Asymptotic Error Estimates Research Articles

Approximation of operators on real line

There are exponential operators which are defined on the whole real line, namely, the Ismail–May operators , connected to , and the Gauss–Weierstrass operators , connected with 1. Here we consider the composition of these two operators and observe that the composition operator is not of exponential type operator. We find moment‐generating functions and moments of these operators and estimate some point‐wise convergence of these operators using the characteristic function. We also estimate quantitative Voronovskaya type asymptotic formula and error estimation. Also, the difference of the composition operators with its components and is estimated in terms of modulus of continuity. Finally, we provide some graphs to see visualize the convergence by considering the function .

Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions

<p>In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.</p>

Efficient computational methods of highly oscillatory Bessel transforms with a singular point of Cauchy type and a nonlinear special oscillator

In this article, we propose two efficient combined methods to compute the integral ∫01f(x)x−cJm(ωxr)dx with a special and nonlinear oscillator xr, where 0<c<1,m≥0,2r∈N+. One is established by combining the complex integration method with the two-point Taylor interpolation method. The other is to combine two complex integration methods. The explicit formula of the integral ∫0+∞1x−cJm(ωxr)dx is expressed in terms of the Meijer G function. Importantly, the rigorous error analysis is performed by a large amount of theoretical analysis. The asymptotic error estimates in inverse powers of ω are achieved. Finally, numerical experiments show the validity of theoretical analysis and the efficiency of the proposed methods. In addition, we make numerical comparisons between the two proposed combined methods and another combined method.

Gaussian quadrature rules for composite highly oscillatory integrals

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. An interesting connection between the quadrature nodes and the Legendre points is proved and it is shown that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature cannot be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.

Asymptotic switched composite adaptive control with application to robotic interaction tasks

AbstractThis article proposes a new switched adaptive control design for uncertain switched systems with composite (time‐driven and state‐dependent) switching and shows its applicability in switched impedance control. A composite switched adaptive control design, consisting of the direct switched adaptive control and the indirect switched adaptive control counterpart, is developed to improve the control performance. Specifically, a new stability condition for composite switching is proposed by making use of differential matrix equations and Sylvester matrix equations, which are a generalization of Lyapunov matrix equations. The design results in a time‐varying multiple Lyapunov function that is decreasing at the switching instants. From the theoretical point of view, the relevance of this work is the construction of the adaptive laws that guarantee asymptotic tracking error and asymptotic estimation for the direct and indirect switched adaptive control loops, respectively. From the practical point of view, the relevance of this work is validated in a new switched impedance control for the robot interaction with uncertain and discontinuous environments.

Efficient and accurate quadrature methods of Fourier integrals with a special oscillator and weak singularities

The recent article (J. Math. Anal. Appl. 494 (2021), Article number: 124448) presented an asymptotic Filon-type method for computing the oscillatory integral with a special oscillator and weak singularities, ∫0bxα(b−x)βf(x)eiωxrdx,−1<α,β≤0,0<b<+∞,ω∈R,r∈N+. In this article, we propose and analyze two different efficient and accurate quadrature methods for this singularly oscillatory integral. First, we give a two-point Taylor interpolation method by using a two-point Taylor polynomial instead of f(x). In addition, we propose a more efficient contour integration method. By exploiting the Taylor polynomial of the function f at x=0, and then based on the additivity of the integration interval, we change the considered integral into two integrals. One integral can be efficiently computed by the contour integration method based on Cauchy Residue Theorem and generalized Gaussian-Laguerre quadrature rule. The other integral can be explicitly calculated by special functions. Specifically, we perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency parameter ω. Ultimately, the proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124448) and the modified Filon-type method. At the same computational cost, the two-point Taylor interpolation method and the asymptotic Filon-type method have a very close accuracy level. Their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. We verify error analyses of the proposed methods by experimental results. Numerical experiments can also verify the efficiency and precision of the proposed methods.

On Approximation Properties of Gamma Type Operator Based on (p,q)-İnteger

In the literature; extensive work on the q and (p,q)-calculus has contributed greatly to describing the different generalizations of many operators involving the q and (p,q)-integers. In this study, we will present to you, that we define Gamma type operator based on (p,q)-integer. We get some direct output including asymptotic formula and error estimation in terms of modulus continuity. In addition, as a result of the research, we estimate the convergence rate of these operators in a weighted space.

