Norm Estimates Research Articles

A modified weak Galerkin finite element method for parabolic equations on anisotropic meshes

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method.

The effects of attention in auditory-visual integration revealed by time-varying networks.

Attention and audiovisual integration are crucial subjects in the field of brain information processing. A large number of previous studies have sought to determine the relationship between them through specific experiments, but failed to reach a unified conclusion. The reported studies explored the relationship through the frameworks of early, late, and parallel integration, though network analysis has been employed sparingly. In this study, we employed time-varying network analysis, which offers a comprehensive and dynamic insight into cognitive processing, to explore the relationship between attention and auditory-visual integration. The combination of high spatial resolution functional magnetic resonance imaging (fMRI) and high temporal resolution electroencephalography (EEG) was used. Firstly, a generalized linear model (GLM) was employed to find the task-related fMRI activations, which was selected as regions of interesting (ROIs) for nodes of time-varying network. Then the electrical activity of the auditory-visual cortex was estimated via the normalized minimum norm estimation (MNE) source localization method. Finally, the time-varying network was constructed using the adaptive directed transfer function (ADTF) technology. Notably, Task-related fMRI activations were mainly observed in the bilateral temporoparietal junction (TPJ), superior temporal gyrus (STG), primary visual and auditory areas. And the time-varying network analysis revealed that V1/A1↔STG occurred before TPJ↔STG. Therefore, the results supported the theory that auditory-visual integration occurred before attention, aligning with the early integration framework.

Curve Fitting Based Minimum Norm Estimation (CFB-MNE) for motor unit spatial localization using high density surface electromyogram signals

The purpose of the study is the real time spatial localization of single motor units using simulated Motor Unit Action Potential dictionary from a cylindrical muscle volume conductor model by means of a state-of-the-art curve fitting method. This Curve Fitting Based Minimum Norm Estimation (CFB-MNE) was made possible by using minimum norm estimation methods to solve an underdetermined inverse problem knowing produced HD-sEMG signals. Specific data extracted from the inverse problem of a pre-determined number of simulated motor units were used to create a 3D curve, which can be used as a fitting curve to anticipate the unknown location of motor units. Results show that the proposed algorithms succeeded in accurately localizing motor units with varying noise levels in a fast duration (less than 100 ms). From all the simulations, a mean root mean square error of maximum 1 mm was recorded for the localization of the depth (less than 1 mm) and a negligible error for the angular position (less than 1°). The proposed work is an innovative algorithm that aids in non-invasively localizing motor units within a muscle in real time. Further efforts are planned to manage MUAP superposition and type. Final applications are related to prosthetic control, rehabilitation guidance or aging monitoring.

Evaluation of Brain Source Localization Methods Based on Test-Retest Reliability With Multiple Session EEG Data.

Various EEG source localization methods have been proposed for functional brain research. The evaluation and comparison of these methods are usually based on simulated data but not real EEG data, as the ground truth of source localization is unknown. In this study, we aim to evaluate source localization methods quantitatively under the real situation. We examined the test-retest reliability of the source signals reconstructed from a public six-session EEG data of 16 subjects performing face recognition tasks by five mainstream methods, including weighted minimum norm estimation (WMN), dynamical Statistical Parametric Mapping (dSPM), Standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA), dipole modeling and linearly constrained minimum variance (LCMV) beamformers. All methods were evaluated in terms of peak localization reliability and amplitude reliability of source signals. In the two brain regions responsible for static face recognition, all methods have promising peak localization reliability, with WMN showing the smallest peak dipole distance between session pairs. The spatial stability of source localization in the familiar face condition is better than those in the unfamiliar face and the scrambled face conditions in the face recognition areas in the right hemisphere. In addition, the test-retest reliability of source amplitude by all methods is good to excellent under the familiar face condition. Stable and reliable results for source localization can be obtained in the presence of evident EEG effects. Due to different levels of a priori knowledge, different source localization methods have different applicable scenarios. These findings provide new evidence for the validity of source localization analysis and a new perspective for the evaluation of source localization methods on real EEG data.

On norms of derivations implemented by self-adjoint operators

In this paper, we concentrate on norms of derivations implemented by self-adjoint operators. We determine the upper and lower norm estimates of derivations implemented by self-adjoint operators.The results show that the knowledge of self-adjoint governs the quantum chemical system in which the eigenvalue and eigenvector of a self-adjoint operator represents the ground state energy and the ground state wave function of the system respectively.

