L1-norm Research Articles

Schwarz-Pick lemma for (α,β)-harmonic functions in the unit disc

We obtain Schwarz–Pick lemma for (α,β)-harmonic functions u in the disc, where α and β are complex parameters satisfying ℜα+ℜβ>−1. We prove sharp estimate of ‖Du(0)‖ for such functions in terms of Lp norm of the boundary function. Also, we give asymptotically sharp estimate of ‖Du(z)‖. This estimate is generalized to higher order derivatives. Our results extend earlier results for α-harmonic and Tα-harmonic functions.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Distributed adaptive robust containment control for reaction–diffusion neural networks with external disturbances under directed graphs

In this paper, the leader–follower robust synchronization issue is mainly addressed for reaction–diffusion neural networks (RDNNs) with multiple leaders and external disturbances under directed graphs. Based on the σ modification approach, we propose a novel distributed adaptive controller by adding a term −θi(ci−1)2 to avoid the phenomenon of parameter drift, that is, the adaptive parameters grow to infinity. Meanwhile, different from the adaptive control algorithm proposed in the undirected graph, we introduce a new function χi(t) to provide additional freedom for the design to achieve robust containment when confronted with external disturbances. Further, the robustness of tracking synchronization with one leader is guaranteed by the proposed adaptive controller when the external disturbances concerning L2 norm are bounded. Finally, relevant numerical simulation graphics are displayed separately to verify the correctness of the related theoretical results.

Bilinear strongly generalized fractional integrals and their commutators over non-homogeneous metric spaces

The aim of this paper is to establish the boundedness of the bilinear strongly generalized fractional integral operator T˜α and its commutator T˜α,b1,b2 generated by b1,b2∈RBMO˜(μ) and T˜α on non-homogeneous metric measure spaces. Under assumption that the dominating function λ and the Lebesgue measure function u(⋅,⋅) defined on X×(0,∞) satisfy certain conditions, the authors prove that the bilinear generalized fractional integral operator T˜α is bounded from product of Lebesgue spaces Lp1(μ)×Lp2(μ) into spaces Lq(μ), bounded from product of Morrey spaces Mq1p1(μ)×Mq2p2(μ) into spaces Mts(μ), and it is also bounded from product of spaces Lp1,u1(μ)×Lp2,u2(μ) into spaces Lq,u(Rn), where u1×u2=u, 1q=1p1+1p2−2α, 1t=1q1+1q2−2α and 1s=1p1+1p2−2α for 1<q1≤p1<1α and 1<q2≤p2<1α. Furthermore, the boundedness of the T˜α,b1,b2 on spaces Lp(μ), on spaces Mqp(μ) and on spaces Lp,u(μ) is obtained.

Accurate detection method of traveling wave shape based on EEMD and L1 norm regularization

Aiming at the problem of waveform distortion of the secondary traveling wave obtained from the primary traveling wave signal transmitted by the voltage traveling wave sensor, an accurate detection method of voltage traveling wave based on ensemble empirical mode decomposition (EEMD) and L1 norm least squares regularization is proposed in this paper. Firstly, the forward transfer function model of the specialized voltage traveling wave sensor is established, and the inversion model is derived to analyze the phase frequency and amplitude frequency characteristics of the secondary traveling wave after transmission. EEMD is used to decompose the secondary traveling wave into the intrinsic modal function (IMF) components of different frequency bands. The L1 norm least square regularization inversion algorithm was used to invert each IMF component. Finally, the IMF' components after inversion are synthesized to obtain the primary traveling wave. Simulation and experimental results demonstrate that the proposed method is not affected by noise, mode aliasing effect and regularization parameter accuracy, and the obtained waveform has high similarity to the simulated waveform, which enables precise identification of fault traveling wave.

Enhancing quantum coherence in multiqubit-interacting system

Abstract We propose a quantum coherence protection scheme for a two-qubit system by considering the exact evolution of multi-interacting qubits in a common reservoir. We find that the l 1 norm of coherence and the relative entropy of coherence can be noticeably enhanced by increasing the number of qubits in the weak/strong system-environment coupling regime, which contrasts sharply with the impact of coupling between qubits. Moreover, in the infinite-time limit, all the coherence measures reach their steady values which are determined only by the number of qubits.

Class-Balanced Regularization for Long-Tailed Recognition

Long-tailed recognition performs poorly on minority classes. The extremely imbalanced distribution of classifier weight norms leads to a decision boundary biased toward majority classes. To address this issue, we propose Class-Balanced Regularization to balance the distribution of classifier weight norms so that the model can make more balanced and reasonable classification decisions. In detail, CBR separately adjusts the regularization factors based on L2 regularization to be correlated with the class sample frequency positively, rather than using a fixed regularization factor. CBR trains balanced classifiers by increasing the L2 norm penalty for majority classes and reducing the penalty for minority classes. Since CBR is mainly used for classification adjustment instead of feature extraction, we adopt a two-stage training algorithm. In the first stage, the network with the traditional empirical risk minimization is trained, and in the second stage, CBR for classifier adjustment is applied. To validate the effectiveness of CBR, we perform extensive experiments on CIFAR10-LT, CIFAR100-LT, and ImageNet-LT datasets. The results demonstrate that CBR significantly improves performance by effectively balancing the distribution of classifier weight norms.

