Spline Approximation Research Articles

A comparison between filtering approach and spline approximation method in smoothing flight data

A comparison between filtering approach and spline approximation method in smoothing flight data

Solving an Inverse Problem for Time-Series-Valued Computer Simulators via Multiple Contour Estimation

Computer simulators are often used as a substitute of complex real-life phenomena, which are either expensive or infeasible to experiment with. This paper focuses on how to efficiently solve the inverse problem for an expensive to evaluate time-series-valued computer simulator. The research is motivated by a hydrological simulator which has to be tuned for generating realistic rainfall–runoff measurements in Athens, Georgia, USA. Assuming that the simulator returns g(x, t) over L time points for a given input x, the proposed methodology begins with a careful construction of a discretization (time-) point set of size $$k \ll L$$ , achieved by adopting a regression spline approximation of the target response series at k optimal knot locations $$\{t_1^*, t_2^*,\ldots ,t_k^*\}$$ . Subsequently, we solve k scalar-valued inverse problems for simulator $$g(x,t_j^*)$$ via the contour estimation method. The proposed approach, named MSCE, also facilitates the uncertainty quantification of the inverse solution. Extensive simulation study is used to demonstrate the performance comparison of the proposed method with the popular competitors for several test-function-based computer simulators and a real-life rainfall–runoff measurement model.

Digital Event-Based Stabilization of Nonlinear Time-Delay Systems

In this paper, the stabilization problem of nonlinear time-delay systems via quantized sampled-data event-based controllers is investigated. Fully nonlinear (i.e., possibly non-affine in the control) systems affected by state delays are studied. Sufficient conditions are provided for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the continuous-time controller at hand, updated through a proposed event-triggered mechanism, ensures the semi-global practical stability property, with arbitrarily small final target ball of the origin, of the related closed-loop system. A spline approximation methodology is used in order to cope with the problem of the possible non-availability in the buffer of suitable past values of the system variables needed for the digital implementation of the controller. The stabilization in the sample-and-hold sense theory is used as a tool to prove the results. In the theory here developed, the case of non-uniform quantization of the input/output channels and the case of aperiodic sampling are both included. The proposed theoretical results are validated through an application concerning the plasma glucose regulation problem in type-2 diabetic patients.

THE STATISTICAL SIMULATION OF RANDOM FIELDS WITH THE GAUSSIAN TYPE CORRELATION FUNCTION BY THE INVESTIGATION OF THE MAGNETOMETRY DATA

In the article, universal methods of statistical modeling (Monte Carlo methods) of geophysical data using the Gaussian correlation function have been developed, which make it possible to solve the problems of generating adequate realizations of random fields on a grid in three-dimensional space of required regularity and detail. Since in geophysics, most of the results of object research are presented in digital form, the accuracy of which depends on various random influences, the problem of the condition of the maps arises in the case when the data cannot be obtained with the specified detail in some observation areas. It is proposed to apply statistical simulation of random fields methods, to solve the problems of conditional maps, supplement the required detail of research results with additional data, to achieve the required accuracy of observations, and other similar problems in geophysics. An algorithm for numerical modeling of realizations of homogeneous isotropic random fields in three-dimensional space with a Gaussian correlation function is formulated on the basis of the theorem on estimation of the mean-square approximation of such random fields by the partial sum of the "spectral decomposition" series. Using the example of data from aeromagnetic surveying in the area of the Ovruch depression, the proposed algorithm for statistical modeling of random fields is implemented in solving the problems of map fitness by supplementing the data with simulated adequate implementations to the required level of detail. When analyzing data by profiles, they are divided into deterministic (trend) and random components. The trend is proposed to approximate by cubic splines and the homogeneous isotropic random component is proposed to modeling on the basis "spectral decomposition" of random fields on 3-D space in the Ovruch depression. According to the algorithm, authors received random component implementations on the study area with twice detail for each profile. When checking their adequacy, authors made the conclusions that the relevant random components histogram has Gaussian distribution. The built variogram of these implementations has the best approximation by theoretical variogram which is connected to the Gaussian type correlation function. As a result of superimposing the simulated array of the random component on the spline approximation of the real data, a more detailed implementation was obtained for the data of geomagnetic observations in the selected area. A comparative analysis of the results of modeling realizations random fields with the Gaussian correlation function with other correlation functions is carried out. Therefore, the method of statistical modeling of realizations of random fields in three-dimensional space with the Gaussian correlation function makes it possible to supplement the results of measurements of the full magnetic field intensity vector with data with a given detail as much as possible.

