Convergence Analysis Research Articles

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time

This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.

A highly accurate numerical method for solving boundary value problem of generalized Bagley‐Torvik equation

A highly accurate numerical method is given for the solution of boundary value problem of generalized Bagley‐Torvik (BgT) equation with Caputo derivative of order by using the collocation‐shooting method (C‐SM). The collocation solution is constructed in the space as piecewise polynomials of degree at most and has continuous derivative on the considered domain. Furthermore, under some smoothness, conditions of the exact solution convergence analysis are given. Numerical investigation is carried on several examples, and besides showing the justification of theoretical results, comparisons with existing numerical methods are also given. Results indicate that C‐SM gives highly accurate results and is efficient both in computational time and storage.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Fractal Self-Similarity in Semantic Convergence: Gradient of Embedding Similarity across Transformer Layers

This paper presents a mathematical analysis of semantic convergence in transformer-based language models, drawing inspiration from the concept of fractal self-similarity. We introduce and prove a novel theorem characterizing the gradient of embedding similarity across layers. Specifically, we establish that there exists a monotonically increasing function that provides a lower bound on the rate at which the average cosine similarity between token embeddings at consecutive layers and the final layer increases. This establishes a fundamental property: semantic alignment of token representations consistently increases through the network, exhibiting a pattern of progressive refinement, analogous to fractal self-similarity. The key challenge addressed is the quantification and generalization of semantic convergence across diverse model architectures and input contexts. To validate our findings, we conduct experiments on BERT and DistilBERT models, analyzing embedding similarities for diverse input types. While our experiments are limited to these models, we empirically demonstrate consistent semantic convergence within these architectures. Quantitatively, we find that the average rates of semantic convergence are approximately 0.0826 for BERT and 0.1855 for DistilBERT. We observe that the rate of convergence varies based on token frequency and model depth, with rare words showing slightly higher similarities (differences of approximately 0.0167 for BERT and 0.0120 for DistilBERT). This work advances our understanding of transformer models’ internal mechanisms and provides a mathematical framework for comparing and optimizing model architectures.

Sparse recovery using expanders via hard thresholding algorithm

Expanders play an important role in combinatorial compressed sensing. Via expanders measurements, we propose the expander normalized heavy ball hard thresholding algorithm (ENHB-HT) based on expander iterative hard thresholding (E-IHT) algorithm. We provide convergence analysis of ENHB-HT, and it turns out that ENHB-HT can recover an s-sparse signal if the measurement matrix A∈{0,1}m×n satisfies some mild conditions. Numerical experiments are simulated to support our two main theorems which describe the convergence rate and the accuracy of the proposed algorithm. Simulations are also performed to compare the performance of ENHB-HT and several existing algorithms under different types of noise, the empirical results demonstrate that our algorithm outperform a few existing ones in the presence of outliers.

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

Distributed Observer for Linear Systems with Multirate Sampled Outputs Involving Multiple Delays

This paper focuses on the design of a continuous distributed observer for linear systems under multirate sampled output measurements involving multiple delays. It is mathematically proved that the continuous distributed observer can achieve estimation in a sensor network environment, where output measurements from each sensor are available at different sampling instants, whether these times are periodic or aperiodic, and despite the presence of multiple time-varying delays. Each sampled and delayed measurement represents a node of the network, necessitating a dedicated observer for each node, which has access to only part of the system’s output and communicates with its neighbors according to a given network graph. The exponential convergence of the error dynamics is ensured by Lyapunov stability analysis, which accounts for the influence of the sampled and delayed measurements at each node. To demonstrate the effectiveness of the proposed observer, simulation tests were conducted on the tracking control of chasing satellites in low Earth orbit (LEO), encompassing both small and large sampling rates and delays. The continuous distributed observer with sampled output measurements exhibited convergence in scenarios with different sampling intervals, even in the presence of time-varying delays, achieving asymptotic omniscience, as demonstrated in the convergence analysis.

