Error Estimates Research Articles

Exponential Convergence and Computational Efficiency of BURA-SD Method for Fractional Diffusion Equations in Polygons

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the hp FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method.

Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order α with α ∈ (0, 1) is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time t = 0. Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the temporal fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate (2 − α) for suitable choice of grading parameter (γ ≥ (2 − α)/α). Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for γ ≥ 2/α. In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.

Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L2- and H1-norms are derived for the problem with initial data u0∈H01(Ω)∩H2(Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H1-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L∞ error estimate is obtained for 2D problems that too with the initial condition is in H01(Ω)∩H2(Ω). All results proved in this paper are valid uniformly as α→1−, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Numerical investigation on sample population requirement for mean particle diameter analysis

The determination of the minimum particle number needed to meet error specifications and confidence levels poses a fundamental challenge in particle size analysis. Conventional models, primarily designed for log-normal distributions, may yield inaccuracies when applied to other distribution functions. This study introduces a numerical approach to explore the necessary sample size for measuring different types of mean diameters. The methodology involves distribution conversion, sample generation, repeated sampling, and error estimation. Specifically, Gates–Gaudin–Schuhmann (GGS) and Rosin–Rammler (RR) distributions serve as representative models in this investigation. The impact of sample size, span ratio, and boundary sizes on relative errors is examined. Additionally, an empirical model is formulated, simplifying the calculation of the requisite sample size when the relative error and mass-based span ratio are provided. The proposed method is also verified using experimental particle size data that follow the RR distribution.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

A hybrid information fusion method for SINS/GNSS integrated navigation system utilizing GRU-aided AKF during GNSS outages

Abstract To enhance the performance of integrated inertial navigation system (INS) and global navigation satellite system (GNSS) during GNSS outages, this paper proposed a fusion positioning method based on predictive observation information and adaptive filter parameter. Combined with an adaptive Kalman filter (AKF) and a Gated Recurrent Unit neural network (NN) that directly relates the inertial measurement unit (IMU) output sequence to the error estimation, the hybrid information fusion system can provide effective corrections to compensate for horizontal position errors under the constraints of complex and dynamic vehicle movement data during GNSS outages. Meanwhile, the designed adaptive parameter of the integrated navigation filter can adjust the credibility of the state prediction section when the GNSS is reconnected, ensuring the system can switch rapidly between the INS/GNSS and INS/NN integrated modes. The performance of the proposed information fusion method has been experimentally validated using IMU and GNSS data collected in a vehicle navigation test conducted on a stretch of expressway. The comparison results indicate that the proposed algorithm has error suppression capabilities under various experimental constraints and demonstrates a degree of extendibility and reusability.

Income inequality, governance quality, and environmental degradation in Asian countries: Does interaction of governance quality matter? Evidence from panel data

The objective of this research is to examine the effects of income inequality, governance quality, and their interaction on environmental quality in Asian countries. Time series data are obtained from 45 Asian countries for the period 1996–2020 for this empirical analysis. The research has performed various econometric tests to ensure the robustness and reliability of the results. We have addressed different econometric issues, such as autocorrelation, heteroskedasticity, and cross-sectional dependence, using the Driscoll-Kraay (DK) standard error estimation and endogeneity issues by the system generalized method of moments (S-GMM). The results of the study revealed that income inequality and governance quality have a positive impact on environmental degradation, while the interaction of governance quality with income inequality has a negative effect on it. In addition, economic growth, population growth, urbanization, and natural resource dependency are found to deteriorate the quality of the environment. The findings of the study offer insightful policies to reduce environmental degradation in Asian countries.

Optical functions of uniaxial rutile and anatase (TiO2) revisited

The optical functions of uniaxial rutile and anatase (TiO2) were determined from 200 to 850 nm (6.2 to 1.46 eV) using several of four optical techniques: (1) standard spectroscopic two-modulator generalized ellipsometry (2-MGE), (2) near-normal-incidence two-modulator generalized ellipsometry microscopy (2-MGEM), (3) Mueller matrix transmission of rutile, and (4) polarized transmission of rutile. The 2-MGE measurements yielded highly accurate values of the dielectric functions and error estimates from 1.46 to 6.2 eV, whereas the polarization-dependent transmission yielded more accurate values of the absorption coefficient below the band edge of rutile. The 2-MGEM also measured the diattenuation, which is related to the birefringence, and other parameters but at near-normal incidence at a single wavelength (577 nm).

Telemetry without collars: performance of fur- and ear-mounted satellite tags for evaluating the movement and behaviour of polar bears

The study of animal movement provides insights into underlying ecological processes and informs analyses of behaviour and resource use, which have implications for species management and conservation. The tools used to study animal movement have evolved over the past decades, allowing for data collection from a variety of species, including those living in remote environments. Satellite-linked radio and GPS collars have been used to study polar bear (Ursus maritimus) ecology and movements throughout the circumpolar Arctic for over 50 years. However, due to morphology and growth constraints, only adult female polar bears can be reliably collared. Collars have proven to be safe, but there has been opposition to their use, resulting in a deficiency in data across much of the species’ range. To bolster knowledge of movement characteristics and behaviours for polar bears other than adult females, while also providing an alternative to collars, we tested the use of fur- and ear-mounted telemetry tags that can be affixed to polar bears of any sex and age. We tested three fur tag designs (SeaTrkr, tribrush and pentagon tags), which we affixed to 15 adult and 1 subadult male polar bears along the coast of Hudson Bay during August–September 2021–2022. Fur tags were compared with ear tags deployed on 42 subadult and adult male polar bears captured on the coast or the sea ice between 2016 and 2022. We used data from the tags to quantify the amount of time subadult and adult males spent resting versus traveling while on land. Our results show the three fur tag designs remained functional for shorter mean durations (SeaTrkr = 58 days; tribrush = 47 days; pentagon = 22 days) than ear tags (121 days), but positional error estimates were comparable among the Argos-equipped tags. The GPS/Iridium-equipped SeaTrkr fur tags provided higher resolution and more frequent location data. Combined, the tags provided sufficient data to model different behavioural states. Furthermore, as hypothesized, subadult and adult male polar bears spent the majority of their time resting while on land, increasing time spent traveling as temperatures cooled. Fur tags show promise as a short-term means of collecting movement data from free-ranging polar bears.