Information geometry approach to parameter estimation in hidden Markov model

We consider the estimation of the transition matrix of a hidden Markovian process by using information geometry with respect to transition matrices. In this paper, only the histogram of k-memory data is used for the estimation. To establish our method, we focus on a partial observation model with the Markovian process and we propose an efficient estimator whose asymptotic estimation error is given as the inverse of projective Fisher information of transition matrices. This estimator is applied to the estimation of the transition matrix of the hidden Markovian process. In this application, we carefully discuss the equivalence problem for hidden Markovian process on the tangent space.

A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions

&lt;p style='text-indent:20px;'&gt;A novel quadrature rule is formed combining Lobatto six point transformed rule and Gauss-Legendre five point transformed rule each having precision nine. The mixed rule so formed is of precision eleven. Through asymptotic error estimation the novelty of the quadrature rule is justified. Some test integrals have been evaluated using the mixed rule and its constituents both in non-adaptive and adaptive modes. The results are found to be quite encouraging for the mixed rule which is in conformation with the theoretical prediction.&lt;/p&gt;

Eigenvalue problems for a class of infinite complex symmetric tridiagonal matrices with related three-term recurrence relation

We consider the eigenvalue problem of a class of infinite complex symmetric tridiagonal matrices whose diagonal and off-diagonal elements diverge in modulus, and which have a compact inverse. We regard the matrix as a linear operator mapping a maximal domain in Hilbert space ℓ2 into ℓ2. This paper aims to extend the work of Ikebe et al. on a class of eigenvalue problems and for which asymptotic error estimates have been obtained. In this paper we focus on the following points: (1) considering what class of zero-finding problems of three-term recurrence relations can be reformulated as eigenvalue problems of the class of infinite tridiagonal matrices stated above; (2) determining a class of matrices for which obtaining good approximate eigenvalues is guaranteed by using those of truncated principal sub-matrices; and (3) determining a class of matrices that permits us the asymptotic error estimates computed as in (2).

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of L_{p,{1 / omega }}-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Model Identification and Adaptive State Observation for a Class of Nonlinear Systems

In this article, we consider the joint problems of state estimation and model identification for a class of continuous-time nonlinear systems in the output-feedback canonical form. An adaptive observer is proposed that combines an extended high-gain observer and a discrete-time identifier. The extended observer provides the identifier with a dataset permitting the identification of the system model and the identifier adapts the extended observer according to the new estimated model. The design of the identifier is approached as a system identification problem and sufficient conditions are presented that, if satisfied, allow different identification algorithms to be used for the adaptation phase. The cases of recursive least-squares and multiresolution black-box identification via wavelet-based identifiers are specifically addressed. Stability results are provided relating the asymptotic estimation error to the prediction capabilities of the identifier. Robustness with respect to additive disturbances affecting the system equations and measurements is also established in terms of an input-to-state stability property relative to the noiseless estimates.

QMC integration errors and quasi-asymptotics

Abstract A crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as O ⁢ ( 1 / N ) {O(1/\sqrt{N})} . The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier 1 / N {1/N} . However, the multiplier ( ln ⁡ N ) d {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor 1 / N {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with N = 2 m {N=2^{m}} with natural m makes the integration error approximately proportional to 1 / N {1/N} . In our numerical experiments, d ≤ 15 {d\leq 15} , and we used N ≤ 2 40 {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d ≤ 2 14 {d\leq 2^{14}} and N ≤ 2 63 {N\leq 2^{63}} .

Using Probabilistic Approach to Evaluate the Total Population Density on Coarse Grids.

Evaluation of the population density in many ecological and biological problems requires a satisfactory degree of accuracy. Insufficient information about the population density, obtained from sampling procedures negatively, impacts on the accuracy of the estimate. When dealing with sparse ecological data, the asymptotic error estimate fails to achieve a reliable degree of accuracy. It is essential to investigate which factors affect the degree of accuracy of numerical integration methods. When the number of traps is less than the recommended threshold, the degree of accuracy will be negatively affected. Therefore, available numerical integration methods cannot guarantee a satisfactory degree of accuracy, and in this sense the error will be probabilistic rather than deterministic. In other words, the probabilistic approach is used instead of the deterministic approach in this instance; by considering the error as a random variable, the chance of obtaining an accurate estimation can be quantified. In the probabilistic approach, we determine a threshold number of grid nodes required to guarantee a desirable level of accuracy with the probability equal to one.

