Norm Estimates Research Articles

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.

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.

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.

Norm Estimates for Remainders of Noncommutative Taylor Approximations for Laplace Transformers Defined by Hyperaccretive Operators

Norm Estimates for Remainders of Noncommutative Taylor Approximations for Laplace Transformers Defined by Hyperaccretive Operators

Some norm estimates for the triangular projection on the space of bounded linear operators between two symmetric sequence spaces

Some norm estimates for the triangular projection on the space of bounded linear operators between two symmetric sequence spaces

Schatten class properties and essential norm estimates of operators on Bergman spaces induced by regular weights of annulus

Schatten class properties and essential norm estimates of operators on Bergman spaces induced by regular weights of annulus

Growth Spaces on Circular Domains Taking Values in a Banach Lattice, Embeddings and Composition Operators

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As an application, we prove a characterization of weighted composition operators.

Effects of experimental, network-based social circle norm feedback on studying behavior and alcohol consumption.

Misrepresentation of peer behavior has often been observed in college students and may lead to over-expression of alcohol consumption and under-expression of studying. While social norm feedback approaches have had mixed success in addressing these misrepresentations and altering behavior, they may have been too unspecific to be effective and did not directly assess individual perception accuracy. We thus investigated how specific, one-time feedback on the behavioral distribution of alcohol consumption or study time of a clearly defined, individually-adjusted social circle would affect the respective norm estimations and behavior of a class of Psychology students (n = 89 in January) across their first year of study. Students overestimated alcohol consumption and partially underestimated studying norms. While social circle feedback on alcohol consumption did not clearly affect both individual estimation accuracy and alcohol consumption, feedback on peers' studying time increased studying with no clear effect on estimation accuracy. This indicates that social circle norm feedback may be suitable to evoke behavioral effects. The correction of the detected inaccuracies did not appear to be a precondition for the feedback to be effective.

Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness

We present a variational approach aimed at enhancing the training of physics-informed neural networks (PINNs) and more general surrogate models for learning partial differential equations (PDE). In particular, we extend our formerly introduced notion of Sobolev cubatures to negative orders, enabling the approximation of negative order Sobolev norms. We mathematically prove the effect of negative order Sobolev cubatures in improving the condition number of discrete PDE learning problems, providing balancing scalars that mitigate numerical stiffness issues caused by loss imbalances. Additionally, we consider polynomial surrogate models (PSMs), which maintain the flexibility of PINN formulations while preserving the convexity structure of the PDE operators. The combination of negative order Sobolev cubatures and PSMs delivers well-conditioned discrete optimization problems, solvable via an exponentially fast convergent gradient descent for λ-convex losses. Our theoretical contributions are supported by numerical experiments, addressing linear and non-linear, forward and inverse PDE problems. These experiments show that the Sobolev cubature-based PSMs emerge as the superior state-of-the-art PINN technique.

EEG/MEG source imaging of deep brain activity within the maximum entropy on the mean framework: Simulations and validation in epilepsy.

Electro/Magneto-EncephaloGraphy (EEG/MEG) source imaging (EMSI) of epileptic activity from deep generators is often challenging due to the higher sensitivity of EEG/MEG to superficial regions and to the spatial configuration of subcortical structures. We previously demonstrated the ability of the coherent Maximum Entropy on the Mean (cMEM) method to accurately localize the superficial cortical generators and their spatial extent. Here, we propose a depth-weighted adaptation of cMEM to localize deep generators more accurately. These methods were evaluated using realistic MEG/high-density EEG (HD-EEG) simulations of epileptic activity and actual MEG/HD-EEG recordings from patients with focal epilepsy. We incorporated depth-weighting within the MEM framework to compensate for its preference for superficial generators. We also included a mesh of both hippocampi, as an additional deep structure in the source model. We generated 5400 realistic simulations of interictal epileptic discharges for MEG and HD-EEG involving a wide range of spatial extents and signal-to-noise ratio (SNR) levels, before investigating EMSI on clinical HD-EEG in 16 patients and MEG in 14 patients. Clinical interictal epileptic discharges were marked by visual inspection. We applied three EMSI methods: cMEM, depth-weighted cMEM and depth-weighted minimum norm estimate (MNE). The ground truth was defined as the true simulated generator or as a drawn region based on clinical information available for patients. For deep sources, depth-weighted cMEM improved the localization when compared to cMEM and depth-weighted MNE, whereas depth-weighted cMEM did not deteriorate localization accuracy for superficial regions. For patients' data, we observed improvement in localization for deep sources, especially for the patients with mesial temporal epilepsy, for which cMEM failed to reconstruct the initial generator in the hippocampus. Depth weighting was more crucial for MEG (gradiometers) than for HD-EEG. Similar findings were found when considering depth weighting for the wavelet extension of MEM. In conclusion, depth-weighted cMEM improved the localization of deep sources without or with minimal deterioration of the localization of the superficial sources. This was demonstrated using extensive simulations with MEG and HD-EEG and clinical MEG and HD-EEG for epilepsy patients.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

Examination of optical soliton solutions for the perturbed Schrödinger–Hirota equation with anti-cubic law in the presence of spatiotemporal dispersion

In the current paper, the perturbed Schrödinger–Hirota equation having anti-cubic nonlinearity is analyzed with the aid of the new Kudryashov scheme. What distinguishes this article from other articles is that it not only attains multifold analytical solutions to the underresearched model but also verifies the impact of the anti-cubic law media on soliton attitude for the first time. The algorithmic rules and solution functions of the presented method have been controlled with symbolic algebraic software, and every outcome has been approved attentively. Then, the given method has been implemented on the model under consideration for the collective test objective. With the conventional norm approximation, the nonlinear partial differential structure of the model under consideration has been turned into the ordinary differential structure by performing the wave transmutation, and then the presented technique has been implemented into the ordinary differential structure of the proposed model. After this process, we have acquired a system of linear algebraic equations and their convenient solutions. Afterward, by attaining the proper solution sets, the soliton solutions of the given model, such as bright, W-shape-like, and dark soliton forms, have been arranged, and some chosen diagrammatic views have been presented.

Adaptive continuation based smooth -norm approximation for compressed sensing MR image reconstruction.

There are a number of algorithms for smooth -norm (SL0) approximation. In most of the cases, sparsity level of the reconstructed signal is controlled by using a decreasing sequence of the modulation parameter values. However, predefined decreasing sequences of the modulation parameter values cannot produce optimal sparsity or best reconstruction performance, because the best choice of the parameter values is often data-dependent and dynamically changes in each iteration. We propose an adaptive compressed sensing magnetic resonance image reconstruction using the SL0 approximation method. The SL0 approach typically involves one-step gradient descent of the SL0 approximating function parameterized with a modulation parameter, followed by a projection step onto the feasible solution set. Since the best choice of the parameter values is often data-dependent and dynamically changes in each iteration, it is preferable to adaptively control the rate of decrease of the parameter values. In order to achieve this, we solve two subproblems in an alternating manner. One is a sparse regularization-based subproblem, which is solved with a precomputed value of the parameter, and the second subproblem is the estimation of the parameter itself using a root finding technique. The advantage of this approach in terms of speed and accuracy is illustrated using a compressed sensing magnetic resonance image reconstruction problem and compared with constant scale factor continuation based SL0-norm and adaptive continuation based -norm minimization approaches. The proposed adaptive estimation is found to be at least twofold faster than automated parameter estimation based iterative shrinkage-thresholding algorithm in terms of CPU time, on an average improvement of reconstruction performance 15% in terms of normalized mean squared error. An adaptive continuation-based SL0 algorithm is presented, with a potential application to compressed sensing (CS)-based MR image reconstruction. It is a data-dependent adaptive continuation method and eliminates the problem of searching for appropriate constant scale factor values to be used in the CS reconstruction of different types of MRI data.

Pre-conditioning CQ algorithm for solving the split feasibility problem and its application to image restoration problem

In this study, we define two pre-conditioning CQ algorithms for the split feasibility problem by using self-adaptive and line-search techniques in the real Hilbert space. One of the self-adaptive techniques is used to obtain optimal step size. These techniques do not require prior knowledge of the operator norm or estimation of the matrix norm. We also establish weak convergence theorems with respect to particular norms. As an application, numerical experiments show that the proposed algorithm is effective in image restoration problem.

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

Decay estimates for Cayley transforms and inverses of semigroup generators via the B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document}-calculus

Let -A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-A$$\\end{document} be the generator of a bounded C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup (e-tA)t≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(e^{-tA})_{t \\ge 0}$$\\end{document} on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform Vω(A):=(A-ωI)(A+ωI)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{\\omega }(A) :=(A-\\omega I) (A+\\omega I)^{-1}$$\\end{document} with ω>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega >0$$\\end{document}. We give a decay estimate for ‖Vω(A)nA-1‖\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert V_{\\omega }(A)^nA^{-1}\\Vert $$\\end{document} when (e-tA)t≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(e^{-tA})_{t \\ge 0}$$\\end{document} is polynomially stable. Considering the case where the parameter ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} varies, we estimate ‖(∏k=1nVωk(A))A-1‖\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert (\\prod _{k=1}^n V_{\\omega _k}(A))A^{-1}\\Vert $$\\end{document} for exponentially stable C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroups (e-tA)t≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(e^{-tA})_{t \\ge 0}$$\\end{document}. Next we show that if the generator -A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-A$$\\end{document} of the bounded C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup has a bounded inverse, then supt≥0‖e-tA-1A-α‖<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sup _{t \\ge 0} \\Vert e^{-tA^{-1}} A^{-\\alpha } \\Vert < \\infty $$\\end{document} for all α>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >0$$\\end{document}. We also present an estimate for the rate of decay of ‖e-tA-1A-1‖\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert e^{-tA^{-1}} A^{-1} \\Vert $$\\end{document}, assuming that (e-tA)t≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(e^{-tA})_{t \\ge 0}$$\\end{document} is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document}-calculus.

Prescribed-time fault-tolerant tracking control for Markov jump nonlinear systems based on saturation dynamics filters

This paper investigates the problem of prescribed-time fault-tolerant tracking control for a class of Markov jump nonlinear systems affected by actuator faults and input saturation. First, a new time-varying constraint function and an error transformation are invoked to transform the original prescribed-time tracking control problem into a tracking control problem with deferred constraint for tracking error. Furthermore, the parameter correction terms are designed, on which a novel Markov jump saturation dynamic filter is constructed to handle the unfavorable effect of input saturation. Meanwhile, the projection operator technique and the norm estimation methodology are integrated to devise adaptive laws to estimate the unknown actuator faults evolving in accordance with the Markov chain. A new prescribed-time fault-tolerant tracking control scheme is put forward that realizes the deferred constraint on the tracking error, which in turn ensures that the tracking error of the system converges to a prescribed accuracy within a prescribed-time. Finally, two simulation examples, one of which implements a prescribed-time fault-tolerant tracking control for rotary steerable drilling tool system, are presented to illustrate the effectiveness and practicality of the developed scheme.

Error estimates for completely discrete FEM in energy‐type and weaker norms

AbstractThe paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.

Geometrically exact 3D beam theory with embedded strong discontinuities for modeling of localized failure in bending

In this work, we propose the elastoplastic model of a three-dimensional (3D) geometrically exact beam, which can capture localized bending deformation leading to a plastic hinge. This is a follow-up work to Tojaga et al. (2023), where we only studied the fracture of fiber-like beam structures that break in mode I or mode II, here extended to handle beam-like behavior with bending failure and the main difficulty of the non-vectorial character of large 3D rotations. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model, which leads to the multiplicative decomposition of the rotation tensor. We shown how to replace this with the corresponding additive decomposition of the rotation vector derivatives, to be performed in the material description in the tangent space of SO(3) manifold in the initial configuration. The plasticity model for which we give a more detailed implementation employs a Rankin-like multi-surface plasticity criteria, where each moment vector component (i.e. torsion or bending) is considered separately. The non-local variational formulation is proposed to deal with the softening phenomena characteristic of localized bending failure, where the fracture energy is introduced as the main parameter for softening. The discrete approximation is built in terms of the embedded-discontinuity finite element method (ED-FEM), which introduces a jump in rotation vector that can be handled at the level of a particular beam element. 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 moments) from local (plastic curvature) variables. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Pre-Schwarzian and Schwarzian norm estimates for subclasses of univalent functions

Pre-Schwarzian and Schwarzian norm estimates for subclasses of univalent functions