Bound Of Norm Research Articles

Sampling discretization of the uniform norm and applications

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any N-dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in N number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on N. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

A Legendre wavelet collocation method for solving neutral delay differential equations

The Legendre wavelet based method has been employed in this paper to investigate neutral delay differential equations. The highest order derivative is approximated by Legendre wavelet using the integral operator technique. Then integrations of Legendre wavelet are used to approximate the lower order derivatives and unknown function. To get an algebraic system of linear or nonlinear equations, approximated values of unknown function and its derivatives are substituted in neutral delay differential equations. On solving the developed system, we get unknown wavelet coefficients and subsequently the approximate solution. To analyze the theoretical usability of the approach, the upper bound of error norm is established. Moreover, the theoretical results are confirmed through few numerical experiments. A comparison of the results of presented method with method available in literature is given to conclude the superiority of the proposed method.

On Bounds for Norms of Sine and Cosine Along a Circle on the Complex Plane

In the paper, the author presents lower and upper bounds for norms of the sine and cosine functions along a circle on the complex plane.

Further results on synchronization of chaotic Lur’e systems based on aperiodic time-triggered intermittent control

Aperiodic time-triggered intermittent control (ATIC) is presented to investigate the exponential synchronization issue of chaotic Lur’e systems (CLSs). In our work, for CLSs, the mathematical expression of ATIC is put forward initially. Different from time-triggered intermittent control (TIC), ATIC allows aperiodic sampling. The aperiodic sampling characteristic of ATIC makes it more flexible and adaptable in complex environments, which greatly increases its applications. Then, for the deduced ATIC error system, a key lemma is established, which demonstrates that, on control or free time intervals, the bound of the error vector norm at any instance can be described by the multiple of its norm at initial instance. Further, based on the lemma and a mixed time-dependent Lyapunov functional (MTLF), the exponential stability theorem is given for the deduced ATIC error system. Consequently, the exponential synchronization criterion with less conservativeness is obtained for CLSs via ATIC. With its aid, the design method of ATIC is proposed. Finally, the validity and superiority of ATIC are verified by a simulation example.

The behavior of the Gauss-Radau upper bound of the error norm in CG

Consider the problem of solving systems of linear algebraic equations varvec{A}varvec{x}=varvec{b} with a real symmetric positive definite matrix varvec{A} using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the varvec{A}-norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of the Gauss-Radau upper bound on the varvec{A}-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate varvec{mu } to the smallest eigenvalue of varvec{A}. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of varvec{mu }. We construct a model problem that is used to demonstrate and study the behavior of the upper bound in dependence of varvec{mu }, and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of varvec{A}. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate varvec{mu }. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.

Robust Local Stabilization of Nonlinear Systems With Controller-Dependent Norm Bounds: A Convex Approach With Input-Output Sampling

This letter presents a framework for synthesizing a robust full-state feedback controller for systems with unknown nonlinearities. Our approach characterizes input-output behavior of the nonlinearities in terms of local norm bounds using available sampled data corresponding to a known region about an equilibrium point. A challenge in this approach is that if the nonlinearities have explicit dependence on the control inputs, an a priori selection of the control input sampling region is required to determine the local norm bounds. This leads to a “chicken and egg” problem, where the local norm bounds are required for controller synthesis, but the region of control inputs needed to be characterized cannot be known prior to synthesis of the controller. To tackle this issue, we constrain the closed-loop control inputs within the sampling region while synthesizing the controller. As the resulting synthesis problem is non-convex, three semi-definite programs (SDPs) are obtained through convex relaxations of the main problem, and an iterative algorithm is constructed using these SDPs for control synthesis. Two numerical examples are included to demonstrate the effectiveness of the proposed algorithm.