Ablation and molten layer flow simulation for plate model of SiO2f/SiO2 composite material using particle method

In this paper, the moving particle semi-implicit method (MPS) is extended from calculating free mobility to simulating the extremely viscous and temperature-dependent molten layer flow of SiO2f/SiO2 composite material under aerodynamic heating conditions, which includes strong heating and shear of incoming flow. A method for applying heat flux and airflow shear, based on the conceptual particle approach, has been established. Heat transfer, melting, solidification, and evaporation behaviors are considered, with temperature-dependent viscosity variations also accounted for. The ablative regression of the plate model is verified using experimental results of the SiO2f/SiO2 composite material, and results from convergence analysis demonstrate the accuracy of the space step size selection. Surface morphology analysis through three-dimensional computation indicates that the extended particle method also accurately describes the surface morphology of SiO2f/SiO2 composite material under aerodynamic heating conditions. Thus, the extended particle method accurately simulates both the ablation process and the surface morphology of the SiO2f/SiO2 composite material. The influences of acceleration and surface tension are discussed. Ablative recession, when subject to acceleration, is smaller than that observed in its absence. When exposed to surface tension, the liquid layer tends to form a spherical shape, and the particles behave as a cohesive unit, resulting in smaller ablative recession than in the absence of surface tension.

A Benchmark for Neutron Kinetics Methods in Hexagonal Geometry

A new benchmark solution has been developed to aid in the development of neutron kinetics solvers for hexagonal geometries, such as those in water-water energetic reactors. This benchmark problem is based on the two-dimensional, two-group, International Atomic Energy Agency–Hex steady-state benchmark problem. Two transient problems are presented: a ramp and a step transient. To create a benchmark-quality solution to this transient problem, a basic neutron kinetics model was added to the computer program LUPINE (Liquid metal–cooled fast reactor Utility for Physics Informed Nuclear Engineering). LUPINE solves neutron kinetics equations in general unstructured mesh. First, the LUPINE kinetics solvers are verified using the TWIGL benchmark problems. Then the methods in LUPINE are used to perform a spatiotemporal convergence analysis to ensure that the solutions are sufficiently converged. Finally, Richardson extrapolation is performed to obtain the reference solutions for these new kinetics benchmark problems.

Concurrent learning for adaptive pontryagin's maximum principle of nonlinear systems with inequality constraints

AbstractIn this article, a finite‐horizon adaptive Pontryagin's maximum principle is presented for nonlinear systems with state inequality constraints. Concurrent learning (CL) technique is adopted to identify the unknown parameters of the dynamic systems. Based on the identification model, a novel adaptive iterative algorithm under the Pontryagin's framework is introduced to learn the finite‐horizon optimal control solution. Convergence analysis of the algorithm is provided by showing that the cost function sequence is monotonically decreasing. Furthermore, we extend the adaptive iterative algorithm to time‐varying nonlinear systems. The new algorithm overcomes the technical obstacles of the existing adaptive/approximate dynamic programming (ADP) approaches to deal with the time‐varying characteristic of Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE), especially when state constraints exist. Simulation examples are carried out to validate the effectiveness of the theoretical results.

Design and application of interval type-2 fuzzy neural network systems optimized with hybrid algorithms

Interval type-2 fuzzy neural network systems (IT2 FNNSs) own the advantages of IT2 fuzzy logic systems (FLSs) and NN. The paper designs a kind of IT2 FNNSs for permanent magnetic drive (PMD) forecasting. For each rules of IT2 FNNSs, whose antecedents, consequents and input measurements are chosen as Gaussian IT2 membership functions (MFs). The proposed hybrid backpropagation (BP) algorithms and recursive least square (RLS) algorithms are adopted to tune all the parameters simultaneously. Simulation instances on the basis of data of permanent magnetic drive (PMD) torque and revolutions per minute (rpm) are adopted for testing the performances of proposed hybrid optimized IT2 FNNSs for forecasting. Convergence analysis illustrates that the IT2 FNNSs have excellent generalization capability compared with both singleton and non-singleton T1 FNNSs.

Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the l∞(L2) and l∞(H1) norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.

