Norm Estimates Research Articles

Adaptive Beamforming with Sidelobe Level Control for Multiband Sparse Linear Array

Multiband antenna arrays have the capability of effectively working in multiple frequency bands and thus significantly simplify the antenna system. To further reduce the system overhead, this paper discusses the joint design of antenna selection and adaptive beamforming for multiband antenna arrays, where the sidelobe level is also controlled so as to alleviate the effect of unknown sporadic interference. Based on the maximum signal-to-interference-plus-noise ratio (SINR) criterion and sidelobe level constraints, the proposed multiband sparse array design is formulated into a nonconvex constrained nonlinear optimization problem with an l0,2-mixed norm regularization. This problem ensures that the same antenna positions are selected at all operating frequencies while the beamformer weights of each frequency are optimized independently. By exploiting the semi-definite relaxation and the reweighted l1,∞-norm approximation, the problem is converted into a series of convex subproblems and is then effectively solved by the proposed iterative reweighted method. Numerical results show that the proposed multiband sparse array significantly reduces the sidelobe levels in all operating frequencies while maintaining the maximum SINR, so it provides superior performance of interference suppression.

MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT OF NUMERICAL SIMULATION OF NUCLEAR CONTAMINATION TREATMENT AND ITS CONVERGENCE ANALYSIS

We consider a nonlinear system with boundary-initial value conditions of convection-diffusion partial differential equations describing nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection-diffusion equations, and the transport of temperature is defined by a heat equation. The transport pressure appears in the concentration equations and heat equation accompanying with Darcy velocity, and controls their processes. The flow equation is solved by the conservative method of mixed volume element and the accuracy of Darcy velocity is improved one order. The method of characteristic mixed volume element is applied to solve the concentrations and the heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm high computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while it has smaller time-truncation error and higher order of accuracy. The mixed volume element method has law of conservation on every element to treat the diffusion and it can obtain numerical solution is of the concentration and adjoint vectors. Using the theory and technique of priori estimate of differential equations, we derive an optimal second order estimate in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve such an international famous problem.

Reduced model for fracture of geometrically exact planar beam: Non‐local variational formulation, ED‐FEM approximation and operator split solution

SummaryIn this work, we present a novel reduced model for computing the solution to a bending fracture problem of the geometrically exact beam. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model. The non‐local variational formulation is proposed to deal with the softening phenomena characteristic of fracture. The fracture energy is introduced as the main fracture parameter, rather than (in general) ill‐defined characteristic length used for previous non‐local models. The discrete approximation is built in terms of the embedded‐discontinuity finite element method (ED‐FEM), which needs no initial crack for optimal performance nor detailed crack kinematics description, such as in the X‐FEM or the phase‐field approach. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best‐approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and stress) from local (internal) variables including the crack opening. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Well‐posedness and decay for 3D non‐isothermal Navier–Stokes–Allen–Cahn system in ℝ3

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in . This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with “good” linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small.

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

Norm Estimates for the Kakeya Maximal Function in High Dimensions with Respect to General Measures

Norm Estimates for the Kakeya Maximal Function in High Dimensions with Respect to General Measures

Differences in generation and maintenance between ictal and interictal generalized spike-and-wave discharges in childhood absence epilepsy: A magnetoencephalography study

PurposeChildhood absence epilepsy (CAE) is characterized by impaired consciousness and distinct electroencephalogram (EEG) patterns. However, interictal epileptiform discharges (IEDs) do not lead to noticeable symptoms. This study examines the disparity between ictal and interictal generalized spike-and-wave discharges (GSWDs) to determine the mechanisms behind CAE and consciousness. MethodsWe enrolled 24 patients with ictal and interictal GSWDs in the study. The magnetoencephalography (MEG) data were recorded before and during GSWDs at a sampling rate of 6000 Hz and analyzed across six frequency bands. The absolute and relative spectral power were estimated with the Minimum Norm Estimate (MNE) combined with the Welch technique. All the statistical analyses were performed using paired-sample tests. ResultsDuring GSWDs, the right lateral occipital cortex indicated a significant difference in the theta band (5–7 Hz) with stronger power (P = 0.027). The interictal group possessed stronger spectral power in the delta band (P < 0.01) and weaker power in the alpha band (P < 0.01) as early as 10 s before GSWDs in absolute and relative spectral power. Additionally, the ictal group revealed enhanced spectral power inside the occipital cortex in the alpha band and stronger spectral power in the right frontal regions within beta (15–29 Hz), gamma 1 (30–59 Hz), and gamma 2 (60–90 Hz) bands. ConclusionsGSWDs seem to change gradually, with local neural activity changing even 10 s before discharge. During GSWDs, visual afferent stimulus insensitivity could be related to the impaired response state in CAE. The inhibitory signal in the low-frequency band can shorten GSWD duration, thereby achieving seizure control through inhibitory effect strengthening.

Tuning Minimum-Norm regularization parameters for optimal MEG connectivity estimation

The accurate characterization of cortical functional connectivity from Magnetoencephalography (MEG) data remains a challenging problem due to the subjective nature of the analysis, which requires several decisions at each step of the analysis pipeline, such as the choice of a source estimation algorithm, a connectivity metric and a cortical parcellation, to name but a few. Recent studies have emphasized the importance of selecting the regularization parameter in minimum norm estimates with caution, as variations in its value can result in significant differences in connectivity estimates. In particular, the amount of regularization that is optimal for MEG source estimation can actually be suboptimal for coherence-based MEG connectivity analysis. In this study, we expand upon previous work by examining a broader range of commonly used connectivity metrics, including the imaginary part of coherence, corrected imaginary part of Phase Locking Value, and weighted Phase Lag Index, within a larger and more realistic simulation scenario. Our results show that the best estimate of connectivity is achieved using a regularization parameter that is 1 or 2 orders of magnitude smaller than the one that yields the best source estimation. This remarkable difference may imply that previous work assessing source-space connectivity using minimum-norm may have benefited from using less regularization, as this may have helped reduce false positives. Importantly, we provide the code for MEG data simulation and analysis, offering the research community a valuable open source tool for informed selections of the regularization parameter when using minimum-norm for source space connectivity analyses.

EEG cortical network reveals the temporo-spatial mechanism of visual search

This study aims to explore a method based on brain networks for implicit attention by using wavelet coherence as feature to identify individual targets in the visual field, find the optimal classification rhythm and time window, and investigate the relationship between the optimal rhythm and N2pc event-related potential. The study uses a weighted minimum norm estimate to locate the sources of the scalp EEG and reconstructs the source time series. The functional connectivity between brain areas during the visual search process is evaluated using wavelet coherence analysis, and a lateral difference network is constructed based on the difference in coherence values between the left and right visual fields. A support vector machine classifier is trained based on the wavelet coherence network features to identify the target in the left or right visual field. We also extract N2pc from the source activity data of the parieto-occipital brain region and record the time period in which N2pc occurred. The study finds that the best classification performance is achieved in the theta rhythm from 200 to 400 ms and achieved an average classification accuracy of 87% (chance level: 51.07%) in a serial search task. And this time window corresponds to the time period when N2pc appeared. The results show that the use of wavelet coherence analysis to evaluate the functional connectivity between brain areas during the visual search process provides a new approach for analyzing brain activity. The study's findings regarding the relationship between the N2pc and theta rhythm and the effectiveness of using wavelet coherence network features based on the theta rhythm for visual search classification contribute to the understanding of the neural mechanisms underlying visual search.

A weak Galerkin finite element method for nonlinear convection-diffusion equation

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.

2D cross-hole electromagnetic inversion algorithms based on regularization algorithms

Abstract The cross-hole electromagnetic (EM) method, which is currently at the forefront of electric logging technology, fundamentally solves the problems of the lateral imaging ability of single-well logging and the lack of detection of inter-well physical properties. However, due to the complexity of underground reservoir distribution and the non-uniqueness problem of geophysical inversion, there remains a lack of practical and effective cross-hole electromagnetic inversion methods. Our goal is to develop an efficient method to reduce the non-uniqueness of the physical property model recovered in the inversion. It is worth noting that the regularization algorithm, as a means to approximately solve inversion problems, can obtain different solutions by changing the form of the regularization function, so as to ensure the stability of inversion results and conform to the smooth or non-smooth characteristics in known geology or geophysics. We adjust the features of the final inversion model in a defined framework by changing the values of the $\alpha $ coefficient in the regularization and using the Lawson norm as a ${l}_p$-norm approximation form for $p \in [ {0,2} ]$. At the same time, the iteratively reweighted least-squares method is used to solve the optimization problem, and the gradient in the Gauss–Newton solution is adjusted successively to ensure that every term in the regularization contributes to the final solution. Compared with the traditional ${l}_2$-norm inversion method, the sparse inversion method can make more effective use of information regarding known physical properties and obtain better inversion results. Then, the effectiveness of our inversion method is verified by model tests and inversion of measured data in a mining area.