Knowledge-Guided Learning for Transceiver Design in Over-the-Air Federated Learning

In this paper, we consider communication-efficient over-the-air federated learning (FL), where multiple edge devices with non-independent and identically distributed datasets perform multiple local iterations in each communication round and then concurrently transmit their updated gradients to an edge server over the same radio channel for global model aggregation using over-the-air computation (AirComp). We derive the upper bound of the time-average norm of the gradients to characterize the convergence of AirComp-assisted FL, which reveals the impact of the model aggregation errors accumulated over all communication rounds on convergence. Based on the convergence analysis, we formulate an optimization problem to minimize the upper bound to enhance the learning performance, followed by proposing an alternating optimization algorithm to facilitate the transceiver design for AirComp-assisted FL. As the alternating optimization algorithm suffers from high computation complexity, we further develop a knowledge-guided learning algorithm that exploits the structure of the analytic expression of the transmit power to achieve computation-efficient transceiver design. Simulation results demonstrate that the proposed knowledge-guided learning algorithm achieves a comparable performance as the alternating optimization algorithm, but with a much lower computation complexity. Moreover, both proposed algorithms outperform the baseline methods in terms of convergence speed and test accuracy.

Inclusion Methods for Multiplication of Three Point Matrices

This study considers inclusion methods for the multiplication of three point matrices. These methods are essential for verification methods such as for a matrix determinant, generalized eigenvalue decomposition, singular value decomposition, square roots and inverse square roots of a matrix, and Sylvester equations. We consider three methods that involve five matrix multiplications and compare the tightness of enclosures and numerical experiments. The proposed methods produce tighter results than the previous method with the same computational cost. We also show methods for obtaining the upper bound of the maximum norm of a product of three point matrices.

A Legendre wavelet collocation method for 1D and 2D coupled time-fractional nonlinear diffusion system

A Legendre wavelet collocation method is proposed for solving a nonlinear coupled time fractional diffusion system. We have formulated a Riemann-Liouville fractional integral operator for Legendre wavelet (RLFIO-L) adopting the definition of Riemann-Liouville fractional integral operator combined with the Laplace transformation. Both the time and space variables are discretized in terms of the Legendre wavelet and RLFIO-L. The nonlinear coupled diffusion system is quasi-linearized by making use of the Newton's method. For theoretical concerns, the upper bound of error norm of the proposed method is estimated. Some numerical experiments are presented to authenticate the computational efficacy of the method.

Tight computationally efficient approximation of matrix norms with applications

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of ``solvable cases,'' most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (``ellitopic'' and ``co-ellitopic,'' respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding ``robust operator norm of uncertain matrix with box uncertainty,'' that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\leq1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of ``uncertain-but-bounded'' noise varying in a box.

Moving devices channel tracking with massive MIMO systems over high-rate internet of things

Channel estimation becomes a challenging issue with the exponentially increasing high-rate Internet-of-Things (IoT) over massive MIMO systems. This paper simultaneously estimates and tracks the channels gains of both inactive and active devices per access slot. Due to the uncertain mobility of IoT devices, the channels age at unpredictable rates; also the additive noise of the received signal and the channel state noise are correlated. This framework is referred to as a practical scenario. The penalty function approach is first used in this paper to derive a novel analytic solution for the optimal tracker of the practical scenario. Several sufficient conditions are presented to demonstrate the optimal tracker’s stability and the upper bound of the estimation error norm when the channel state is unstable. Then, three new suboptimal trackers are developed to alleviate the computational burden. Additionally, due to the unpredictability of channel evolution, a novel suboptimal tracker based on adaptive matrices is proposed to improve the channel estimation efficiency. An upper bound on the error covariance is computed in a recursive form to determine the tracker gain. A sufficient condition is established on the adaptive matrices ensuring the tracker’s feasibility. The lower bound on the optimal tracker’s achievable sum-rate is analytically computed. While changes in the device’s velocity reduce estimation accuracy, simulation results demonstrate that the designed trackers are less affected compared with the previous trackers.

Numerical simulation for generalized space-time fractional Klein–Gordon equations via Gegenbauer wavelet

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

Sine transform based preconditioning techniques for space fractional diffusion equations

AbstractWe study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi‐dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew‐symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh‐independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique.

