Convergence Of Approximations Research Articles

The Research of Short-term Electric Load Forecasting based on Machine Learning Algorithm

The ability of electric load forecasting become the key to measuring electric planning and dispatching, and improving the accuracy of electric load forecasting has become one hot spot topic of scholars in recent years. The traditional electric load forecasting algorithm mainly include statistical learning algorithm. The electric load forecasting algorithm based on traditional forecasting algorithm, machine learning algorithm and neural network algorithm greatly improve the convergence approximation effect and accuracy. In recent years, the electric load forecasting algorithms based on the deep learning algorithm has greatly reduced the error and improved the measurement accuracy and robustness, such as RNN, GRU, LSTM, DBN, TCN, and so on, as well as the combination algorithm based on deep learning. This paper introduces the classical algorithm of deep learning in electric load forecasting. The main purpose is to explore the error and fitting degree of algorithms established by different algorithms, and provide reference for the selection of forecasting algorithms for various types of electric load forecasting.

Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.

A Comparison of the Adomian Decomposition Method and Variational Iteration Method for a Two-Dimensional Nonlinear Wave Equation

This paper studies a two-dimensional wave equation in the presence of power and derivative nonlinearity subject to suitable initial conditions. It aims to construct the exact-analytical solution of a two-dimensional nonlinear wave equation using two different semi-analytical methods and to compare the obtained results. First, a well-known Adomian decomposition method (ADM) based on operators is employed. Many researchers use the ADM to investigate several scientific applications, and this method straightforwardly attacks the problem without using linearization, discretization, perturbation, or any other restrictive assumption that may change the physical behavior of the problem. Second, the variational iteration method (VIM) also provides rapid convergent successive approximations of the closed-form solutions if it exists; otherwise, it provides an approximation of a high degree accuracy level even in case of few obtained iterations. The obtained results are drawn graphically and presented in the tables. Both methods provide almost the same solutions in each nonlinearity case, and VIM has computational advantages over ADM in computation size. Further, ADM and VIM provide a series of solutions that converge in a very small time domain, which limits these methods. Keywords: Adomian decomposition method, variational iteration method, approximations, partial differential equations, two-dimensional wave equation. https://doi.org/10.55463/issn.1674-2974.50.2.3

Application of Continued Fraction in Pell's Equation

This paper uses a continued fraction to explain various approaches to solving Diophantine equations. It first examines the fundamental characteristics of continued fractions, such as convergent and approximations to real numbers. Using continued fractions, we can solve the Pell's equation. Certain theorems have also been discussed for how to expand quadratic irrational integers into periodic continued fractions. Finally, the relationship between convergent and best approximations and use of continuous fraction in calendar construction has-been investigated. The analytical theory of continued fractions is a significant generalization of continued fractions and represents a large field for current and future research.

Reference Vector Assisted Candidate Search with Aggregated Surrogate for Computationally Expensive Many Objective Optimization Problems

Pareto-optimal sets of multiobjective optimization problems with black-box and computationally expensive objective functions are generally hard to locate within a limited computational budget, and this situation gets even worse when more than three objectives are involved. To this end, we present a novel surrogate-assisted many-objective optimization algorithm RECAS. Unlike most prior studies, the proposed algorithm is a non–evolutionary-based method, and it iteratively determines new points for expensive evaluation via a series of independent reference vector assisted candidate searches. Furthermore, to make the number of surrogates to be maintained independent of the number of objectives, in each candidate search, RECAS constructs a surrogate model in an aggregated manner to approximate the quality assessment indicator of each point rather than a certain objective function. Under some mild assumptions, this study proves that RECAS converges almost surely to the Pareto-optimal front. In the numerical experiments, the effectiveness and reliability of RECAS are examined on both DTLZ and WFG test suites with the number of objectives varying from 2 to 10. Compared with six state-of-the-art many-objective optimization algorithms, RECAS generally performs better in maintaining convergent and well-spread approximation of the Pareto-optimal front. Finally, the good performance of RECAS on two watershed simulation model calibration problems indicates its great potential in handling real-world applications. History: Accepted by Antonio Frangioni, Area Editor for Design and Analysis of Algorithms–Continuous. Funding: This work was supported by the National University of Singapore start-up grant to C.A. Shoemaker [Grant R-266-000-109-133]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1260 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.7243971 ].