Regularization Technique for an Inverse Space-Fractional Backward Heat Conduction Problem

This manuscript deals with a regularization technique for a generalized space-fractional backward heat conduction problem (BHCP) which is well-known to be extremely ill-posed. The presented technique is developed based on the Meyer wavelets in retrieving the solution of the presented space-fractional BHCP. Some sharp optimal estimates of the Holder-Logarithmic type are theoretically derived by imposing an a-priori bound assumption via the Sobolev scale. The existence, uniqueness and stability of the considered problem are rigorously investigated. The asymptotic error estimates for both linear and non-linear problems are all the same. Finally, the performance of the proposed technique is demonstrated through one- and two-dimensional prototype examples that validate our theoretical analysis. Furthermore, comparative results verify that the proposed method is more effective than the other existing methods in the literature.

Simultaneous Approximation for Beta-Baskakov -Stancu Operators

The paper is some proposal of advance properties of Beta-Baskakov-Stancu operators consisting hypergeo-metric series function. The central moments, Voronovskaya type asymptotic formula and error estimates applying some characteristic properties are obtained for mentioned operators.

Using penalized likelihood to select parameters in a random coefficients multinomial logit model

This paper is about estimating a random coefficients logit model in which the distribution of each coefficient is characterized by finitely many parameters, some of which may be zero. The paper gives conditions under which, with probability approaching 1 as the sample size increases, penalized maximum likelihood (PML) estimation with the adaptive LASSO (AL) penalty distinguishes correctly between zero and non-zero parameters. The paper also gives conditions under which PML reduces the asymptotic mean-square estimation error of any continuously differentiable function of the model’s parameters. The paper describes a method for computing PML estimates and presents the results of Monte Carlo experiments that illustrate their performance. It also presents the results of PML estimation of a random coefficients logit model of choice among brands of butter and margarine in the British groceries market.

Bootstrap methods for inference in the Parks model

Abstract This paper shows how to bootstrap hypothesis tests in the context of the Parks’s (1967) Feasible Generalized Least Squares estimator. It then demonstrates that the bootstrap outperforms FGLS(Parks)’s top competitor. The FGLS(Parks) estimator has been a workhorse for the analysis of panel data and seemingly unrelated regression equation systems because it allows the incorporation of cross-sectional correlation together with heteroskedasticity and serial correlation. Unfortunately, the associated, asymptotic standard error estimates are biased downward, often severely. To address this problem, Beck and Katz (1995) developed an approach that uses the Prais-Winsten estimator together with “panel corrected standard errors” (PCSE). While PCSE produces standard error estimates that are less biased than FGLS(Parks), it forces the user to sacrifice efficiency for accuracy in hypothesis testing. The PCSE approach has been, and continues to be, widely used. This paper develops an alternative: a nonparametric bootstrapping procedure to be used in conjunction with the FGLS(Parks) estimator. We demonstrate its effectiveness using an experimental approach that creates artificial panel datasets modelled after actual panel datasets. Our approach provides a superior alternative to existing estimation options by allowing researchers to retain the efficiency of the FGLS(Parks) estimator while producing more accurate hypothesis test results than the PCSE.

Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator

In this work, we perform a complete asymptotic analysis and the construction of affordable quadrature rules for a class of oscillatory infinite Bessel transform with a general oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points, we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter $$\omega $$. The resulting asymptotic expansions clarify the large $$\omega $$ behavior of the transform and provide powerful tools for designing quadrature rules and conducting error analysis. As a consequence, efficient and affordable new modified Filon-type methods for computing the transform numerically are proposed. Particularly, we carry out the rigorous error analysis and obtain asymptotic error estimates in inverse powers of $$\omega $$. Numerical examples can confirm our analysis. The accuracy can be improved greatly by either adding more derivatives interpolation at endpoints or adding more interior nodes. Moreover, only using a small number of nodes and multiplicities, we can obtain the required accuracy level. For fixed number of nodes and multiplicities, the higher accuracy can be achieved with the larger values of $$\omega $$.

Thermal Monitoring of Buildings by Aggregated Temperature Estimation

Abstract Thermal monitoring is important not only to ensure the comfort of building inhabitants but also to reduce energy consumption and greenhouse gas emissions. It can be, however, a challenging task because of limited computational and sensing resources at hand. This paper provides an efficient technique to estimate average (or mean operative) temperatures of rooms in a building. The proposed average observer is of minimum order whose parameters are chosen to minimize the asymptotic estimation error. The results show the effectiveness of the approach in the estimation-based temperature regulation.