Sensor array design of optically pumped magnetometers for accurately estimating source currents

An optically pumped magnetometer (OPM) is a new generation of magnetoencephalography (MEG) devices that is small, light, and works at room temperature. Due to these characteristics, OPMs enable flexible and wearable MEG systems. On the other hand, if we have a limited number of OPM sensors, we need to carefully design their sensor arrays depending on our purposes and regions of interests (ROIs). In this study, we propose a method that designs OPM sensor arrays for accurately estimating the cortical currents at the ROIs. Based on the resolution matrix of minimum norm estimate (MNE), our method sequentially determines the position of each sensor to optimize its inverse filter pointing to the ROIs and suppressing the signal leakage from the other areas. We call this method the Sensor array Optimization based on Resolution Matrix (SORM). We conducted simple and realistic simulation tests to evaluate its characteristics and efficacy for real OPM-MEG data. SORM designed the sensor arrays so that their leadfield matrices had high effective ranks as well as high sensitivities to ROIs. Although SORM is based on MNE, the sensor arrays designed by SORM were effective not only when we estimated the cortical currents by MNE but also when we did so by other methods. With real OPM-MEG data we confirmed its validity for real data. These analyses suggest that SORM is especially useful when we want to accurately estimate ROIs’ activities with a limited number of OPM sensors, such as brain-machine interfaces and diagnosing brain diseases.

Anti-saturation fault-tolerant control for Markov jump nonlinear systems with unknown control coefficients and unmodeled dynamics

In this paper, the problem of anti-saturation fault-tolerant control is investigated for a class of Markov jump nonlinear systems subject to input saturation. The nonlinear system under investigation contains unknown control coefficients, unmodeled dynamics and faults, which evolve according to the Markov chain. A new saturation dynamic filter is designed to reduce the effects of input saturation. The appropriate Lyapunov function is constructed by utilizing available dynamic signals to deal with unmodeled dynamics. The Nussbaum technique is combined with the norm estimation method, so that the challenges brought by faults and unknown parameters of Markov switching are effectively overcome. In addition, based on the command filtering method, the anti-saturation fault-tolerant controller is designed to ensure that all signals of the closed-loop system are bounded in probability. Finally, two examples are given to validate the effectiveness of the developed method.

Numerical modeling of wave propagation phenomena in thermo-poroelastic media via discontinuous Galerkin methods

We present and analyze a high-order discontinuous Galerkin method for the space discretization of the wave propagation model in thermo-poroelastic media. The proposed scheme supports general polytopal grids. Stability analysis and hp-version error estimates in suitable energy norms are derived for the semi-discrete problem. The fully-discrete scheme is then obtained based on employing an implicit Newmark-β time integration scheme. A wide set of numerical simulations is reported, both for the verification of the theoretical estimates and for examples of physical interest. A comparison with the results of the poroelastic model is provided too, highlighting the differences between the predictive capabilities of the two models.

Preferred vector machine for forest fire detection

Machine learning-based fire detection/recognition is very popular in forest-monitoring systems. However, without considering the prior knowledge, e.g., equal attention on both classes of the fire and non-fire samples, fire miss-detected phenomena frequently appeared in the current methods. In this work, considering model’s interpretability and the limited data for model-training, we propose a novel pixel-precision method, termed as PreVM (Preferred Vector Machine). To guarantee high fire detection rate under precise control, a new L0 norm constraint is introduced to the fire class. Computationally, instead of the traditional L1 re-weighted techniques in L0 norm approximation, this L0 constraint can be converted into linear inequality and incorporated into the process of parameter selection. To further speed up model-training and reduce error warning rate, we also present a kernel-based L1 norm PreVM (L1-PreVM). Theoretically, we firstly prove the existence of dual representation for the general Lp (p≥1) norm regularization problems in RKHS (Reproducing Kernel Hilbert Space). Then, we provide a mathematical evidence for L1 norm kernelization to conquer the case when feature samples do not appear in pairs. The work also includes an extensive experimentation on the real forest fire images and videos. Compared with the-state-of-art methods, the results show that our PreVM is capable of simultaneously achieving higher fire detection rates and lower error warning rates, and L1-PreVM is also superior in real-time detection.

MEG language mapping using a novel automatic ECD algorithm in comparison with MNE, dSPM, and DICS beamformer.