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★ ★ The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_2549) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578).

In the article, we focus on the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition on a cylindrical symmetric field. The ill-posedness of this problem is proved. We introduce the modified Landweber iteration method(MLIM), the Truncated singular value decomposition(TSVD) method for solving it and propose a new regularization method, named as the TSVD-modified Landweber iteration method(TMLIM). The error estimates between the exact solution and the regularized approximate solution are presented by using two regularization parameter selection rules. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the regularization methods.

Accelerometer static state detection (SSD)-assisted GNSS/accelerometer bridge monitoring algorithm

Abstract In the field of structural health monitoring (SHM), a loosely coupled (LC) Kalman filtering algorithm that accounts for baseline drift errors is commonly used to integrate GNSS data with accelerometer data. In the LC algorithm, the baseline drift errors are considered unknown parameters that need to be estimated. In scenario of continuous float solutions, the estimation of baseline drift error is often inaccurate, leading to the divergence of monitoring results. Theoretically, as a type of motion sensor, accelerometers are expected to qualitatively determine the priori state of bridges, whether dynamic or static. Utilizing the inherent characteristics of accelerometers and the principle of zero-velocity detection in integrated navigation, we originally propose a bridge static state detection (SSD) method based on low-cost accelerometer, and introduces this prior SSD information as a constraint in GNSS/accelerometer LC algorithm, called SSD-LC bridge monitoring algorithm. Through a simulation platform and real-world bridge monitored tests, the effectiveness of our proposed SSD method has been verified. Furthermore, our proposed SSD-LC bridge monitoring algorithm can effectively mitigate the divergence problem in baseline drift estimation that occurs with continuous GNSS float solutions in traditional algorithms, which can effectively avoid misjudgments and false alarms in bridge monitoring during GNSS anomalies.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Random Stepped Frequency ISAR 2D Joint Imaging and Autofocusing by Using 2D-AFCIFSBL

With the increasingly complex electromagnetic environment faced by radar, random stepped frequency (RSF) has garnered widespread attention owing to its remarkable Electronic Counter-Countermeasure (ECCM) characteristic, and it has been universally applied in inverse synthetic aperture radar (ISAR) in recent years. However, if the phase error induced by the translational motion of the target in RSF ISAR is not precisely compensated, the imaging result will be defocused. To address this challenge, a novel 2D method based on sparse Bayesian learning, denoted as 2D-autofocusing complex-value inverse-free SBL (2D-AFCIFSBL), is proposed to accomplish joint ISAR imaging and autofocusing for RSF ISAR. First of all, to integrate autofocusing into the ISAR imaging process, phase error estimation is incorporated into the imaging model. Then, we increase the speed of Bayesian inference by relaxing the evidence lower bound (ELBO) to avoid matrix inversion, and we further convert the iterative process into a matrix form to improve the computational efficiency. Finally, the 2D phase error is estimated through maximum likelihood estimation (MLE) in the image reconstruction iteration. Experimental results on both simulated and measured datasets have substantiated the effectiveness and computational efficiency of the proposed 2D joint imaging and autofocusing method.

Analytical quaternion-based bias estimation algorithm for fast and accurate stationary gyro-compassing

This work introduces a novel approach to Strapdown Inertial Navigation System (SINS) alignment, distinct from recursive methods like Kalman filtering. The proposed methodology expedites bias error calculations by utilizing quaternion-based analytical relationships, which bypasses the slow convergence behavior associated with recursive algorithms, particularly in azimuth angle error estimation. In addition, the proposed approach demonstrates comparable accuracy to traditional fine alignment methods. Simulations and experiments validate that in contrast to the 10-min time requirement of traditional fine alignment methods (for azimuth angle estimation in stationary conditions), the proposed approach achieves the same accuracy within 20 s. However, limitations exist as the algorithm is applicable only in stationary conditions, and necessitating a high-grade IMU capable of measuring the earth’s rotation rate.

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

Multistep Iterative Methods for Solving Equations in Banach Space

The novelty of this article lies in the fact that we extend the use of a multistep method for developing a sequence whose limit solves a Banach space-valued equation. We suggest the error estimates, local convergence, and semi-local convergence, a radius of convergence, the uniqueness of the required solution that can be computed under ω-continuity, and conditions on the first derivative, which is on the method. But, earlier studies used high-order derivatives, even though those derivatives do not appear in the body structure of the proposed method. In addition to this, they did not propose computable estimates and semi-local convergence. We checked the applicability of our study to three real-life problems for semi-local convergence and two problems chosen for local convergence. Based on the obtained results, we conclude that our approach improves its applicability and makes it suitable for challenges in applied science.

Approximation error estimates by noise‐injected neural networks

One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments.