradient method.

Optimal approximate minimization of one-letter weighted finite automata

Abstract In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory and bounds on the quality of the approximation in the spectral norm and $\ell ^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

Investigating the impact of the regularization parameter on EEG resting-state source reconstruction and functional connectivity using real and simulated data

Accurate EEG source localization is crucial for mapping resting-state network dynamics and it plays a key role in estimating source-level functional connectivity. However, EEG source estimation techniques encounter numerous methodological challenges, with a key one being the selection of the regularization parameter in minimum norm estimation. This choice is particularly intricate because the optimal amount of regularization for EEG source estimation may not align with the requirements of EEG connectivityanalysis, highlighting a nuanced trade-off. In this study, we employed a methodological approach to determine the optimal regularization coefficient that yields the most effective reconstruction outcomes across all simulations involving varying signal-to-noise ratios for synthetic EEG signals. To this aim, we considered three resting state networks: the Motor Network, the Visual Network, and the Dorsal Attention Network. The performance was assessed using three metrics, at different regularization parameters: the Region Localization Error, source extension, and source fragmentation. The results were validated using real functional connectivity data. We show that the best estimate of functional connectivity is obtained using 10−2, while 10−1 has to be preferred when source localization only is at target.

Dispersive estimates for Maxwell's equations in the exterior of a sphere

The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in L1−L∞ operator norm in terms of time t and powers of h. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator L1−L∞ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.

Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

In this paper, we apply the weak Galerkin (WG) finite element method to solve the singularly perturbed fourth-order boundary value problem in a 2D domain. A Shishkin mesh is used to ensure that the method exhibits uniform convergence, regardless of the singular perturbation parameter. Asymptotically optimal order error estimate in a H2 discrete norm is established for the corresponding WG solutions. Numerical tests are provided to verify the convergence theory.

Koopman modeling for optimal control of the perimeter of multi-region urban traffic networks

The purpose of perimeter control is to regulate the transfer flow between the perimeters of the urban traffic network, so that the vehicle aggregation in each region is maintained at a desired level. It is regarded as one of the most important methods to solve traffic congestion in urban road networks. Accurate mathematical modeling of perimeter controlled traffic dynamics remains a challenge due to its high nonlinearity and uncertainty. Machine learning methods have been used to learn traffic dynamic models for perimeter control. However, these existing techniques lack generalization and interpretability. In this paper, we propose a Koopman modeling approach (i.e., a two-stage method) that uses a new eigenfunction of the Koopman operator based on a novel L0 norm approximation to construct an interpretable finite-dimensional approximation of the Koopman operator, which is a linear operator that describes how eigenfunctions evolve along the trajectory of a specific nonlinear dynamical system. The main advantage of utilizing the Koopman operator is that it can characterize the nonlinear system in a global linear lifted feature space. Furthermore, an algorithm based on the Koopman modeling method and model predictive control is proposed for real-time perimeter control of urban road networks. Some experiments are carried out to demonstrate the effectiveness of the proposed algorithm.

A robust force identification method based on L1+2-norm approximation theory

Obtaining the real force of the structure is the basic condition for efficient vibration reduction design. However, the uneasiness and instability of the inverse problem make the force identification still challenging. On the basis of L2-norm regularization, we propose a new method for force identification based on L1+2-norm approximation theory. By introducing the L1-norm penalty operator, a joint norm approximation optimization model is constructed to limit the non-zero force and punish the force value which approaching to zero, so that it can adapt to the identification of various types of forces. Then the optimal force solution is obtained by using the Primal-dual interior point method. Finally, the effectiveness and accuracy of the proposed method are verified by numerical simulation and experiment of asymmetric plate and double-layer plate structures. Different from previous studies, the L1+2-norm method in this work has high robustness, considers the characteristics of sparse regularization and L2-norm regularization, and can adapt to the identification of continuous forces and sparse forces at the same time, which has potential engineering application value.

Integrating Electroencephalography Source Localization and Residual Convolutional Neural Network for Advanced Stroke Rehabilitation.

Motor impairments caused by stroke significantly affect daily activities and reduce quality of life, highlighting the need for effective rehabilitation strategies. This study presents a novel approach to classifying motor tasks using EEG data from acute stroke patients, focusing on left-hand motor imagery, right-hand motor imagery, and rest states. By using advanced source localization techniques, such as Minimum Norm Estimation (MNE), dipole fitting, and beamforming, integrated with a customized Residual Convolutional Neural Network (ResNetCNN) architecture, we achieved superior spatial pattern recognition in EEG data. Our approach yielded classification accuracies of 91.03% with dipole fitting, 89.07% with MNE, and 87.17% with beamforming, markedly surpassing the 55.57% to 72.21% range of traditional sensor domain methods. These results highlight the efficacy of transitioning from sensor to source domain in capturing precise brain activity. The enhanced accuracy and reliability of our method hold significant potential for advancing brain-computer interfaces (BCIs) in neurorehabilitation. This study emphasizes the importance of using advanced EEG classification techniques to provide clinicians with precise tools for developing individualized therapy plans, potentially leading to substantial improvements in motor function recovery and overall patient outcomes. Future work will focus on integrating these techniques into practical BCI systems and assessing their long-term impact on stroke rehabilitation.