Multilayer medium behavior simulation in the presence of delamination defects

The article is devoted to the study of the steady harmonic oscillation excitation processes by a surface source in a multilayer linearly elastic layered half-plane with an internal defect. The application of the work results is associated with the design and assessment of the multilayer structures state, such as geological structures, non-rigid pavements, layered composites. The study is based on the use of the boundary element method and the construction of boundary equations with respect to displacements at the defect boundary. Using the integral Fourier transform, fundamental solutions for a multilayer half-plane satisfying the stress-free conditions at the upper boundary of the domain are constructed. A feature of the construction is the good computational ability of the proposed algorithm with a large number of layers. The basic one is the construction of a solution for a homogeneous half-plane. This allows you to implement parallel computing. In the numerical implementation, a spline approximation of the defect boundary is used. Linear boundary elements are applied. As an example, a comparative analysis of the amplitude-frequency characteristics on the surface of the half-plane is given for cases of a limited defect, its absence and contact of layers without friction on the entire boundary. This allows us to draw conclusions about the possibility of identifying internal damage at the boundaries of layers by monitoring the wave field on the surface of a multilayer structure.

Iterative Bernstein splines technique applied to fractional order differential equations

<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

Solution of Integral Equations Using Local Splines of the Second Order

Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of integral equations of the second kind with a weak singularity. To construct the numerical schemes, local polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The results of the numerical experiments are given. This methodology has many applications in problems in Applied and Theoretical Mechanics

Methods of transient em data compression

A transient electromagnetic signal is recorded in an earth formation while drilling at multiple times. Data for subsequent interpretation is transmitted via a hydrochannel with a narrow bandwidth, for that reason the amount of transmitted information is limited. Therefore, data compression is an indispensable element of processing. It is important to determine what parameters and how many of them are necessary to describe the measured EMF curve. The idea is to transfer to the surface only significant parameters allowing to restore the recorded curve with a given accuracy. The paper proposes simple and fast data compression methods based on the spline approximation of the signal. Also, an orthogonal basis in the measurement space is used to describe each signal by a linear combination of vectors from this basis. We analyze transient curves and consider various magnetic field components. Numerical experiments show how many parameters are required to describe and reconstruct the signal of a deep-reading electromagnetic tool.

On the Control of Numerical Results in the Problems of Identification of Dynamic Energy Objects

The article proposes an approach to solving actual problem of the obtained results’ control in the construction and implementation of al-gorithms for identifying energy objects based on integral dynamic models. The considered method based on the use of quadrature algo-rithms and splines for approximation of the kernel, with the transition to solving equations with a degenerate kernel using recurrent formulas, showed sufficiently high efficiency in solving the problems of obtain-ing high performance at provided control over the accuracy of the inte-gral dynamic model parameters’ calculation, as well as providing re-sistance to the experimental data errors. The proposed method makes it possible to solve the problem of accumulating calculations, which, in turn, leads the numerical implementation algorithm to a form in which it is feasible to obtain solutions in real-time. The obtained integrated models have a sufficient level of adequacy and can be usedin integrat-ed computing control systems for energy objects

The Local Nonpolynomial Splines and Solution of Integro-Differential Equations

The application of the local polynomial splines to the solution of integro-differential equations was regarded in the author’s previous papers. In a recent paper, we introduced the application of the local nonpolynomial splines to the solution of integro-differential equations. These splines allow us to approximate functions with a presribed order of approximation. In this paper, we apply the splines to the solution of the integro-differential equations with a smooth kernel. Applying the trigonometric or exponential spline approximations of the fifth order of approximation, we obtain an approximate solution of the integro-differential equation at the set of nodes. The advantages of using such splines include the ability to determine not only the values of the desired function at the grid nodes, but also the first derivative at the grid nodes. The obtained values can be connected by lines using the splines. Thus, after interpolation, we can obtain the value of the solution at any point of the considered interval. Several numerical examples are given.

