Spline Approximation Research Articles

The Hermite-Birkhoff Problem and Local Spline Approximation

This paper discusses the use of local spline approximations to solve the Hermite-Birkhoff problem. The solution to a specific problem using polynomial and non-polynomial local splines of the third order of approximation is considered. Here we discuss the case when the values of the function 𝑢(𝑥) and its derivative 𝑢’(𝑥) are given at the nodes of the grid in an alternative way: … , 𝑢(𝑥𝑗), 𝑢′(𝑥𝑗+1), 𝑢(𝑥𝑗+2), … . Note that when using polynomial and non-polynomial spline approximations, it is possible to obtain acceptable solutions in several interesting cases that are impossible when we use the classical approach. In the case of using local basis splines, many previously unsolvable problems turn out to be solvable. The results of the numerical experiments are presented.

Intractability results for integration in tensor product spaces

We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error, i.e., the error for N=0, where ε belongs to (0,1). We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality.Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Local spline refinement driven by fault jump estimates for scattered data approximation

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

Reliability-based G1 continuous arc spline approximation

This paper introduces an algorithm for approximating a set of data points with G1 continuous arcs, leveraging covariance data associated with the points. Prior approaches to arc spline approximation typically assumed equal contribution from all data points, resulting in potential algorithmic instability when outliers are present. To address this challenge, we propose a robust method for arc spline approximation, taking into account the 2D covariance of each data point. Beginning with the definition of models and parameters for single-arc approximation, we extend the framework to support multiple-arc approximation for broader applicability. Finally, we validate the proposed algorithm using both synthetic noisy data and real-world data collected through vehicle experiments conducted in Sejong City, South Korea.

A Shape-Preserving Variational Spline Approximation Problem for Hole Filling in Generalized Offset Surfaces

In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces.

Second-order water wave properties over Roseau’s three-parameter bottom topography

Summary In his book Asymptotic Wave Theory (North-Holland 1976), Roseau develops the only known exact solution for linear wave propagation over a shoal. Remarkably, almost half a century later, the only properties of that solution which appear to have been used to validate more wide-ranging models are the limiting values of wave height in the far field. The primary aim of this work is to remedy this situation by providing a robust computational setting where the solution may be fully evaluated randomly both on the surface and the interior, speedily and to a high degree of accuracy. Such ‘benchmark’ solutions will facilitate a much more rigorous approach to validation of more general alternative approximation models. One such has some of its numerical results here examined and subsequently largely vindicated by the present calculations. Computation of the absolutely convergent solution integrals is supported by cubic spline approximation of an integrand component which satisfies a certain difference equation and the work is extended to cover bottom profiles with overhang including the extreme case where the overhang degenerates to a semi-infinite flat plate. The further development of solutions at second-order reveals details of mean Eulerian current and the attendant Stokes drift enabling mass transport computations; all illustrated through the entire depth with a range of contouring and arrows plots. The force field that drives a second harmonic component is calculated and shows excellent agreement with its equivalent in the more general modal expansion model applied to a non-identical but somewhat similar profile. This further validates the modelling approach and application described in that work.

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

SPLINE OPTIMIZATION OF SOFT CONNECTIVES IN MACHINE LEARNING MODELS

In this study, the problem of limited accuracy of machine learning models using soft logical connectives is investigated. Such connectives have shown their effectiveness in models with fuzzy initial data. On the one hand, the fundamental disadvantage of soft connectives is their non-associativity. On the other hand, the disadvantages of the currently used soft connectives include the loss of monotonicity and the inability to control several factors simultaneously. All these problems have been solved by the authors. We have proposed an approximation of the signum function by a ?1-smooth spline. The first part of the spline is responsible for the curvature of the connective at the diagonal and is adjustable. The second part of the spline is the solution to the optimization problem. We minimized the difference between the connective and the associative connective, the latter in our study was the minimum function. In the resulting solution, the rate of deviation reduction is the highest among known connectives. We have achieved not only a small deviation from associativity, but also the presence of a large domain of exact associativity. This area is up to a third of the volume of all triples of arguments. A comparative analysis of the currently used soft connectives with the constructed model was carried out. It was shown that the spline approximation is able to reduce the influence of all negative factors and is more flexible in setting. Moreover, the constructed spline model allows numerous modifications depending on the factor that requires the most attention for different tasks.

