Convergence Estimates Research Articles

Variable-Performance Nonlinear Feedback Filter Based on Dynamic Surface Stabilization Approach

The proposed surface stabilization technique based filter feedbacks the filtering error with its nonlinear variable gain algorithm in order to recover the source signal information rejecting the high-frequency component (e.g., low-pass filtering), thereby guaranteeing closed-loop stability. Thus, resultant implementation is possible without signal dynamics information. The result provides two contributions: first, the improvement of the transient filtering performance through the closed-form feedback gain magnification mechanism and second, the recovery of the exponential filtering performance due to the surface stabilization mechanism, including the exponential convergent disturbance estimator originated from the nonlinear disturbance observer. The experimental results verify the effectiveness of the proposed technique, which improves the 500-W three-phase motor control performance.

Distributed Mean-Field Density Estimation for Large-Scale Systems

This article studies how to estimate the mean-field density of large-scale systems in a distributed manner. Such problems are motivated by the recent swarm control technique that uses mean-field approximations to represent the collective effect of the swarm, wherein the mean-field density (especially its gradient) is usually used in feedback control design. In the first part, we formulate the density estimation problem as a filtering problem of the associated mean-field partial differential equation (PDE), for which we employ Kernel density estimation to construct noisy observations and use filtering theory of PDE systems to design an optimal (centralized) density filter. It turns out that the covariance operator of observation noise depends on the unknown density. Hence, we use approximations for the covariance operator to obtain a suboptimal density filter, and prove that both the density estimates and their gradient are convergent and remain close to the optimal one using the notion of input-to-state stability (ISS). In the second part, we continue to study how to decentralize the density filter such that each agent can estimate the mean-field density based on only its own position and local information exchange with neighbors. We prove that the local density filter is also convergent and remains close to the centralized one in the sense of ISS. Simulation results suggest that the centralized suboptimal density filter is able to generate convergent density estimates, and the local density filter is able to converge and remain close to the centralized filter.

A locking-free staggered DG method for the Biot system of poroelasticity on general polygonal meshes

AbstractIn this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with a stress–displacement–rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approximation developed in Zhao and Park (2020, A staggered cell-centered DG method for linear elasticity on polygonal meshes, SIAM J. Sci. Comput.42, A2158–A2181). The proposed method is locking-free and can handle highly distorted grids, possibly including hanging nodes, which is desirable for practical applications. We prove the convergence estimates for the semidiscrete scheme and fully discrete scheme for all the variables in their natural norms. In particular, the stability and convergence analyses do not need a uniformly positive storativity coefficient. Moreover, to reduce the size of the global system, we propose a five-field-formulation-based fixed stress splitting scheme, where the linear convergence of the scheme is proved. Several numerical experiments are carried out to confirm the optimal convergence rates and the locking-free property of the proposed method.

On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers

In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.

Estimating the spatial correlation and convergence of China's healthcare resources allocation: evidence from the Yangtze River Delta Region

BackgroundChina’s imbalanced allocation of healthcare resources mainly arises from urban–rural and intercity differences, the solution of which has been the goal of reforms during the past decades. Estimating the spatial correlation and convergence could help to understand the impact of China’s fast-evolving medical market and the latest healthcare reforms.MethodsThe entropy weight method was used to construct a healthcare resource supply index (HRS) by using data of 41cities in a cluster in the Yangtze River Delta (YRD) from 2007 to 2019. The Dagum Gini coefficient, kernel density estimation, Moran's I, and LISA cluster map were used to characterize the spatiotemporal evolution and agglomeration of healthcare resources, and then a spatial panel model was used to perform β convergence estimation by incorporating the spatial effect, city heterogeneity, and healthcare reforms.ResultsHealthcare resources supply in the YRD region increases significantly and converges rapidly. There is a significant spatial correlation and agglomeration between provinces and cities, and a significant spatial spillover effect is also found in β convergence. No evidence is found that the latest healthcare reforms have an impact on the balanced allocation and convergence of healthcare resources.ConclusionChina’s long-term investment in past decades has yielded a more balanced allocation and intercity convergence of healthcare resources. However, the latest healthcare reforms do not contribute to the balanced allocation of healthcare resources from the supply-side, and demand-side analysis is needed in the future studies.

An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay

A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion–reaction equations with a fixed time delay. The temporal discretization for the variable-order fractional derivative is performed by the L1-approximation. An appropriate basis function in terms of Legendre polynomials is used to construct the Galerkin spectral method for the spatial discretization of the second-order spatial operator. The main advantage of the proposed approach is that the implementation of the iterative process is avoided for the nonlinear term in the variable fractional-order problem. Convergence and stability estimates for the constructed scheme are proved theoretically by discrete energy estimates. Some numerical experiments are finally provided to demonstrate the efficiency and accuracy of the theoretical findings.

Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

An Accurate Near‐Field Distance Estimation Differential Algorithm

The triangular geometry is the basis of near field array accurate distance estimation algorithms. The Fisher expression of traditional distance estimation is derived by utilizing the Taylor series. To improve convergence rate and estimation accuracy, a novel iterative distance estimation algorithm is proposed with differential equations based on the triangular geometry. Firstly, its convergence performance is analysed in detail. Secondly, the selection of the initial value and the number of iterations are respectively studied. Thirdly, compared with the traditional estimation algorithms by utilizing the Fisher approximation, the proposed algorithm has a higher convergence rate and estimation accuracy. Moreover, its pseudocode is presented. Finally, the experiment results and performance analysis are provided to verify the effectiveness of the proposed algorithm.

A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains

Abstract In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov–Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class of processes is provided to point that such estimates considering several transition steps may be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.

Modal analysis of elastic vibrations of incompressible materials using a pressure-stabilized finite element method

This paper describes a modal analysis technique to approximate the vibrations of incompressible elastic solids using a stabilized finite element method to approximate the associated eigenvalue problem. It is explained why residual based formulations are not appropriate in this case, and a formulation involving only the pressure gradient is employed. The effect of the stabilization term compared to a Galerkin approach is detailed, both in the derivation of the approximate formulation and in the error estimate provided. • Stabilized finite element method for vibrations of incompressible elastic solids. • Adequacy to solve the eigenvalue problem arising in modal analysis. • Modal analysis of the problem resulting after space discretization. • Stability and convergence estimates for the approximate modal analysis. • Numerical results confirm the accurate and robust behavior of the formulation.

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

In this paper, we develop and analyze a fully discrete mixed finite element method for unsteady inductionless magnetohydrodynamics problem. An Euler semi-implicit scheme is proposed with a mixed variational formulation based on the variables (u,p,j,ϕ), in which the Navier–Stokes equations are approximated by stable finite elements and the current density is discretized by the divergence-conforming finite element. This scheme has the feature that the discrete current density keeps charge conservation property. It is shown that both the continuous problem and its fully discrete Euler semi-implicit scheme are well-posed. We prove unconditionally convergence and error estimates for velocity, pressure and current density. Finally, numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials

In the present work, we consider a class of fractional delay differential equations of order ρ with Atangana-Baleanu fractional derivatives in the Caputo sense. We convert our fractional delay problem to Volterra integral equation and used them to establish the local existence theorem using the Arzela-Ascoli theorem and Schauder’s fixed point theorem. After that, the contraction mapping theorem was used to prove the global existence and uniqueness theorem. For a numerical solution, we develop the Galerkin algorithm based on shifted Legendre polynomials to solve the utilized fractional delay problem. This algorithm is based on reducing such problems to those of solving a system of algebraic equations, also we state and prove the convergence and error estimate theorems. Four numerical examples, two linear and two nonlinear are presented to test the efficiency and accuracy of the proposed algorithm with some tables and figures to compare our results with the exact solutions. Several, conclusions, recommendations, inductions, and highlights were formulated in the last chapter to extinguish the perfection of the work.

Determination of a spacewise-dependent heat source by a logarithmic-type regularization method

This paper is devoted to an inverse problem of determining a spacewise-dependent heat source in a linear parabolic equation from the measurement at the final moment. A new regularization method called the logarithmic-type regularization method is proposed to determine the unknown source. Then the convergence estimates of regularization solutions are obtained under the a-priori and a-posteriori regularization parameter selections. Finally, some numerical examples are presented to verify the validity of the logarithmic-type regularization method.

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

In this study, we obtain a numerical solution for Fisher's equation using a numerical experiment with three different cases. The three cases correspond to different coefficients for the reaction term. We use three numerical methods namely; Forward-Time Central Space (FTCS) scheme, a Nonstandard Finite Difference (NSFD) scheme, and the Explicit Exponential Finite Difference (EEFD) scheme. We first study the properties of the schemes such as positivity, boundedness, and stability and obtain convergence estimates. We then obtain values of L1 and L∞ errors in order to obtain an estimate of the optimal time step size at a given value of spatial step size. We determine if the optimal time step size is influenced by the choice of the numerical methods or the coefficient of reaction term used. Finally, we compute the rate of convergence in time using L1 and L∞ errors for all three methods for the three cases.

