Approximation Properties Research Articles

A POSTERIORI ERROR ESTIMATES FOR APPROXIMATE SOLUTIONS OF THE OBSTACLE PROBLEM FOR THE 𝑝-LAPLACIAN

The paper is concerned with a functional identity and estimates which are fulfilled for the measures of deviations from exact solutions of the obstacle problem for the 𝑝-Laplacian. They hold true for any functions from the corresponding (energy) functional class, which contains the generalised solution of the problem as well. We do not use any special properties of approximations or numerical methods nor information of the exact configuration of the coincidence set. The right-hand side of the identities and estimates contains only known functions and can be explicitly calculated, and the left side represents a certain measure of the deviation of the approximate solution from the exact solution. The right-hand side of the identity and estimates contains only known functions and can be explicitly calculated, while and the left side represents a certain measure of the deviation of the approximate solution from the exact one. The obtained functional relations allow to estimate the error of of any approximate solutions of the problem regardless of the method of their obtaining. In addition, they enable to compare the exact solutions of problems with different data. The latter provides the possibility to estimate the errors of mathematical models.

Approximation Properties for Blended B-Splines on Unstructured Quadrilateral Mesh

Approximation Properties for Blended B-Splines on Unstructured Quadrilateral Mesh

Lipschitz‐free spaces over strongly countable‐dimensional spaces and approximation properties

AbstractLet be a compact, metrisable and strongly countable‐dimensional topological space. Let be the set of all metrics on compatible with its topology, and equip with the topology of uniform convergence, where the metrics are regarded as functions on . We prove that the set of metrics for which the Lipschitz‐free space has the metric approximation property is residual in .

Higher order approximation of functions by modified Goodman-Sharma operators

Here we study the approximation properties of a modified Goodman-Sharma operator recently considered by Acu and Agrawal in [1]. This operator is linear but not positive. It has the advantage of a higher order of approximation of functions compared with the Goodman-Sharma operator. We prove direct and strong converse theorems in terms of a related K-functional.

Output feedback control for non-strict feedback nonlinear systems: an adaptive fuzzy backstepping scheme

The adaptive fuzzy tracking control design issue is considered in this paper for a class of non-strict feedback nonlinear systems subject to external disturbances. A fuzzy high-gain state observer is constructed to reconstruct the unmeasurable states of the system by leveraging the universal approximation property of fuzzy logic systems for arbitrary unknown nonlinear functions. Within the framework of backstepping control design and in conjunction with adaptive techniques, an adaptive fuzzy control approach is presented. Subsequently, to work out the inherent ‘complexity explosion’ problem of the proposed control approach, dynamic surface control technique is introduced, leading to a simplified adaptive fuzzy output feedback control scheme. Stability analysis shows that the two proposed control schemes can ensure that all signals of the closed-loop system are semi-globally uniformly ultimately bounded, meanwhile the system tracking error can converge to a small neighborhood around the origin. Ultimately, a numerical simulation example is provided to validate the feasibility of the proposed control scheme.

A Call for a Different Plant Modelling Paradigm and a New Generation of Software

This paper is an invitation to change the plant modeling paradigm to achieve much easier convergence and ensure model consistency between different incarnations of the plant model, while retaining accuracy on par with the rigorous models. Rigorous physical properties [property/mole] are replaced by [property/mass] approximation at local conditions, making bulk properties much less sensitive to changes in stream composition. This eliminates the need for stream mole fractions and stream compositions can be described by linear component mass flows. Detailed process flow diagram is a common topology for all models. Different incarnations of node models are employed at different levels of abstraction (planning, scheduling, optimization, control, design), ensuring inheritance, consistency and increasing accuracy of solutions as plant model instances move from mass to mass-energy, to mass-energy-and-approximate-stream-properties. After the plant model with approximate properties is solved, they are updated via rigorous thermodynamic methods and the plant model is resolved until convergence.

On the turnpike to design of deep neural networks: Explicit depth bounds

It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable answer to the question of how many layers should be considered in a DNN. The underlying assumption is that the number of neurons per layer—i.e., the width of the DNN—is kept constant. Pursuing a different route than the classical analysis of approximation properties of sigmoidal functions, we prove explicit bounds on the required depths of DNNs based on asymptotic reachability assumptions and a dissipativity-inducing choice of the regularization terms in the training problem. Numerical results obtained for the two spiral task data set for classification indicate that the proposed constructive estimates can provide non-conservative depth bounds.

Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties and graded meshes

The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their C0-continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements in the physical domain and to omit redundant degrees of freedom in the vicinity of the singularity. Numerical results validate the effectiveness of hierarchical mesh grading for simulating eigenfunctions of low and high regularity.

The exponential integrability properties and strong convergence rate of an explicit Euler's method for highly nonlinear stochastic differential equations

In this paper, we employ the perturbation theory and the exponential integrability properties of the numerical approximations to obtain the strong convergence rate for a type of truncated Euler's method. Some numerical examples are given to validate the theoretical result.

Piecewise rank‐one approximation of vector fields with measure derivatives

AbstractThis work addresses the question of density of piecewise constant (resp. rigid) functions in the space of vector‐valued functions with bounded variation (resp. deformation) with respect to the strict convergence. Such an approximation property cannot hold when considering the usual total variation in the space of measures associated to the standard Frobenius norm in the space of matrices. It turns out that oscillation and concentration phenomena interact in such a way that the Frobenius norm has to be homogenized into a (resp. symmetric) Schatten‐1 norm that coincides with the Euclidean norm on rank‐one (resp. symmetric) matrices. By means of explicit constructions consisting of the superposition of sequential laminates, the validity of an optimal approximation property is established at the expense of endowing the space of measures with a total variation associated with the homogenized norm in the space of matrices.