Research on 3D Visualization of Drone Scenes Based on Neural Radiance Fields

Neural Radiance Fields (NeRFs), as an innovative method employing neural networks for the implicit representation of 3D scenes, have been able to synthesize images from arbitrary viewpoints and successfully apply them to the visualization of objects and room-level scenes (&lt;50 m2). However, due to the capacity limitations of neural networks, the rendering of drone-captured scenes (&gt;10,000 m2) often appears blurry and lacks detail. Merely increasing the model’s capacity or the number of sample points can significantly raise training costs. Existing space contraction methods, designed for forward-facing trajectory or the 360° object-centric trajectory, are not suitable for the unique trajectories of drone footage. Furthermore, anomalies and cloud fog artifacts, resulting from complex lighting conditions and sparse data acquisition, can significantly degrade the quality of rendering. To address these challenges, we propose a framework specifically designed for drone-captured scenes. Within this framework, while using a feature grid and multi-layer perceptron (MLP) to jointly represent 3D scenes, we introduce a Space Boundary Compression method and a Ground-Optimized Sampling strategy to streamline spatial structure and enhance sampling performance. Moreover, we propose an anti-aliasing neural rendering model based on Cluster Sampling and Integrated Hash Encoding to optimize distant details and incorporate an L1 norm penalty for outliers, as well as entropy regularization loss to reduce fluffy artifacts. To verify the effectiveness of the algorithm, experiments were conducted on four drone-captured scenes. The results show that, with only a single GPU and less than two hours of training time, photorealistic visualization can be achieved, significantly improving upon the performance of the existing NeRF approaches.

Some Lp norm involving the polar derivative of a polynomial

Let [Formula: see text], [Formula: see text], is a polynomial of degree [Formula: see text] having all zeros in [Formula: see text], [Formula: see text], then for [Formula: see text] with [Formula: see text], [Formula: see text] and for each [Formula: see text] with [Formula: see text]; it was proved by (N. A. Rather et al., Some [Formula: see text] inequalities involving the polar derivative of a polynomial, Appl. Math. E-Notes 14 (2014) 116–122) [Formula: see text] [Formula: see text] In this paper, we establish some refinements and generalizations of above and some known polynomial inequalities concerning the polar derivative of a polynomial.

Poisson image deblurring with frame-based nonconvex regularization

Poisson image deblurring, which aims to restore the latent image from its blurred and noisy observation, has drawn significant attention in image processing. Due to its ill-posed nature, enhancing image quality often involves incorporating a well-defined prior to effectively regularize the ill-posed inverse problem. Building upon the framelet system, we propose a frame-based nonconvex regularization method for Poisson image deblurring. The method is formulated by combining a data-fitting term with the difference of two norms, namely ℓ1 and ℓ2, on the latent image. We solve the optimization problem by combining the difference of convex functions algorithm (DCA) with the alternating direction method of multipliers (ADMM), establishing its convergence. We further employ a simulated annealing procedure and show that proposed algorithm almost certainly converges to a global minimum. Two different approaches are employed to handle the frame-based ℓ1 norm within the ADMM framework. In particular, the frame-based nonconvex regularization method is also considered for the blind Poisson problem. An effective recovery model and its algorithm are presented. Experimental results demonstrate the effectiveness of our proposed models compared with other models in terms of quantitative metrics and visual quality.

A variational model for cartoon-texture decomposition of a color image

Image decomposition is one of the most important tasks in image processing. In this paper, we develop a novel saturation-value total variation model for color image decomposition. The approach is to propose a variational model containing an energy functional to derive the cartoon and texture decomposition in the saturation-value color space so that the color can be kept in the resulting cartoon component. The proposed idea is different from existing color image decomposition model where color information usually exists in both cartoon and texture components. In the variational model, saturation-value total variation is incorporated for regularizing the cartoon component of a color image, meanwhile, L1 norm is considered for modeling the texture component in the saturation-value color space. Theoretically, we develop a smooth approximation approach to study the existence of the solution of the proposed optimization problem. We formulate an effective and efficient algorithm to solve the proposed minimization problem based on the framework of alternating direction method of multipliers. Numerical examples are presented to demonstrate that the performance of the proposed model is better than that of other testing methods for several testing color images.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

Optimal long time error estimates of a second-order decoupled finite element method for the Stokes–Darcy problem

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δt≤C(physical parameters), the optimal long time error estimates are obtained both in the L2 norm and in the H1 norm. Numerical results are provided to illustrate the convergence rate O(Δt2) of the temporal approximation.