Single-Index Mixed-Effects Model for Asymmetric Bivariate Clustered Data

Studies/trials assessing status and progression of periodontal disease (PD) usually focus on quantifying the relationship between the clustered (tooth within subjects) bivariate endpoints, such as probed pocket depth (PPD), and clinical attachment level (CAL) with the covariates. Although assumptions of multivariate normality can be invoked for the random terms (random effects and errors) under a linear mixed model (LMM) framework, violations of those assumptions may lead to imprecise inference. Furthermore, the response-covariate relationship may not be linear, as assumed under a LMM fit, and the regression estimates obtained therein do not provide an overall summary of the risk of PD, as obtained from the covariates. Motivated by a PD study on Gullah-speaking African-American Type-2 diabetics, we cast the asymmetric clustered bivariate (PPD and CAL) responses into a non-linear mixed model framework, where both random terms follow the multivariate asymmetric Laplace distribution (ALD). In order to provide a one-number risk summary, the possible non-linearity in the relationship is modeled via a single-index model, powered by polynomial spline approximations for index functions, and the normal mixture expression for ALD. To proceed with a maximum-likelihood inferential setup, we devise an elegant EM-type algorithm. Moreover, the large sample theoretical properties are established under some mild conditions. Simulation studies using synthetic data generated under a variety of scenarios were used to study the finite-sample properties of our estimators, and demonstrate that our proposed model and estimation algorithm can efficiently handle asymmetric, heavy-tailed data, with outliers. Finally, we illustrate our proposed methodology via application to the motivating PD study.

Map-aiding of the additional secondary factors isosurface by the spline approximation method as a condition of improving the e-Loran observations accuracy

The benefit of the Loran ground-based radio navigation system as an alternative to global navigation satellite systems is argued. The motivation of searching for a duplicate variant is caused by the necessity to ensure the reliability of positioning as a subject of countering spoofing and jamming risks when implementing the concept of cybernetic awareness on water transport. Hypothetically, the possibility of using e-Loran on the Northern Sea Route as a backup navigation system is considered. The experience of operating pilot projects of the enhanced Loran in the issue of the predicted potential of positioning accuracy is analyzed. The optimal method of compensation for the system error of geolocation based on the use of additional secondary factors is investigated. Taking into account the principle of the d-Loran functional, based on comparing the measured values of the radio signal passage delay with published analogues for transmitting clarifications to marine consumers, the necessity of using integrity as a criterion for a confidence assessment of the navigation information processing reliability has been determined. A critical analysis of the attempts effectiveness to use the linear interpolation method to calculate intermediate representative values of the delay in the radio signal transmission in order to simulate the map of additional secondary factors is performed. The hypothesis of the prospects for the isosurface corrections synthesis based on B-spline approximation is put forward. The South Korean experiment of synthesizing a chart of additional secondary factors with measured indicators of the radio signal propagation time delay in nanoseconds from the Pohang transmitting station has been repeated at the correct algorithmic level. In order to demonstrate the practical feasibility of the spline algorithm, a computer visualization of a cartographic fragment of additional secondary factors of Yongil Bay is performed. It is suggested that three-dimensional representation of the additional secondary factor for situational perception by the Officer on watch of map-aided correction as a process in alternative positioning in order to increase the reliability of location control through visual evaluation of proper use of corrections of the e-Loran differential variant. The prospect of using the developed package of applied Pascal-programs with implementations on display-type monitors as intellectual support for decision-making by the navigator in a posteriori assessment of the observation accuracy due to the visual representation of the correction field is noted.

A new approach for fast field calculation in electrostatic electron lens design and optimization

In electron optics, calculation of the electric field plays a major role in all computations and simulations. Accurate field calculation methods such as the finite element method (FEM), boundary element method and finite difference method, have been used for years. However, such methods are computationally very expensive and make the computer simulation challenging or even infeasible when trying to apply automated design of electrostatic lens systems with many free parameters. Hence, for years, electron optics scientists have been searching for a fast and accurate method of field calculation to tackle the aforementioned problem in the design and optimization of electrostatic electron lens systems. This paper presents a novel method for fast electric field calculation in electrostatic electron lens systems with reasonably high accuracy to enable the electron-optical designers to design and optimize an electrostatic lens system with many free parameters in a reasonably short time. The essence of the method is to express the off-axis potential in an axially symmetrical coordinate system in terms of derivatives of the axial potential up to the fourth order, and equate this to the potential of the electrode at that axial position. Doing this for a limited number of axial positions, we get a set of equations that can be solved to obtain the axial potential, necessary for calculating the lens properties. We name this method the fourth-order electrode method because we take the axial derivatives up to the fourth order. To solve the equations, a quintic spline approximation of the axial potential is calculated by solving three sets of linear equations simultaneously. The sets of equations are extracted from the Laplace equation and the fundamental equations that describe a quintic spline. The accuracy and speed of this method is compared with other field calculation methods, such as the FEM and second order electrode method (SOEM). The new field calculation method is implemented in design/optimization of electrostatic lens systems by using a genetic algorithm based optimization program for electrostatic lens systems developed by the authors. The effectiveness of this new field calculation method in optimizing optical parameters of electrostatic lens systems is compared with FEM and SOEM and the results are presented. It should be noted that the formulation is derived for general axis symmetrical electrostatic electron lens systems, however the examples shown in this paper are with cylindrical electrodes due to the simplicity of the implementation in the software.