Approximation by Sampling Durrmeyer Operators in Weighted Space of Functions

The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling type approximation methods and presenting well definiteness of the operators on weighted spaces of functions, we examine pointwise and uniform convergence of the family of operators and determine the rate of convergence via weighted modulus of continuity. A quantitative Voronovskaja theorem is also proved in order to obtain rate of pointwise convergence and upper estimate for this convergence. The last section is devoted to some numerical evaluations of sampling Durrmeyer operators with suitable kernels.

Fixed-time observer-based homogeneous controller with state-dependent exponent for fault tolerant control of an underwater vehicle

This paper is concerned with designing a fault tolerant control system in the presence of unknown vehicle dynamics and saturation and rate limits of the thrusters for an underwater remotely operated vehicle. For this, a novel variable-exponent finite-time controller based on homogeneity is proposed for the underwater vehicle. The error-dependent exponent is introduced to provide better performance in transient and obtain faster converge to the desired states. Stability analyses are carried out for the controller and it is proven that system states will converge to the origin in finite time. In addition, phase plane analyses are performed and it is shown that system trajectories converge to the origin faster under proposed controller compared to controller with constant exponents. A fixed-time extended state observer with a new fault detection and identification unit is introduced which provides faster estimation of failures in transients despite the presence of uncertainties and disturbances. State estimation error dynamics is proven to be fixed-time convergent and fault estimation error is globally uniformly ultimately bounded. Simulations are carried out and comparisons are made with several FTCSs. The results show that the performance of the proposed algorithm is superior to previous works both in transients and upon the failures.

Comments on the mass sheet degeneracy in cosmography analyses

We make a number of comments regarding modeling degeneracies in strong lensing measurements of the Hubble parameter H 0. The first point concerns the impact of weak lensing associated with different segments of the line of sight. We show that external convergence terms associated with the lens-source and observer-lens segments need to be included in cosmographic modeling, in addition to the usual observer-source term, to avoid systematic bias in the inferred value of H 0. Specifically, we show how an incomplete account of some line of sight terms biases stellar kinematics as well as ray tracing simulation methods to alleviate the mass sheet degeneracy.The second point concerns the use of imaging data for multiple strongly-lensed sources in a given system.We show that the mass sheet degeneracy is not fully resolved by the availability of multiple sources: some degeneracy remains because of differential external convergence between the different sources.Similarly, differential external convergence also complicates the use of multiple sources in addressing the approximate mass sheet degeneracy associated with a local (“internal”) core component in lens galaxies. This internal-external degeneracy is amplified by the non-monotonicity of the angular diameter distance as a function of redshift.For a rough assessment of the weak lensing effects, we provide estimates of external convergence using the nonlinear matter power spectrum, paying attention to non-equal time correlators.

A time-varying impact of ageing population on the economy of Malaysia

Purpose By 2040, 14.5% of Malaysia’s population will consist of those aged 65 years and above. The purpose of this paper is to examine the time-varying impact of an ageing population on the economy of Malaysia.Design/methodology/approach The relationship between the ageing population and economic performance was examined from the period 1971–2019. The time-varying rolling convergence estimation of 40 + k observations sample frame of the trace statistics was analysed.Findings The ageing population had affected the economic activities in Malaysia over all sampling time frame from 1971 to 2010, 1972–2011, …to 1980–2019. The growth in total population and economic activities also caused a significant increase in the ageing population in the long run. An improved economic performance signifies the affordability for better healthcare services and improvement in medical science technology to treat diseases.Originality/value Its ageing population has gradually slow down economic activities in Malaysia. Now is the time to be prepared and address an ageing workforce issues such as productivity, retirement policy, savings behaviour and life-long learning capabilities.Peer review The peer review history for this article is available at https://publons.com/publon/10.1108/IJSE-04-2021-0234.

Optimal rate convergence analysis of a numerical scheme for the ternary Cahn–Hilliard system with a Flory–Huggins–deGennes energy potential

We present an error analysis for a fully discrete finite difference scheme for the three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn–Hilliard system with a Flory–Huggins–deGennes free energy potential. The numerical scheme was recently proposed, and the positivity-preserving property and unconditional energy stability were theoretically established. In this paper, we rigorously prove first order convergence in time and second order convergence in space for the numerical scheme, in the LΔt∞(0,T;Hh−1)∩LΔt2(0,T;Hh1) norm. Many highly non-standard estimates have to be involved, due to the nonlinear and singular nature of the surface diffusion coefficients. The combination of (i) a higher order asymptotic expansion of the numerical solution (up to second order temporal accuracy); (ii) a rough error estimate (to establish the LΔt∞ bound for the phase variables); (iii) and a refined error estimate have to be carried out to conclude such a convergence result. To our knowledge, it will be the first work to provide an optimal rate convergence estimate for a ternary phase field system with singular energy coefficients.