Multi-Pursuer and One-Evader Evasion Differential Game with Integral Constraints for an Infinite System of Binary Differential Equations

In the Hilbert space l2, a differential evasion game involving multiple pursuers is considered. Integral constraints are imposed on player control functions. The pursuers are tasked with bringing the state of a system back to the origin of l2, while the evader simultaneously tries to avoid it. It is assumed that the energy of the evader is greater than the total energy of the pursuers. In this paper, we contribute to the solution of the differential evasion game with multiple pursuers by building an exact strategy for the evader.

A Fast IAA–Based SR–STAP Method for Airborne Radar

Space–time adaptive processing (STAP) is an effective technology in clutter suppression and moving target detection for airborne radar. When working in the heterogeneous environment, the number of training samples that satisfy independent and identically distributed (IID) conditions is insufficient, making it difficult to ensure the estimation accuracy of the clutter plus noise covariance matrix for traditional STAP methods. Sparse recovery–based STAP (SR–STAP) methods have received widespread attention in the past few years. The accurate estimation of the clutter plus noise covariance matrix can be achieved using only a few training samples. The iterative adaptive approach (IAA) can quickly and accurately estimate the power spectrum, but applying this method directly to the STAP method cannot produce good performance. In this paper, a fast IAA–based SR–STAP method is proposed. Based on the weighted l1 problem, the IAA spectrum is used as a weighted term to obtain a good approximation. In order to obtain an analytical solution, we use the weighted l2 norm to approximate the weighted l1 norm without loss of performance. Compared with the IAA–STAP method, the proposed method is more robust to errors. Moreover, the proposed method has a fast computational speed. The effectiveness of the proposed method is demonstrated by simulations.

A-priori evaluation of data-driven models for large-eddy simulations in Rayleigh–Bénard convection

Natural convection is a commonly occurring heat-transfer problem in many industrial flows and its prediction with conventional large eddy simulations (LES) at higher Rayleigh numbers using progressively coarser grids leads to increasingly inaccurate estimates of important performance indicators, such as Nusselt number (Nu). Thus, to improve the heat transfer predictions, we utilize Gene Expression Programming (GEP) to develop sub-grid scale (SGS) stress and SGS heat-flux models simultaneously for LES. The models’ development is performed using reference direct numerical simulation data of a typical natural convection case, i.e. the Rayleigh–Bénard convection (RBC). The training frameworks are built by using alignment as the evaluation metrics of the basis functions when comparing to the Gaussian-filtered SGS stress and SGS heat-flux. The trained models in isotropic form, by utilizing the L2 norm of the grid cell ΔL2 as the length scale, demonstrate good performance in the bulk region, but less improved performance in the near wall region. It is shown, that for LES of wall-bounded flow, the GEP models in anisotropic form, i.e. using different grid length scale for the different spatial directions, are required to obtain generalized models suitable for different regions. Consequently, the a-priori results demonstrate a significant improvement in the prediction of both instantaneous and mean quantities for a wide range of filter widths.

Turbulent image deblurring using a deblurred blur kernel

Abstract In the context of addressing a noisy turbulence-degraded image, it is common to use a denoising low-pass filter before implementing a deblurring algorithm. However, this filter not only suppresses noise but also induces a certain degree of blur into the degraded image. This blur effect causes a blurred estimate of the true blur kernel and ultimately leads to a distorted estimate of the latent clear image. To tackle this issue, this paper presents an innovative single-image deblurring method. It integrates a dedicated blur kernel deblurring step to mitigate the effects of the denoising filter. The L0 norm and L2 norm serve as the respective constraints for latent clear image and blur kernel. Experimental results on both synthetic and real-world turbulence-degraded images demonstrate the effectiveness and efficiency of the proposed method.

Unconditional stability and error estimates of the modified characteristics FEMs for the Micropolar Navier–Stokes Equations

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ-independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L2 norm and H1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis.

Smooth hard shrinkage operator for tensor completion based on self-adaptive transforms

The tensor completion(TC) problem has attracted wide attention due to its significant practical significance and broad application background. Various approaches have been used to solve this problem, including approximating the rank function through its convex envelope, the tensor nuclear norm. However, due to the gap between the rank function and its convex envelope, this approach often fails to achieve satisfactory recovery. Recent developments have highlighted that non-explicit non-convex functions, especially the one induced by the Smooth Hard(SH) shrinkage operator, can approximate the l0 norm more effectively compared to other methods, and this function facilitates the reconstruction of noisy MRI data with minimal sampling rate requirements. In our paper, we implement the SH shrinkage operator within the TC framework, achieving accurate tensor data recovery at lower sampling rates. Moreover, we adopt a method that adaptively generates transformation matrices based on specific features of the data, which, compared to traditional TC models, can better exploit the inherent low-rank properties of tensor data. Additionally, our approach integrates a reweighting strategy to efficiently merge tensors recovered from various dimensions. Finally, a feasible minimization algorithm is proposed, which combines the non-explicit non-convex regularizer and self-adaptive transforms. It is proven that the constructed sequence of the proposed algorithm is a Cauchy sequence and can converge to a stable point. Extensive experimental results demonstrate that our proposed approach can achieve accurate tensor recovery at lower sampling rates compared to other methods.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L1 space is presented for the studied models using the proposed scheme.