New error analysis and recovery technique of a class of fully discrete finite element methods for the dynamical inductionless MHD equations

In this paper, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic (MHD) equations. The methods use the semi-implicit backward Euler scheme in time and use the standard inf–sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L∞(0,T;L2(Ω))-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor–Hood element supplies the optimal accuracy for the velocity in L2(0,T;H1(Ω))-norm and the pressure in L1(0,T;L2(Ω))-norm. Moreover, we propose a simple recovery technique to obtain a new numerical current density of first-order higher accuracy in the spatial direction by re-solving a mixed Poisson equation. Numerical experiments are performed to verify the theoretical analysis.

Transformation of Two-Dimensional Roesser Model Into Two-Dimensional Decoupled Model: Application to Model Reduction via Balanced Framework

Two transformations are presented in this research; the first approach takes a generic two-dimensional (2-D) framework and transforms it into a diagonalized 2-D framework without resorting to minimal rank-decomposition criteria. Afterward, it converts the diagonalized system into two subsystems (i.e., one-dimensional (1-D) cascaded systems). Moreover, the proposed transformation retains the symmetrical structure of the two decomposed subsystems. Likewise, the second transformation converts 1-D submodels to the balanced structure used for model reduction operations. The proposed model reduction approach efficiently performs the model reduction task on a given 2-D diagonalized model by utilizing optimal Hankel norm approximation. The numerical results demonstrate that the proposed transformations efficiently convert the standard 2-D framework to a diagonalized model form and develop a stable reduced-order model.

REGULARIZED JACOBI-TYPE ADMM-METHOD FOR FINDING SOLUTIONS TO GENERALIZED NASH EQUILIBRIUM PROBLEM

In this paper, we extended the well-known alternating direction method of multipliers (ADMM) for optimization problems to generalized Nash equilibrium problems (GNEP) with shared constraints. We developed an ADMM-type algorithm with fixed regularization to tackle the problem (GNEP) where an upper estimate for the operator norm is not known and then we apply a multiplier-penalty in order to get rid of the joint constraints. We equipped the Hilbert space with an appropriate weighted scalar product and it turns out to be weakly convergent under a lipschitz and monotonicity assumption. A proximal term is then added to improve the convergence properties. Furthermore, a comparative analysis of quasi-variational inequality method, interior point method, penalty method and the proposed method are discussed.

Schwarzian Norm Estimates for Some Classes of Analytic Functions

Schwarzian Norm Estimates for Some Classes of Analytic Functions

A nodally bound-preserving finite element method

Abstract This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.

Maximal regularity of parabolic equations associated with a discrete Laplacian

Let Δd be the discrete Laplacian defined on Zd by settingΔdf(n→)=∑j=1d−[f(n→+e→j)+f(n→−e→j)−2f(n→)],n→∈Zd, where {e→j:j=1,…,d} is the standard basis for Rd.In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation{ut+Δdu=f,t∈[0,T)u(0,⋅)=0, where T∈(0,∞).

An upwind finite volume element method on a Shishkin mesh for singularly perturbed convection–diffusion problems

In this paper, we construct and analyze an upwind finite volume element method on a Shishkin mesh for singularly perturbed convection–diffusion problems. We prove the stability of the method under the assumption of the convection and reaction term coefficients. The error estimate in the energy norm is presented on the Shishkin mesh, and the optimal error bound O(N−1(lnN)3/2) is obtained. This error bound is uniformly valid with respect to the singular perturbation parameter. Numerical examples are provided to illustrate our theoretical results.

Wavelet estimation of norms for a probability density with negatively dependent biased data

This article considers wavelet estimations for density functions in Besov spaces based on negatively dependent biased random samples. More precisely, wavelet estimators are defined and their L p ( 1 ≤ p < ∞ ) convergence rates are provided when the parameter μ is known and unknown, respectively.

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.