Good Approximation Properties Research Articles

Numerical homogenization of spatial network models

We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.

Parameterized Domain Mapping for Order Tracking of Rotating Machinery

Because of the periodicity of vibration signals from the cyclic motion of rotating machinery, many spectrum-based methods are widely applied. However, frequency distortion often occurs under non-stationary conditions, which weakens the effectiveness of spectrum-based methods. Order tracking can overcome the frequency distortion problem, but the instantaneous angular speed is hard to detect, especially under strong noise or with close-spaced frequencies. In this paper, we map the signal into a new domain called the pseudo time domain to eliminate the non-stationarity based on the mechanism of rotating machinery. A parameterized domain mapping function (PDMF) is used to represent the relationship between the time and pseudo time domains. Instead of conducting order tracking after detecting the instantaneous angular speed, we directly optimize the parameter set of the PDMF to construct an appropriate pseudo time domain. A new spectrum concentration indicator considering the continuity of the spectrum is built as the optimization objective. The Legendre polynomial, which shows good approximation property, is applied as the general kernel function of PDMF. Numerical and experimental verification shows good anti-noise performance and the ability to deal with complex and close-spaced frequencies under speed variation or speed fluctuation conditions.

On shift selection for Krylov subspace based model order reduction

Mechanical systems are often modeled with the multibody system method or the finite element method and numerically described with systems of differential equations. Increasing demands on detail and the resulting high complexity of these systems make the use of model order reduction inevitable. Frequently, moment matching based on Krylov subspaces is used for the reduction. There, the transfer functions of the full system and of the reduced system are matched at distinct frequency shifts. The selection of these shifts, however, is not trivial. In this contribution we suggest an algorithm that evaluates an increasing number of shifts iteratively until a reduced model that approximates the full model in a subspace with very low approximation error is found. Thereafter, the projection matrix that spans this subspace is decomposed with singular value decomposition and only most important directions are retained. In this way, small reduced models with good approximation properties that do not exceed a predefined error bound can be found or low-error models for a given reduced order can be generated. The evaluation of more shifts than necessary and further reduction by means of singular value decomposition is the novelty of this contribution. In this paper, this novel approach is extensively studied and, furthermore, applied to the numerical example of an industrial helicopter model.

Approximation Properties of the Blending-Type Bernstein–Durrmeyer Operators

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.

Mini-Workshop: Mathematics of Dissipation – Dynamics, Data and Control

Dissipation of energy — as well as its sibling the increase of entropy — are fundamental facts inherent to any physical system. The concept of dissipativity has been extended to a more general system theoretic setting via port-Hamiltonian systems and this framework is a driver of innovations in many of areas of science and technology. The particular strength of the approach lies in the modularity of modeling, the strong geometric, analytic and algebraic properties and the very good approximation properties.

Mean curvature motion of point cloud varifolds

This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate tangent plane for the points in the cloud. The core ingredient of the mean curvature motion model is the regularization of the first variation of the varifoldviaconvolution with kernels with small stencil. Consistency with the evolution velocity for a smooth surface is proven provided that a sufficiently small stencil and a regular sampling are considered. Furthermore, an implicit and a semi-implicit time discretization are derived. The implicit scheme comes with discrete barrier properties known for the smooth, continuous evolution, whereas the semi-implicit still ensures in all our numerical experiments very good approximation properties while being easy to implement. It is shown that the proposed method is robust with respect to noise and recovers the evolution of smooth curves as well as the formation of singularities such as triple points in 2D or minimal cones in 3D.

Adaptive Neural Network Global Nonsingular Fast Terminal Sliding Mode Control for a Real Time Ground Simulation of Aerodynamic Heating Produced by Hypersonic Vehicles

This paper presents a strategy for a thermal-structural test with quartz lamp heaters (TSTQLH), combined with an ultra-local model, a closed-loop controller, a linear extended state observer (LESO), and an auxiliary controller. The TSTQLH is a real time ground simulation of aerodynamic heating for hypersonic vehicles to optimize their thermal protection systems (TPS). However, lack of a system dynamic model for the TSTQLH results in inaccurate tracking of aerodynamic heating. In addition, during the control process, the TSTQLH has internal uncertainties of resistance and external disturbances. Therefore, it is necessary to establish a mathematical model between controllable α(t) and measurable T1(t). An ultra-local model of model-free control plays a crucial role in simplifying system complexity and reducing high-order terms due to high nonlinearities and strong couplings in the system dynamic model, and a global nonsingular fast terminal sliding mode control (GNFTSMC) is added to an ultra-local model, which is used to guarantee great tracking performance in the sliding phase and fast convergence to the equilibrium state in finite time. Moreover, the LESO is used mainly to estimate all disturbances in real time, and an adaptive neural network (ANN) shows a good approximation property in compensation for estimation errors by using a cubic B-spline function. The fitted curve of the wall temperature in the time sequence represents a reference temperature trajectory from the surface contour of an X-43A’s wing. The comparative results validate that the proposed control strategy possesses strong robustness to track the reference temperature trajectory.