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

Geometrically designed variable knot splines in generalized (non-)linear models

In this paper we extend the GeDS methodology, recently developed by Kaishev et al. [18] for the Normal univariate spline regression case, to the more general GNM/GLM context. Our approach is to view the (non-)linear predictor as a spline with free knots which are estimated, along with the regression coefficients and the degree of the spline, using a two stage algorithm. In stage A, a linear (degree one) free-knot spline is fitted to the data applying iteratively re-weighted least squares. In stage B, a Schoenberg variation diminishing spline approximation to the fit from stage A is constructed, thus simultaneously producing spline fits of second, third and higher degrees. We demonstrate, based on a thorough numerical investigation that the nice properties of the Normal GeDS methodology carry over to its GNM extension and GeDS favourably compares with other existing spline methods.The proposed GeDS GNM/GLM methodology is extended to the multivariate case of more than one independent variable by utilizing tensor product splines and their related shape preserving variation diminishing property.

FOURIER ANALYSIS IN SPACES OF SHIFTS

In this paper, we develop a continual analog of decomposition over orthogonal bases in spaces generated by equidistant shifts of a single function. By doing so, we obtain an explicit expression for the best approximation by spaces of shifts in \(L_2(\mathbb {R})\). The result is formulated in terms of classical Fourier transform and tends to have various applications in approximation by spaces of shifts and, in particular, in spline approximation.

Shape preserving approximation for unbounded multivariate functions

The purpose of this research is to present a new method for creating polynomials and spline approximation to unbounded multivariate functions in L ρ , β ([–1, 1] I ) using ordinary modulus of smoothness, Sendov-Popov modulus and Ditzian-Totik modulus.

Spline quasi‐interpolation in the Bernstein basis and its application to digital elevation models

A nonstandard low‐cost spline approximation method for approximating bivariate functions is constructed. It is applied for Digital Elevation approximation and then its accuracy in the downscaling process is studied.

Spline approximation of multivalued functions in linear structures routing

Objectives. The theory and methods of spline approximation of plane curves given by a sequence of points are currently undergoing rapid development. Despite fundamental differences between used splines and those considered in the theory and its applications, results published earlier demonstrate the possibility of using spline approximation when designing routes of linear structures. The main difference here consists in the impossibility of assuming in advance the number of spline elements when designing the routes. Here, in contrast to widely use polynomial splines, the repeating element is the link “segment of a straight line + arc of a circle” or “segment of a straight line + arc of a clothoid + arc of a circle + arc of a clothoid.” Previously, a two-stage scheme consisting of a determination of the number of elements of the desired spline and subsequent optimization of its parameters was proposed. Although an algorithm for solving the problem in relation to the design of a longitudinal profile has been implemented and published, this is not suitable for designing a route plan, since, unlike a profile, a route plan is generally a multivalued function. The present paper aims to generalize the algorithm for the case of spline approximation of multivalued functions making allowance for the design features of the routes of linear structures.Methods. At the first stage, a novel mathematical model is developed to apply the dynamic programming method taking into account the constraints on the desired spline parameters. At the second stage, nonlinear programming is used. In this case, it is possible to analytically calculate the derivatives of the objective function with respect to the spline parameters in the absence of its analytical expression through these parameters.Results. An algorithm developed for approximating multivalued functions given by a discrete series of points using a spline consisting of arcs of circles conjugated by line segments for solving the first stage of the problem is presented. An additional nonlinear programming algorithm was also used to optimize the parameters of the resulting spline as an initial approximation. However, in the present paper, the first stage is considered only, since the complex algorithm of the second stage and its justification require separate consideration.Conclusions. The presented two-stage spline approximation scheme with an unknown number of spline elements is also suitable for approximating multivalued functions given by a sequence of points on a plane, in particular, for designing a route plan for linear structures.

