Approximation Theorem Research Articles

Generating probability distributions on intervals and spheres with application to finite element method

This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval [a,b], which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This type of probability has scaling and symmetry feature, and sufficient conditions under which a density function can be generated through discrete polynomial spectrum are given in this work followed by concrete examples. The density function ρ obtained in this way has the advantage of being positive definite. Computer based numerical simulation shows that the theoretically verified criteria for probability distribution are almost optimal with respect to our testing examples. After the establishment of an approximation theorem in L1 space, we propose a probabilistic Galerkin scheme that can be either continuous or discontinuous, which is potentially useful to asymptotically solve some PDEs on the sphere locally and globally.

A note on the Choquet type operators

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear Korovkin type approximation theorem associated to Bernstein-Kantorovich-Choquet operators is proved. The paper also includes a large generalization of H\"{o}lder's inequality within the framework of monotone and sublinear operators acting on spaces of continuous functions.

Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

An approximate Spielman-Teng theorem for the least singular value sn(Mn) of a random n × n square matrix Mn is a statement of the following form: there exist constants C, c > 0 such that for all η > 0, Pr(sn(Mn) ≤ η) ≲ nCη + exp(−nc). The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for {0, 1}-valued matrices, each of whose rows is an independent vector with exactly n/2 zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a ‘truly combinatorial’ setting.

The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications

We root this tribute to Nicholas Yannelis in Chapter II of his 1983 Rochester Ph.D. dissertation, and in his 1983 paper with Prabhakar: this work strengthens the lower semicontinuity assumption of Michael’s continuous selection theorem to open lower sections, and leads to correspondences defined on a paracompact space with values on a Hausdorff linear topological space. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and apply our result to four domains of Yannelis’ contributions: Berge’s maximum theorem, the Gale–Nikaido–Debreu lemma, the Sonnenschein–Shafer non-transitive setting, and the Anderson–Khan–Rashid approximate existence theorem. The last also resonates with Chapter VI of Yannelis’ dissertation, and allows a more general framing of the pioneering application of the paracompactness condition to his current and ongoing work in mathematical economics.

Stein’s method for the Poisson–Dirichlet distribution and the Ewens sampling formula, with applications to Wright–Fisher models

We provide a general theorem bounding the error in the approximation of a random measure of interest—for example, the empirical population measure of types in a Wright–Fisher model—and a Dirichlet process, which is a measure having Poisson–Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson–Dirichlet distribution. We apply the result to bound the error in the approximation of the stationary distribution of types in the finite Wright–Fisher model with infinite-alleles mutation structure (not necessarily parent independent) by the Poisson–Dirichlet distribution. An important consequence of our result is an explicit upper bound on the total variation distance between the random partition generated by sampling from a finite Wright–Fisher stationary distribution, and the Ewens sampling formula. The bound is small if the sample size n is much smaller than N1/6log(N)−1/2, where N is the total population size. Our analysis requires a result of separate interest, giving an explicit bound on the second moment of the number of types of a finite Wright–Fisher stationary distribution. The general approximation result follows from a new development of Stein’s method for the Dirichlet process, which follows by viewing the Dirichlet process as the stationary distribution of a Fleming–Viot process, and then applying Barbour’s generator approach.

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

In the present paper, we introduce a new kind of convergence, called the statistical relative uniform convergence, for a double sequence of functions at a point, where the relative uniform convergence of the set of the neighborhoods of the given point is considered. By the use of the statistical relative uniform convergence, we investigate a Korovkin type approximation theorem which makes the proposed method stronger than the ones studied before. After that, we give an example using this new type of convergence. We also study the rate of convergence of the proposed convergence.

Weighted (λ,μ)-statistical convergence and statistical summability methods of double sequences of fuzzy numbers with application to Korovkin-type fuzzy approximation theorem

The paper aims to investigate different types of weighted statistical convergence and weighted statistical summability of double sequences of fuzzy numbers. Relations connecting statistical convergence and statistical summability have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. Relevant inclusion relations and related results concerning the new fuzzy summability methods have also been examined in detail. Finally, a Korovkin-type approximation theorem for double sequences of fuzzy positive linear operators has been established using the most generalized version of the summability method and showed its efficiency by means of a concrete example.

Korovkin type approximation via triangular A−statistical convergence on an infinite interval

In the present paper, using the triangular $A-$statistical convergence for double sequences, which is an interesting convergence method, we prove a Korovkin-type approximation theorem for positive linear operators on the space of all real-valued continuous functions on $\left[ 0,\infty \right)\times \left[ 0,\infty \right) $ with the property that have a finite limit at the infinity. Moreover, we present the rate of convergence via modulus of continuity. Finally, we give some further developments.

Deep Learning Analysis of Ultrasonic Guided Waves for Cortical Bone Characterization.