A generalization of modified $$\alpha $$-Bernstein operators and its related estimations and errors

AbstractIn the present article, we introduce a novel generalization of modified Bernstein operators which is again a positive linear operator. We show the necessary and sufficient condition for the convergence of these operators. We also study some other approximation properties of these operators using standard tools of approximation theory.

Adaptive NN Force Loading Control of Electro-Hydraulic Load Simulator

To address the issues of derivative explosion in traditional backstepping control and the strong nonlinearity of hydraulic systems, this paper develops an adaptive neural network control method tailored for electro-hydraulic load simulators. Neural networks are employed to handle external disturbances, modeling uncertainties, and the derivatives of virtual control inputs. First, the precise state-space equations of the system are derived. Next, the approximation property of neural networks is used to design an adaptive backstepping controller, and the symmetric barrier Lyapunov function is used to prove the boundedness of the controller and control parameters. Finally, experiments are conducted to verify the effectiveness and reliability of the control algorithm. The results demonstrate that the proposed control algorithm exhibits excellent tracking performance and effectively reduces control errors.

On density order of quasi-projection operators and dual wavelet frames

From two functions in the Hilbert space L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} whose Fourier transform has certain decay, one defines a quasi-projection operator. In this paper, we prove necessary and sufficient conditions on those functions in order to the associated quasi-projection operator provides a desired approximation order and density order. To give our conditions we will use the classical notion of approximate continuity. As a consequence, we obtain approximation properties of dual wavelet frame constructed by Mixed Oblique Extension Principle. We show our results in the context of reducing subspaces of L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} with a dilation given by an expansive linear map preserving the integer lattice.

Type-2 fuzzy adaptive robust fault-tolerant control for manipulators with disturbance observer

A configuration of interval type-2 (IT2) fuzzy adaptive fault-tolerant control strategy is designed for the position trajectory tracking of cooperated robot manipulators carrying a common object. First, a non-zero time-varying parameter is installed in IT2 FLS, then a property of universal approximation of IT2 fuzzy logic system (FLS) is proposed, where the non-zero time-varying parameter has the relation with the approximation precision of IT2 FLS. Second, for the unknown compounded disturbance, a disturbance observer is designed to estimate it and is integrated into the IT2 adaptive control. To improve the approximation accuracies, the novel IT2 FLSs with non-zero parameter are utilized to compensate for the uncertain nonlinear terms with including the uncertainties and fault components in the system, the fault-tolerant control with some adaptive laws are established using the stability theory of barrier Lyapunov function (BLF), where the approximation accuracies can be modified online using the provided adaptive laws with neglecting the number of fuzzy rules, the computation burden of the proposed control is greatly reduced because of a small number of updated laws. Finally, the simulation results are compared with those of several existing control methods, which confirms the validity of the designed control approach.

Multivariate Approximation Using Symmetrized and Perturbed Hyperbolic Tangent-Activated Multidimensional Convolution-Type Operators

In this article, we introduce, for the first time, multivariate symmetrized and perturbed hyperbolic tangent-activated convolution-type operators in three forms. We present their approximation properties, that is, their quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with the multivariate global smoothness preservation of these operators. We present, in detail, the related multivariate iterative approximation, as well as, multivariate simultaneous approximation, and their combinations. Using differentiability in our research, we produce higher rates of approximation, and multivariate simultaneous global smoothness preservation is also achieved.

Approximation Properties of Chlodovsky-Type Two-Dimensional Bernstein Operators Based on (p, q)-Integers

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the (p,q)-integer. By leveraging the inherent symmetry properties of (p,q)-integers, we examine the approximation properties of our new operator with the help of a Korovkin-type theorem. Further, we present the local approximation properties and establish the rates of convergence utilizing the modulus of continuity and the Lipschitz-type maximal function. Additionally, a Voronovskaja-type theorem is provided for these operators. We also investigate the weighted approximation properties and estimate the rate of convergence in the same space. Finally, illustrative graphics generated with Maple demonstrate the convergence rate of these operators to certain functions. The optimization of approximation speeds by these symmetric operators during system control provides significant improvements in stability and performance. Consequently, the control and modeling of dynamic systems become more efficient and effective through these symmetry-oriented innovative methods. These advancements in the fields of modeling fractional differential equations and control theory offer substantial benefits to both modeling and optimization processes, expanding the range of applications within these areas.

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

A two level approach for simulating Bose–Einstein condensates by Localized Orthogonal Decomposition

In this work, we consider the numerical computation of ground states and dynamics of single-component Bose–Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.

Mathematical and numerical analysis of reduced order interface conditions and augmented finite elements for mixed dimensional problems

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

Parameter-coupled state space models based on quasi-Gaussian fuzzy approximation

The accuracy of a fuzzy system’s approximation is closely tied to the performance of fuzzy control systems design, while this system’s interpretability depends on the description of a mechanical model using human language. This research introduces a quasi-Gaussian membership function characterized by a pair of parameters to achieve the sensitivity of a triangular membership function along with the interpretability of Gaussian membership functions. Consequently, a two-dimensional (2-D) quasi-Gaussian membership function is derived, and a method for establishing quasi-Gaussian fuzzy systems (QGFS) using a rectangular grid is proposed. After validating the approximation properties using the sine function for the one-dimensional (1-D) and 2-D QGFS, the systems are applied to approximate the depyrogenation tunnel, a significant piece of equipment in the pharmaceutical industry with various mechanical designs. Validation results indicate that the 1-D and 2-D QGFS can achieve an approximation error varying within a ± 5% range. Meanwhile, the 1-D and 2-D QGFSs are applied to mechanical models of the depyrogenation tunnel with satisfactory final approximation results. Lastly, the 2-D QGFS is capable of demonstrating an excellent description of models with coupled parameters.