High-Dimensional Statistics: Non-Parametric Generalized Functional Partially Linear Single-Index Model

We study the non-parametric estimation of partially linear generalized single-index functional models, where the systematic component of the model has a flexible functional semi-parametric form with a general link function. We suggest an efficient and practical approach to estimate (I) the single-index link function, (II) the single-index coefficients as well as (III) the non-parametric functional component of the model. The estimation procedure is developed by applying quasi-likelihood, polynomial splines and kernel smoothings. We then derive the asymptotic properties, with rates, of the estimators of each component of the model. Their asymptotic normality is also established. By making use of the splines approximation and the Fisher scoring algorithm, we show that our approach has numerical advantages in terms of the practical efficiency and the computational stability. A computational study on data is provided to illustrate the good practical behavior of our methodology.

Local Interpolation Splines and Solution of Integro-Differential Equations of Mechanic’s Problems

Integro-differential equations are encountered when solving various problems of mechanics. Although Integro-Differential equations are encountered frequently in mathematical analysis of mechanical problems, very few of these equations will ever give us analytic solutions in a closed form. So that construction of numerical methods is the only way to find the approximate solution. This paper discusses the calculation schemes for solving integro-differential equations using local polynomial spline approximations of the Lagrangian type of the fourth and fifth orders of approximation. The features of solving integro-differential equations with the first derivative and the Fredholm and Volterra integrals of the second kind are discussed. Using the proposed spline approximations, formulas for numerical differentiation are obtained. These formulas are used to approximate the first derivative of a function. The numerical experiments are presented.

High-degree splines from discrete Fourier transforms: Robust methods to obtain the boundary conditions

Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches.In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of low or high degree from equidistant interpolation nodes based on the discrete Fourier transform is analyzed. The accuracy of this method greatly depends on the accuracy of the boundary conditions. An algorithm for the computation of the boundary conditions can be found in Beaudoin (1998), and Beaudoin and Beauchemin (2003). However, this algorithm lacks robustness since the approximation of the boundary conditions is strongly dependant on the choice of θ arbitrary parameters, θ being the degree of the spline.The goal of this paper is therefore to propose two new robust algorithms, independent of arbitrary parameters, for the computation of the boundary conditions in order to obtain accurate splines of any degree. Numerical results will be presented to show the efficiency of these new approaches.

High‐fidelity tensor‐decomposition based matrix formation for isogeometric buckling analysis of laminated shells with solid‐shell formulation

AbstractThe computation cost of matrix formation in isogeometric analysis can be drastically reduced by employing tensor decomposition, but the performance like accuracy and robustness still depends on the approach used to realize the low‐rank approximation for the nonpolynomial integral kernels. In the buckling analysis of laminated shells with solid‐shell model, the integral kernels are not smooth in the thickness direction due to their material and stress distribution and have a high complexity because the curvilinear coordinate systems are involved in the representation of anisotropic constitutive relations, which makes it more difficult to guarantee the decomposition accuracy. This paper proposes a robust high‐fidelity tensor‐decomposition based matrix formation method for stiffness matrix and geometric stiffness matrix of the isogeometric buckling analysis problem. The proposed method avoids using a spline approximation and employs the hierarchical block‐wise decomposition approach (HBD) to obtain a determinate decomposition result, which saves more time without loss of accuracy and produces less canonical ranks with a reliable procedure. Besides, the matrix calculation is partly independent of the number of layers, showing a higher efficiency when the number of layers is larger. Finally, several experiments with various Gaussian curvatures are implemented to validate the proposed method.