Ultrasonic guided waves (UGWs) propagating in the long cortical bone can be measured via the axial transmission method. The characterization of long cortical bone using UGW is a multiparameter inverse problem. The optimal solution of the inverse problem often includes a complex solving process. Deep neural networks (DNNs) are essentially powerful multiparameter predictors based on universal approximation theorem, which are suitable for solving parameter predictions in the inverse problem by constructing the mapping relationship between UGW and cortical bone material parameters. In this study, we investigate the feasibility of applying the multichannel crossed convolutional neural network (MCC-CNN) to simultaneously estimate cortical thickness and bulk velocities (longitudinal and transverse). Unlike the multiparameter estimation in most previous studies, the technique mentioned in this work avoids solving a multiparameter optimization problem directly. The finite-difference time-domain (FDTD) method is performed to obtain the simulated UGW array signals for training the MCC-CNN. The network that is exclusively trained on simulated data sets can predict cortical parameters from the experimental UGW data. The proposed method is confirmed by using FDTD simulation signals and experimental data obtained from four bone-mimicking plates and from ten ex vivo bovine cortical bones. The estimated root-mean-squared error (RMSE) in the simulated test data for the longitudinal bulk velocity ( VL ), transverse bulk velocity ( VT ), and cortical thickness (Th) is 97 m/s, 53 m/s, and 0.089 mm, respectively. The predicted RMSE in the bone-mimicking phantom experiments for VL|| , VT|| , and Th is 120 m/s, 80 m/s, and 0.14 mm, respectively. The experimental dispersion trajectories are matched with the theoretical dispersion curves calculated by the predicted parameters in ex vivo bovine cortical bone experiments. Our proposed method demonstrates a feasible approach for the accurate evaluation of long cortical bones based on UGW.

Summability approximation theorems of double random variables

In 1971, Chow and Teicher presented a probability approximation Theorem via summability methods. The goal of this article is to extend Chow and Teicher results to higher dimension. To achieve this we consider multidimensional totally monotonic and independent identically random variables. Using these notions we present a series of approximation type theorems.

Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

In this paper, we introduce a novel extension of the Bernstein-Kantorovich-Stancu type operator of degree n with the help of multiple shape parameters. Voronovskaja and Grüss-Voronovskaja type approximation theorems are examined via Ditzian-Totik moduli of smoothness. We investigate basic statistical convergence properties with respect to a non-negative regular summability matrix. Moreover, using Ditzian-Totik moduli, local and global approximation properties associated to the proposed operator have been established. Finally, several illustrative examples are presented to demonstrate the efficiency, applicability and validity of the operator. The graphical and numerical results verify that the proposed operator gives better approximation as well as expand the previous Bernstein-Kantorovich type modifications including single parameter.

On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

The present paper seeks to illustrate approximation theorems to the fixed point for generalized α -nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and refine many of the recently reported results in the literature.

A lognormal model for evaluating maximum residue levels of pesticides in crops.

To evaluate pesticide regulatory standards in agricultural crops, we introduced a regulatory modeling framework that can flexibly evaluate a population's aggregate exposure risk via maximum residue levels (MRLs) under good agricultural practice (GAP). Based on the structure of the aggregate exposure model and the nature of variable distributions, we optimized the framework to achieve a simplified mathematical expression based on lognormal variables including the lognormal sum approximation and lognormal product theorem. The proposed model was validated using Monte Carlo simulation, which demonstrates a good match for both head and tail ends of the distribution (e.g., the maximum error=2.01% at the 99th percentile). In comparison with the point estimate approach (i.e., theoretical maximum daily intake, TMDI), the proposed model produced higher simulated daily intake (SDI) values based on empirical and precautionary assumptions. For example, the values at the 75th percentile of the SDI distributions simulated from the European Union (EU) MRLs of 13 common pesticides in 12 common crops were equal to the estimated TMDI values, and the SDI values at the 99th percentile were over 1.6-times the corresponding TMDI values. Furthermore, the model was refined by incorporating the lognormal distributions of biometric variables (i.e., food intake rate, processing factor, and body weight) and varying the unit-to-unit variability factor (VF) of the pesticide residues in crops. This ensures that our proposed model is flexible across a broad spectrum of pesticide residues. Overall, our results show that the SDI is significantly reduced, which may better reflect reality. In addition, using a point estimate or lognormal PF distribution is effective as risk assessments typically focus on the upper end of the distribution.

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors $x_i, i=1,\dots,m$ (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

Why neural networks apply to scientific computing?

In recent years, neural networks have become an increasingly powerful tool in scientific computing. The universal approximation theorem asserts that a neural network may be constructed to approximate any given continuous function at desired accuracy. The backpropagation algorithm further allows efficient optimization of the parameters in training a neural network. Powered by GPU's, effective computations for scientific and engineering problems are thereby enabled. In addition, we show that finite element shape functions may also be approximated by neural networks.

A note on simultaneous approximation on Vitushkin sets

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.

Gaussian process approximations for multicolor Pólya urn models

AbstractMotivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Approximation theorems of a solution of amperometric enzymatic reactions based on Green\u2019s fixed point normal-S iteration

In this paper, the authors present a strategy based on fixed point iterative methods to solve a nonlinear dynamical problem in a form of Green’s function with boundary value problems. First, the authors construct the sequence named Green’s normal-S iteration to show that the sequence converges strongly to a fixed point, this sequence was constructed based on the kinetics of the amperometric enzyme problem. Finally, the authors show numerical examples to analyze the solution of that problem.

Исследование некоторых классов почти периодических на бесконечности функций

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an ε-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

BEST APPROXIMATIONS FOR MULTIMAPS ON ABSTRACT CONVEX SPACES

In this article we derive some best approximation theorems for multimaps in abstract convex metric spaces. We are based on generalized KKM maps due to Kassay-Kolumban, Chang-Zhang, and studied by Park, Kim-Park, Park-Lee, and Lee. Our main results are extensions of a recent work of Mitrovic-Hussain-Sen-Radenovic on G-convex metric spaces to partial KKM metric spaces. We also recall known works related to single-valued maps, and introduce new partial KKM metric spaces which can be applied our new results.