Weighted isogeometric collocation based on Spline Projectors

In this paper, we propose a novel version of weighted isogeometric collocation that is especially suited for adaptive THB-spline refinement.It is well known that the choice of the collocation nodes is crucial to ensure stability and good approximation properties, especially when adaptive refinement is performed. In order to address this issue, we make use of a particular class of locally supported quasi-interpolant schemes to propose the new method of Weighted Isogeometric Collocation based on Spline Projectors (WICSP).We show that WICSP performs well in the case of tensor-product spline discretizations, both with respect to the rate of convergence and computational complexity. In particular, we observe experimentally an optimal rate of convergence for odd degree basis functions and obtain a dimension-independent computational complexity O(np) for matrix-free applications, similar to other approaches such as weighted quadrature (Calabrò et al., 2017) and clustered collocation (Montardini et al., 2017).We explore how these results extend to the case of adaptively refined THB-spline discretizations. We observe that WICSP is compatible with THB-spline refinement, exhibiting good accuracy and low computational costs. In fact, we get a complexity of O(np2d) for matrix assembly and O(npd) for the matrix-free case. This compares well with the available methods for isogeometric THB-spline discretizations.

Analysis of non-local multicontinuum upscaling for dual continuum model

In this paper, we develop and analyze a rigorous multiscale upscaling method for dual continuum model, which serves as a powerful tool in subsurface formation applications. Our proposed method is capable of identifying different continua and capturing non-local transfer and effective properties in the computational domain via constructing localized multiscale basis functions. The construction of the basis functions consists of solving local problems defined on oversampling computational region, subject to the energy minimizing constraints that the mean values of the local solution are zero in all continua except for the one targeted. The basis functions constructed are shown to have good approximation properties. It is shown that the method has a coarse mesh dependent convergence. We present some numerical examples to illustrate the performance of the proposed method.

Jackson-type theorem on approximation by non-stationary periodic wavelets

In this paper, sufficient conditions on non-stationary periodic wavelet systems to provide good approximation properties of wavelet expansions are established. The approximation error is estimated in terms of modulus of continuity in [Formula: see text]-spaces.

Applications of Optimal Spline Approximations for the Solution of Nonlinear Time-Fractional Initial Value Problems

Nonlinear fractional differential equations are widely used to model real-life phenomena. For this reason, there is a need for efficient numerical methods to solve such problems. In this respect, collocation methods are particularly attractive for their ability to deal with the nonlocal behavior of the fractional derivative. Among the variety of collocation methods, methods based on spline approximations are preferable since the approximations can be represented by local bases, thereby reducing the computational load. In this paper, we use a collocation method based on spline quasi-interpolant operators to solve nonlinear time-fractional initial value problems. The numerical tests we performed show that the method has good approximation properties.

Adsorption and the Chemical Reaction N2O4 ↔ 2NO2 in the Presence of N2 in a Gas Phase Connected with a Carbon Nanotube.

The paper shows, by molecular simulations, that a CNT (9,9) carbon nanotube allows very efficient separation of nitrogen oxides (NOx) from N2, that has in good approximation properties of the complete air mixture. Gibbs ensemble Monte Carlo simulations are used to describe the adsorption. The permanent chemical reaction between N2O4 and NO2, which occurs simultaneously to adsorption, is treated by the reactive Monte Carlo simulation. A very high selectivity has been found. For a low pressure and at T = 298 K, an adsorption/reaction selectivity between NOx and N2 can reach values up to 3 × 103.

RI-IGABEM based on generalized-[formula omitted] method in 2D and 3D elastodynamic problems

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on the generalized-α method is proposed to solve 2D and 3D elastodynamic problems of homogeneous and inhomogeneous materials. First of all, the elastostatics Kelvin fundamental solution is used as the fundamental solution of the problem. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral caused by the material heterogeneity and the inertia term into an equivalent boundary integral by means of applied points. In addition, using a simple transformation method, the rigid-body technique is applied to solve the strongly singular integrals, and the Telles scheme and the power series expansion method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, the generalized-α method is adopted to solve the time domain problem, which can improve the stability of numerical results by effectively filtering out the false response of high frequency and minimizing the attenuation of low frequency response. A number of 2D and 3D examples, such as those with homogeneous materials, functionally gradient materials, and material defects and inclusions, are used to demonstrate the ability of the scheme to simulate the elastodynamic problems.