The single equivalent current dipole (sECD) is the standard clinical procedure for presurgical language mapping in epilepsy using magnetoencephalography (MEG). However, the sECD approach has not been widely used in clinical assessments, mainly because it requires subjective judgements in selecting several critical parameters. To address this limitation, we developed an automatic sECD algorithm (AsECDa) for language mapping. The localization accuracy of the AsECDa was evaluated using synthetic MEG data. Subsequently, the reliability and efficiency of AsECDa were compared to three other common source localization methods using MEG data recorded during two sessions of a receptive language task in 21 epilepsy patients. These methods include minimum norm estimation (MNE), dynamic statistical parametric mapping (dSPM), and dynamic imaging of coherent sources (DICS) beamformer. For the synthetic single dipole MEG data with a typical signal-to-noise ratio, the average localization error of AsECDa was less than 2 mm for simulated superficial and deep dipoles. For the patient data, AsECDa showed better test-retest reliability (TRR) of the language laterality index (LI) than MNE, dSPM, and DICS beamformer. Specifically, the LI calculated with AsECDa revealed excellent TRR between the two MEG sessions across all patients (Cor = 0.80), while the LI for MNE, dSPM, DICS-event-related desynchronization (ERD) in the alpha band, and DICS-ERD in the low beta band ranged lower (Cor = 0.71, 0.64, 0.54, and 0.48, respectively). Furthermore, AsECDa identified 38% of patients with atypical language lateralization (i.e., right lateralization or bilateral), compared to 73%, 68%, 55%, and 50% identified by DICS-ERD in the low beta band, DICS-ERD in the alpha band, MNE, and dSPM, respectively. Compared to other methods, AsECDa's results were more consistent with previous studies that reported atypical language lateralization in 20-30% of epilepsy patients. Our study suggests that AsECDa is a promising approach for presurgical language mapping, and its fully automated nature makes it easy to implement and reliable for clinical evaluations.

A modified weak Galerkin method for (curl)-elliptic problem

In this paper, we design and analyze a modified weak Galerkin (MWG) finite element method for H ( curl ) -elliptic problem. We first introduce a new discrete weak curl operator and the MWG finite element space. The modified weak Galerkin method does not require the penalty parameter by comparing with traditional DG methods. We prove an optimal error estimate in energy norm. At last, we provide the numerical results to confirm these theoretical results.

The K-functional and saturated linear approximation processes

It is shown that already the knowledge of the saturation class , {{mathfrak {F}}}(J_t, L^p) of a saturated convolution approximation process , {J_t} on , L^p({{mathbb {R}}}^n),, 1le p< infty , completely determines its norm approximation behavior. This is achieved by using the , K-functional , K(t,f;L^p,{{mathfrak {F}}}(J_t, L^p)) as a comparison scale, which relates on the one hand the approximation process and on the other appropriate moduli of smoothness. This implies that simultaneously one gets the so-called direct and inverse theorems. There are open problems if the saturation order is slightly perturbed, e.g., by a , |log t|^lambda -factor, , lambda >0. Proofs are mainly based on the Fourier transformation.

Robust Beamforming Design for Cache-Enabled C-RAN With Fronthaul Multicast

In this article, we study a robust beamforming design problem in the cached-enable cloud radio access network, where the imperfect channel state information of the access links are considered. Two transmission schemes, nonsimultaneous transmission (NST) scheme and simultaneous transmission (ST) scheme, are developed. To improve the system performance, the cooperative transmission technique is applied. In NST scheme, the base stations (BSs) obtain the uncached required files from the central processor (CP) and then transmit all required files to users. Based on this, we design an effective transmission model based on cache files and the qualities of the fronthaul link channels, and formulate a transmission delay minimization problem. Then, an alternative optimization algorithm is developed to solve it. In ST scheme, the BSs obtain the uncached files from the CP and transmit the cached files to users simultaneously. We divide two cases according to the access link transmission rate and cached files. For each case, we formulate an optimization problem by jointly designing the fronthaul multicast beamforming and access link transmission beamforming. Next, an iterative algorithm is developed via the L0 norm approximation, successive convex approximation and semidefinite relaxation techniques. Finally, simulation results show the proposed schemes can reduce the transmission delay better than the conventional multicast schemes.

The Stević-Sharma operator on the Lipschitz space into the logarithmic Bloch space

In this paper, we study the boundedness and compactness of the Stević-Sharma operator on the Lipschitz space into the logarithmic Bloch space. Also, we give an estimate for the essential norm of the above operator.

L2 estimates for weak Galerkin finite element methods for second-order wave equations with polygonal meshes

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L2 norm is shown to hold as O(hk+1+τ2), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in U, where U is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used to obtain either necessary or sufficient conditions for the univalence of a function f. Because of the computational difficulty, the pre-Schwarzian norm has received more attention than the Schwarzian norm. It has applications in the theory of hypergeometric functions, conformal mappings, Teichmüller spaces, and univalent functions. In this paper, we find sharp norm estimates of the pre-Schwarzian derivatives of certain subfamilies of analytic functions involving some conic-like image domains. These results may also be extended to the families of strongly starlike, convex, as well as to functions with symmetric and conjugate symmetric points.

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Generalized weak Galerkin finite element methods for biharmonic equations

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H2 norm and a L2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.

On Matrices with Only One Non-SDD Row

The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension.

Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in -norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in -norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.