L1‐type finite element method for time‐fractional diffusion‐wave equations on nonuniform grids

AbstractAs is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.

키워드 네트워크 분석을 활용한 인공지능 융합 영재교육 연구 동향 분석

Objectives The purpose of this study is to analyze trends in artificial intelligence convergence gifted education in elementary and middle schools that have been studied with a focus on artificial intelligence, and to derive important issues, topics, trends, and trends in related research. Methods Research trends were analyzed for 75 literatures related to artificial intelligence and artificial intelligence convergence gifted education research and education programs. Analysis data was collected through Academic Research Information Service (RISS). The collected data was refined using Excel and text, and major keywords were identified, analyzed, and visualized through frequency analysis and keyword network analysis. CONCOR cluster analysis was used to determine the centrality of keywords. Results First, since 2017, research on artificial intelligence convergence gifted education has been steadily expanding in terms of content and methods. Second, the results of frequency analysis of the main keywords in research related to artificial intelligence convergence gifted education showed ‘artificial intelligence’, ‘artificial intelligence education’, ‘gifted education’, ‘computational thinking ability’, and ‘machine learning’.Third, as a result of TF-IDF weight analysis, ‘artificial intelligence’, ‘artificial intelligence education’, ‘gifted education’, ‘computational thinking ability’, ‘science gifted’, ‘artificial intelligence education program’, and ‘information gifted’ were found. In general, it was similar to the results of the frequency analysis of major keywords, but it was confirmed that there was a difference in the rankings. Keywords with relatively high weight compared to their frequency of occurrence are ‘science gifted’, ‘creativity’, and ‘artificial intelligence education program’.Fourth, through semantic network and CONCOR analysis, it can be confirmed that artificial intelligence convergence gifted education research is being conducted focusing on ‘educational content’, ‘educational effect’, ‘teaching method’, and ‘educational target’. Conclusions This study derived the main topics and flows of research based on the results of keyword network analysis of domestic artificial intelligence convergence gifted education-related research.

Club Convergence in Productivity among European Regions: Recent Evidence and Policy Implications

We use data on 155 NUTS regions, with the objective of i) generating TFP series over 1996-2018, and ii) testing for club-convergence. We also employ a logistic regression to identify the significant club-membership conditioning factors. We contribute to the literature on two main grounds. Firstly, we are the first to extract TFP series while applying a methodology that allows for heterogeneous production functions and accounts for cross-section dependence and nonstationarity. Secondly, we are the first to apply a TFP-club convergence analysis in the context of European regions over a large period of time that witnessed several shocks affecting the European integration. Our analysis reveals that: (i) most of the regions of the top TFP convergence clubs are located in the "old" EU-member states, with few regions situated in countries that joined the EU after 2004; (ii) several regions in the South are trailing with the risk of falling in a "productivity trap"; (iii) regions situated in new EU-member countries registered remarkable TFP growth rates; (iv) larger regional research and development resources, and higher employment in high-technology sectors are associated with a greater probability of being member of top-ranked TFP convergence clubs. Our findings suggest that the needs of the lagging regions should be prioritized in the agenda of the EU. In this respect, we suggest several recommendations.

Impact of mixed boundary conditions and nonsmooth data on layer‐originated nonpremixed combustion problems: Higher‐order convergence analysis

Impact of mixed boundary conditions and nonsmooth data on layer‐originated nonpremixed combustion problems: Higher‐order convergence analysis

Fifth-Kind Chebyshev Spectral Collocation Treatment for the Volterra Integro-Differential Equation of the Third Kind

In this research work, we offer an efficient spectral numerical scheme for handling the Volterra integro-differential of the third kind via Chebyshev polynomials of the fifth kind. The celebrated fundamentals and relations were stated with some details. With the aid of the spectral collocation method and the properties of Chebyshev polynomials, the singular Volterra integro-differential equation with its initial condition was transformed into a system of algebraic equations. The convergence and error analyses were discussed in depth. The validity and applicability of the method were tested and verified through three numerical examples, with the absolute errors reported in tables an graphs.

Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces

In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.