Аппроксимация продолжительности жизни в России на основе интервальных данных и минимаксной модели

Article provides automated processing of interval data on dynamics life expectancy in Russians over a long period of time (more than 100 years). Software product for approximation of dynamic processes in the demographic sphere that occur under unstable trend conditions is developed. Software product is based at the author's model of spline approximation the interval data by linear polynomials. The algorithm of program uses method for finding the moments of joining linear splines using reasonable properties of solving minimax problem for dynamic series whose values are interval data: the lower limit of the interval is fixed life expectancy of men, the upper limit is fixed life expectancy of women. As result of computational experiments, the jump-type curve approximation obtained. The jump-type curve preserves important properties of modeled series and has good approximation properties. The special feature of developed software product is the ability to get an approximating function in a matter of seconds, which has high accuracy and allows you to analyze profile of demographic situation in Russia, taking into account life expectancy of men and women over a long period. Were identified key turning points (mid 20th century and beginning of 21st century, when there was decline in life expectancy of men and his essential distance from a more stable indicator for women), after which dynamics of indicators have shown an increasing trend of both indices and reducing the gap between them.

RI-IGABEM in inhomogeneous heat conduction problems

The isogeometric analysis boundary element method (IGABEM) has great potential in the simulation of heat conduction problems due to its exact geometric representation and good approximation properties. In this paper, the radial integration IGABEM (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source. Similar to traditional BEM, the domain integrals cannot be avoided since the foundational solution for the Laplace equation is used to derive integral equation. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. In order to validate the accuracy and convergence of the RI-IGABEM in the analysis of the single or multiple boundary heat conduction problems, several 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number of applied points, the order of basis functions, and the position of internal applied points.

RI-IGABEM based on PIM in transient heat conduction problems of FGMs

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on precise integration method (PIM) is proposed to solve transient heat conduction problems of functionally gradient materials (FGMs) with heat source Similar to traditional boundary element method (BEM), in which the fundamental solution for the Laplace equation is used to derive the boundary-domain integral equations, the domain integrals caused by the heat source and the thermal conductivity varying with coordinates and the initial temperature appear in the transient heat conduction problems. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, in order to improve the stability of numerical results, the PIM is adopted to solve the time domain problem. In order to validate the accuracy and convergence of the RI-IGABEM 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number and the position of applied points, the order of basis functions, the number of model refinements and the length of time step.

On the Rates of Asymptotic Normality for Bernstein Polynomial Estimators in a Triangular Array

It is well known that the empirical distribution function has superior properties as an estimator of the underlying distribution function F. However, considering its jump discontinuities, the estimator is limited when F is continuous. Mixtures of the binomial probabilities relying on Bernstein polynomials lead to good approximation properties for the resulting estimator of F. In this paper, we establish the rates of (pointwise) asymptotic normality for Bernstein estimators by the Berry-Esseen Theorem in the case that the observations are in a triangular array. Particularly, the (asymptotic) absence of the boundary bias and the asymptotic behaviors of the variance are investigated. Besides, numerical simulations are presented to verify the validity of our main results.

Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches—Picard’s and Newton’s methods—are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.

Spline moment models for the one-dimensional Boltzmann–Bhatnagar–Gross–Krook equation

We introduce Spline Moment Equations (SMEs) for kinetic equations using a new weighted spline ansatz of the distribution function and investigate the ansatz, the model, and its performance by simulating the one-dimensional Boltzmann–Bhatnagar–Gross–Krook equation. The new basis is composed of weighted constrained splines for the approximation of distribution functions that preserves mass, momentum, and energy. This basis is then used to derive moment equations using a Galerkin approach for a shifted and scaled Boltzmann–Bhatnagar–Gross–Krook equation, to allow for an accurate and efficient discretization in velocity space with an adaptive grid. The equations are given in a compact analytical form, and we show that the hyperbolicity properties are similar to the well-known Grad moment model. The model is investigated numerically using the shock tube, the symmetric two-beam test, and a stationary shock structure test case. All tests reveal the good approximation properties of the new SME model when the parameters of the spline basis functions are chosen properly. The new SME model outperforms existing moment models and results in a smaller error while using a small number of variables for efficient computations.

A Robust Moving Total Least-Squares Fitting Method for Measurement Data

The moving least-squares (MLS) and moving total least-squares (MTLS) methods have been widely used for fitting measurement data. They can be used to achieve good approximation properties. However, these two methods are susceptible to outliers due to the way of determining local approximate coefficients, which leads to distorted estimation. To reduce the influence of outliers and random errors of all variables without adding small weights or setting the threshold subjectively, we present a robust MTLS (RMTLS) method, in which an improved least trimmed squares (ILTS) method is used for obtaining the local approximants of the influence domain. The ILTS method divides the nodes in the influence domain into a certain number of subsamples, achieves the local approximants by the total least-squares (TLS) method with compact support weight function, and trims the node with the largest orthogonal residual from each subsample, respectively. The remaining nodes from the subsamples are used to determine the local coefficients. The measurement experiment and numerical simulations are provided to demonstrate the robustness and accuracy of the presented method in comparison with the MLS and MTLS methods.