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

In this paper, we consider the analysis of a nudging based algorithm for data assimilation for the three-dimensional Boussinesq system, which we call the AOT system. A rigorous analysis of this algorithm for well-posed dissipative partial differential equations was first provided by Azouani, Olson and Titi (J Nonlinear Sci 24:277–304, 2014); thus justifying our terminology for the associated nudging system. We provide a sufficient condition, based solely on the observed velocity data obtained from a Leray-Hopf weak solution, for the global well-posedness, regularity and most crucially, the asymptotic tracking property of solutions of the associated (three-dimensional) nudging system. It is to be noted that neither regularity nor any knowledge of a uniform \({\mathbb {H}}^1\)-norm bound is a priori assumed on the solution of the original three-dimensional Boussinesq system from which the observations are obtained. As a corollary of our result, we obtain a novel observable regularity criterion based on finitely many observational data. Our condition also guarantees the construction of the Lipschitz continuous determining map, which is known to play a crucial role in the construction of the so-called determining form and in statistical data assimilation.

Sharp weighted estimates for strong-sparse operators

We prove the sharp weighted-L2 bounds for the strong-sparse operators introduced in [3]. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weight helps to prove the sharpness of the trivial upper bound of the operator norm.

Privacy-preserving Speech-based Depression Diagnosis via Federated Learning.

Mental health disorders, such as depression, affect a large and growing number of populations worldwide, and they may cause severe emotional, behavioral and physical health problems if left untreated. As depression affects a patient's speech characteristics, recent studies have proposed to leverage deep-learning-powered speech analysis models for depression diagnosis, which often require centralized learning on the collected voice data. However, this centralized training requiring data to be stored at a server raises the risks of severe voice data breaches, and people may not be willing to share their speech data with third parties due to privacy concerns. To address these issues, in this paper, we demonstrate for the first time that speech-based depression diagnosis models can be trained in a privacy-preserving way using federated learning, which enables collaborative model training while keeping the private speech data decentralized on clients' devices. To ensure the model's robustness under attacks, we also integrate different FL defenses into the system, such as norm bounding, differential privacy, and secure aggregation mechanisms. Extensive experiments under various FL settings on the DAIC-WOZ dataset show that our FL model can achieve high performance without sacrificing much utility compared with centralized-learning approaches while ensuring users' speech data privacy. Clinical Relevance- The experiments were conducted on publicly available clinical datasets. No humans or animals were involved.

Bounding the Frobenius norm of a q-deformed commutator

For two n×n complex matrices A and B, we define the q-deformed commutator as [A,B]q:=AB−qBA for a real parameter q. In this paper, we investigate a generalization of the Böttcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the q-deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative q. In particular, denoting the Frobenius norm by ||.||F, when A or B is normal, we prove the following inequality to be true and sharp: ||[A,B]q||F2≤(1+q2)||A||F2||B||F2 for positive q. Also, we conjecture that the same bound is true for positive q when A or B is traceless. For negative q, we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when A or B is traceless. All conjectures are supported with numerics and proved for n=2.

Global Existence of Solutions of a Loglog Energy-Supercritical Klein–Gordon Equation

We prove global existence of the solutions of the loglog energy-supercritical Klein–Gordon equation $$ \partial _{tt} u - \triangle u + u = -|u|^{\frac{4}{n-2}} u \log ^{\gamma } \left( \log ( 10 + |u|^{2} ) \right) $$ , with $$n \in \{ 3,4,5 \}$$ , $$ 0< \gamma < \gamma _{n} $$ , and data $$(u_{0},u_{1}) \in H^{k} \times H^{k-1}$$ for $$k > 1$$ (resp. $$ \frac{7}{3}> k > 1)\,\, if n \in \{ 3,4 \}$$ (resp. $$n=5$$ ). The proof is by contradiction. Assuming that blow-up occurs at a maximal time of existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm, which eventually leads to a contradiction with the blow-up assumption.

Schur Complement-Based Infinity Norm Bounds for the Inverse of GDSDD Matrices

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

Error bounds for linear complementarity problems of $ SD{{D}_{1}} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices

&lt;abstract&gt;&lt;p&gt;An upper bound of the infinity norm for the inverse of $ SD{D_1} $ matrix is presented. We apply the new bound to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of $ SD{D_1} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices. In addition, a new lower bound for the smallest singular value is also given. Numerical examples show the validity of the results.&lt;/p&gt;&lt;/abstract&gt;