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s in (0,1), in bounded, curvilinear polygonal domains varOmega subset {{mathbb {R}}}^2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm {mathbb {H}}^s(varOmega ). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations invarOmega . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in varOmega , exponential convergence rates for solutions uin {mathbb {H}}^s(varOmega ) of mathcal {L}^s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towardspartial varOmega . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of mathcal {L}^{-s} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations invarOmega . The present analysis for either approach extends to (polygonal subsets {widetilde{mathcal {M}}} of) analytic, compact 2-manifolds {mathcal {M}}, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain varOmega subset {{mathbb {R}}}^2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is strictly weaker than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert–Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the kernel Gaussian–Sinkhorn divergence, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert–Schmidt norm convergence, we obtain dimension-independent convergence rates for finite sample approximations of the kernel Gaussian–Sinkhorn divergence, of the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent, is exponentially faster than the worst case scenario in the literature.

A random walk through Canadian contributions on empirical processes and their applications in probability and statistics

AbstractIn this article, we present a review of important results and statistical applications obtained or generalized by Canadian pioneers and their collaborators, for empirical processes of independent and identically distributed observations, pseudo‐observations, and time series. In particular, we consider weak convergence and strong approximations results, as well as tests for model adequacy such as tests of independence, tests of goodness‐of‐fit, tests of change point, and tests of serial dependence for time series. We also consider applications of empirical processes of interacting particle systems for the approximation of measure‐valued processes.

Inversion of chlorophyll content under the stress of leaf mite for jujube based on model PSO-ELM method.

During the growth season, jujube trees are susceptible to infestation by the leaf mite, which reduces the fruit quality and productivity. Traditional monitoring techniques for mites are time-consuming, difficult, subjective, and result in a time lag. In this study, the method based on a particle swarm optimization (PSO) algorithm extreme learning machine for estimation of leaf chlorophyll content (SPAD) under leaf mite infestation in jujube was proposed. Initially, image data and SPAD values for jujube orchards under four severities of leaf mite infestation were collected for analysis. Six vegetation indices and SPAD value were chosen for correlation analysis to establish the estimation model for SPAD and the vegetation indices. To address the influence of colinearity between spectral bands, the feature band with the highest correlation coefficient was retrieved first using the successive projection algorithm. In the modeling process, the PSO correlation coefficient was initialized with the convergent optimal approximation of the fitness function value; the root mean square error (RMSE) of the predicted and measured values was derived as an indicator of PSO goodness-of-fit to solve the problems of ELM model weights, threshold randomness, and uncertainty of network parameters; and finally, an iterative update method was used to determine the particle fitness value to optimize the minimum error or iteration number. The results reflected that significant differences were observed in the spectral reflectance of the jujube canopy corresponding with the severity of leaf mite infestation, and the infestation severity was negatively correlated with the SPAD value of jujube leaves. The selected vegetation indices NDVI, RVI, PhRI, and MCARI were positively correlated with SPAD, whereas TCARI and GI were negatively correlated with SPAD. The accuracy of the optimized PSO-ELM model (R2 = 0.856, RMSE = 0.796) was superior to that of the ELM model alone (R2 = 0.748, RMSE = 1.689). The PSO-ELM model for remote sensing estimation of relative leaf chlorophyll content of jujube shows high fault tolerance and improved data-processing efficiency. The results provide a reference for the utility of UAV remote sensing for monitoring leaf mite infestation of jujube.

A domain decomposition method of Schwarz waveform relaxation type for singularly perturbed nonlinear parabolic problems

We introduce a domain decomposition method of discrete Schwarz waveform relaxation (DSWR) type for a singularly perturbed nonlinear parabolic problem. The method utilizes Shishkin transition parameter for a space–time decomposition of the computational domain. In each subdomain, the problem is discretized using the central differencing and backward difference schemes on a uniform mesh in space and time directions, respectively. Further, the exchange of information between the subdomains is done through the Dirichlet data that leads to optimal convergence. We analyse the convergence of the developed method and show that the method converges very fast for small perturbation parameter and provides uniformly convergent approximations to the solution of the nonlinear problem. Finally, with some numerical experiments, we illustrate our theoretical results.

Narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

This article develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved,and numerical experiments are provided along with some computer implementation details.
 For more information see https://ejde.math.txstate.edu/conf-proc/26/f1/abstr.html

An exponential convergence approximation to singularly perturbed problems by Log orthogonal functions

An exponential convergence approximation to singularly perturbed problems by Log orthogonal functions

Model for Assessing the Level of Knowledge Convergence in Multinational Projects

In modern conditions, knowledge management acquires a new meaning and becomes one of the decisive factors for success in the project implementation. Knowledge transfer is significantly complicated in international projects. This requires an in-depth analysis of different participants` project management systems, identifying their differences and determining the ability to converge (convergence) through knowledge transfer. The paper proposes the model for assessing the convergence level of project management systems, which includes a fuzzy assessment of the factors influencing the ability of the system to transfer knowledge, as well as assessing the rate of convergence (approximation) in projects. The results of the study shows that the proposed methods allow identifying “bottlenecks” of knowledge transfer processes in multinational projects and determining a strategy to increase the level of knowledge systems convergence at the project initial stage. Evaluation of the accuracy and reliability of the proposed methods prove the adequacy of their applications for forecasting new project convergence level.

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

This paper is devoted to the implementation of fractional differential transform method (FDTM) for the evaluation of new approximations for the fractional models of the system of multi‐pantograph differential equations. Some new properties of FDTM have also been proposed for the fractional integro‐differential equations. It is noteworthy that the integro‐differential equations are studied for integers in the literature, and we have proposed the solutions of proportional delayed integro‐differential equations for fractional order. The evaluated results are verified numerically by considering some test examples of fractional integro‐differential equations and fractional models of the system of multi‐pantograph differential equations. The outcomes conclude that evaluated series solutions via FDTM promise more appropriate approximations possess very negligible error and represent the convergent approximations. Thus, evaluated series solutions are comparatively more accurate and require less computational time.

Numerical solution of singularly perturbed Fredholm integro-differential equations by homogeneous second order difference method

This work presents a computational approximate to solve singularly perturbed Fredholm integro-differential equation with the reduced second type Fredholm equation. This problem is discretized by a finite difference approximate, which generates second-order uniformly convergent numerical approximations to the solution. Parameter-uniform approximations are generated using Shishkin type meshes. The performance of the numerical scheme is tested which supports the effectiveness of the technique.

Self-guided quantum state tomography for limited resources

Quantum state tomography is a process for estimating an unknown quantum state; which is innately probabilistic. The exponential growth of unknown parameters to be estimated is a fundamental difficulty in realizing quantum state tomography for higher dimensions. Iterative optimization algorithms like self-guided quantum tomography have been effective in robust and accurate ascertaining a quantum state even with exponential growth in Hilbert space. We propose a faster convergent simultaneous perturbation stochastic approximation algorithm which is more practical in a resource-deprived situation for determining the underlying quantum states by incorporating the Barzilai–Borwein two-point step size gradient method with minimal loss of accuracy.

Kernel center adaptation in the reproducing kernel Hilbert space embedding method

SummaryThe performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state‐space. In this article, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems by exploiting the fact that the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self‐organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.

Pathwise Convergent Approximation for the Fractional SDEs

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

Approximation by linear combinations of translates of a single function

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the $L^p$-approximation convergence rate by these methods, when $n \to \infty$, for $1 \leq p \leq \infty$. We also generalize these results to classes of multivariate functions defined the convolution with the tensor product of a single function. In the case $p=2$, for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of $n$ translates of arbitrary function.

Graph Convergence and Iterative Approximation of Solution of a Set‐Valued Variational Inclusions

In this article, first, we introduce a class of proximal‐point mapping associated with generalized αiβj‐Hp(.,.,…)‐accretive mapping. Further, we discuss the graph convergence of generalized αiβj‐Hp(.,.,…)‐accretive mapping. As an application, we consider a set‐valued variational inclusion problem (SVIP) in real Banach spaces. Furthermore, we propose an iterative scheme involving the above class of proximal‐point mapping to find a solution of SVIP and discuss its convergence under some convenient assumptions. An example is constructed and demonstrated few graphics in support of our main results.