The problem of implementing the concept of maximum integration of heterogeneous data in practical navigation

The topic of the work is the research of the problem of the practical realization of opportunistic navigation, which actually consists in the fact that the “signals of opportunity” are not optimized for global navigation according to their original purpose. A hypothesis about the possibility of strategic application of the navigation concept based on the principles of data integration with the objective existence of tactically surmountable difficulties is put forward. In the conditions of vulnerability of the global satellite system, an approach to use an additional source of combined signals in order to continuously ensure accurate geolocation of water transport vehicles is proposed. The comparative assessment confirms the comparability of the positioning error of alternative navigation with the data of classical satellite observations of mobile objects. An assumption about the effectiveness of spline modeling of the cartographic standard of informativeness for the practical organization of the basic principle of opportunistic positioning by analogy with the paradigm of correlation-extreme navigation is made. A computer visualization of a fixed fragment of the ocean gravitational field based on spline approximation is performed in order to optimize the problem of simultaneous localization and three-dimensional representation of the ship movement in a graphical environment unaffiliated with foreign software. A hybrid algorithm for mathematical synthesis of the navigational isosurface has been tested as a harmonized support for the navigation staff to implement the “terrain-referenced procedure” in the landscape coordinate space. The variants of the “ambiguity resolution” problem implementation in the issue of maximum integration of heterogeneous navigation data are considered. The probability of practical use of the opportunistic navigation postulates in marine applications, taking into account the “navigation gap” factor for leveling local satellite incapacity with a real possibility of countering malicious interception of ship control, is estimated. The expediency of using the factor of observational reckoning of the vessel path as an additional condition for navigation safety in the aspect of packet processing of the “signals of opportunity” slot is substantiated.

The use of complex structure splines in roadway design

Objectives. The aim of the work is to develop the theory of spline-approximation of a sequence of points on a plane for using compound splines with a complex structure. In contrast to a simple spline (e.g., polynomial), a compound spline contains repeating bundles of several elements. Such problems typically arise in the design of traces for railroads and highways. The plan (projection on the horizontal plane) of such a trace is a curve consisting of a repeating bundle of elements “line + clothoid + circle + clothoid ...,” which ensures continuity not only of curve and tangent but also of curvature. The number of spline elements, which is unknown, should be determined in the process of solving the design problem. An algorithm for solving the problem with respect to the spline, which consists of arcs conjugated by straight lines, was implemented and published in an earlier work. The approximating spline in the general case is a multivalued function, whose ordinates may be limited. Another significant factor that complicates the problem is the presence of clothoids that are not expressed analytically (in a formula). The algorithm for determining the number of elements of a spline with clothoids and constructing an initial approximation was also published earlier. The present work considers the next stage of solving the spline approximation problem: optimization using a nonlinear programming spline obtained at the first stage by means of the dynamic programming method.Methods. A new mathematical model in the form of a modified Lagrange function is used together with a special nonlinear programming algorithm to optimize spline parameters. In this case, it is possible to calculate the derivatives of the objective function by the spline parameters in the absence of its analytical expression through these parameters.Results. A mathematical model and algorithm for optimization of compound spline parameters comprising arcs of circles conjugated by clothoids and lines have been developed.Conclusions. The previously proposed two-step scheme for designing paths of linear structures is also suitable for the utilization of compound splines with clothoids.

An entropy-based approach for a robust least squares spline approximation

We consider the weighted least squares spline approximation of a noisy dataset. By interpreting the weights as a probability distribution, we maximize the associated entropy subject to the constraint that the mean squared error is prescribed to a desired (small) value. Acting on this error yields a robust regression method that automatically detects and removes outliers from the data during the fitting procedure, by assigning them a very small weight. We discuss the use of both spline functions and spline curves. A number of numerical illustrations have been included to disclose the potentialities of the maximal-entropy approach in different application fields.

A Fast and Simple Machine Vision Framework for Approximating the Volume of Axi-Symmetric Objects Using Shadow Ray Casting

The volume measurement using machine vision system is contactless techniques that play an important role in industries now a day. Basically, three-dimensional reconstruction is required to determine a depth using a special lighting system or multiple cameras. This increases the complexity of the measurement system. A fast and simple machine vision framework called RayVol for estimating the volume of axisymmetric objects in near real-time using a single camera and simple illumination is presented. The RayVol framework employs a shadow casting method to reconstruct the 3D shape of the object by tracing rays from the object’s shadow pixels to the light source location. The result of this technique shows a significant accuracy improvement from the area-projection method. A virtual slice representing the cross-section of an object is reconstructed using a cubic spline approximation from baseline points derived from the boundary pixels of the object image and a shadow casting method. The volume estimation was calculated by restricted integration using the Riemann sum estimation algorithm, and the closed area of the virtual slices was calculated using the shoestring algorithm. Mangoes were used as a case study of the RayVol framework. The volume estimation provides the correlation coefficient of 0.9849 between the developed system and the water replacement method.

Cubic Polishing Spline-Based Algorithms for Industrial Image Processing

Industrial image processing is a major technology in the fourth industrial revolution. In recent years, the spline method has gained more and more attention in image processing. To explore an efficient reconstruction algorithm, this paper proposes novel cubic polishing spline image processing algorithms. The cubic polishing spline is a high precision spline algorithm that brings more degrees of freedom by adding nodes to better balance the local and global aspects. First, the direct and indirect cubic polishing spline transforms are used for the image interpolation filter. Then, we construct the least squares cubic polishing spline and pyramid approximation method with application to the image reconstruction and image scaling (image upscaling or downscaling). Next, the frequency domain cubic polishing spline lowpass filter and highpass filter are proposed, and the proposed schemes are applied to image smoothing and image sharpening respectively. Finally, the PSNR and SSIM and RMSE evaluation metrics are used to illustrate the reconstruction effects, and experiments show that the algorithm proposed in this paper can achieve better reconstruction results with better fidelity and competitiveness compared with other schemes.

Mathematical approximator based on basic spline approximation

The article considers the problem of constructing a mathematical piecewise linear approximator based on approximation by basis splines. An algorithm has been developed designed to implement a class of special functions and create parallel microprogramming for the algorithm for calculating spline functions and, on their basis, parallel operation of a microprogram automaton with associative memory.

ВИЗНАЧЕННЯ КУСКОВО-ЛІНІЙНОГО ТРЕНДУ НЕСТАЦІОНАРНОГО ЧАСОВОГО РЯДУ НА ОСНОВІ ІНТЕЛЕКТУАЛЬНОГО АНАЛІЗУ ДАНИХ. II. МАШИННI ЕКСПЕРИМЕНТИ ТА РОЗВ’ЯЗАННЯ ПРАКТИЧНОЇ ЗАДАЧІ

The article describes the results of the approbation of the method of constructing a piecewise-linear trend, which can have breaks at the switching points as well as be continuous at these points, i.e., represent a linear flow. An example of applying the method for constructing a linear regression with switches, which has two independent variables with a trend, is considered. The problem of spline approximation of the time series of logarithms of the number of infected people with COVID-19 in Ukraine is stated and solved. Keywords: trend, regression, switch point, spline, real-time calculation.

On the robustification of digital event-based stabilizers for nonlinear time-delay systems

In this paper, the robust stabilization problem by means of quantized sampled-data event-based (QSE) controllers is investigated for nonlinear systems affected by state delays and unknown disturbances. In particular, a methodology for the design of robust QSE stabilizers is provided for control-affine nonlinear systems affected by unknown actuation disturbances and unknown measurement errors. Firstly, the notion of Steepest Descent Feedback (SDF), continuous or not, is suitably revised in order to deal with the robustification of event-based controllers. Then, Input-to-State Stability (ISS) redesign methodologies are used to provide the robustification term which is added to the SDF at hand in order to arbitrarily attenuate the effects of unknown external disturbances affecting the considered control scheme. A spline approximation approach is used in order to cope with the problem of the possible non-availability in the buffer of suitable past values of the system state required for the correct application of the proposed robust QSE controller. It is proved that there exist a suitably fast sampling and an accurate quantization of the input/output channels such that: the robust QSE implementation of SDFs, continuous or not, ensures the semi-global practical stability of the related closed-loop system, regardless of the above disturbances, provided that the observation errors affects marginally the new added control term. The stabilization in the sample-and-hold sense theory is used as a tool to prove the results. The provided results include the case of non-uniform quantization of the input/output channels and the case of aperiodic sampling. Applications are presented in order to validate the results.