Theory Of Spline Functions Research Articles

Optimal Vibration Fields in Problems of Modeling Dynamic States of Technical Objects

Introduction. Vibration interaction control is timely in production processes related to liquid and bulk media, systems of solids experiencing kinematic or force disturbances. At the same time, there is no single methodological basis for the formation of vibrational interactions. The issues of constructing optimal vibration fields of technical objects have not been addressed. The objective of the study is to develop a structural approach to the development of mathematical models in the problems of formation, evaluation, and correction of vibration fields of technical objects under conditions of intense force and kinematic loads. The task is to build vibration fields that are optimal in terms of the set of requirements, with the possibility of selecting the criterion of optimality of the vibration field of a technical object.Materials and Methods. A structural approach was used as the basic methodology. It was based on a comparison of mechanical vibratory systems used as computational schemes of technical objects, and structural schemes of automatic control systems, which are equivalent in dynamic terms. Lagrange formalism, elements of operational calculus based on Laplace integral transformations, sections of vibration theories, algebraic methods, and the theory of spline functions were used for structural mathematical modeling.Results. An approach to the selection of criteria for the optimality of vibration fields based on minimizing the residual of vibration fields for various required conditions was proposed. The problem was considered within the framework of a mechanical vibratory system formed by solids. It was shown that the optimal vibration field was determined by an external disturbance and was to satisfy condition Ay̅ = b. There, A — matrix mapping the operator of conditions to the shape of the vibration field at control points; b — vector of values of vibration field characteristics; “–” above y meant the vibration amplitude of the steady-state component of the coordinate. To evaluate the field with account for noisy or unreliable requirements for dynamic characteristics, the smoothing parameter was used, indicating the priority of the criterion of optimality of the vibration field shape. The construction of a field for a mechanical vibratory system showed that the value of the vibration amplitudes of generalized coordinates remained constant when the frequency of external kinematic disturbances changed. Two approaches to the correction of the field optimality criteria were considered: equalization of the vibration amplitudes of the coordinates of a technical object and the selection of an energy operator.Discussion and Conclusion. The development of the applied theory of optimal vibration fields involved, firstly, the correlation of the energy operator and the operator of the requirements for the shape of the vibration field in the theory of abstract splines. The second pair of comparable elements was the criterion of optimality of the vibration field and a system of requirements for the characteristics of the field at control points. The structural theory of optimal vibration fields improved in this way will find application in various industries. Accurate calculations in the formation, assessment, and correction of the states of systems under vibration loading are required in the tasks of increasing the durability of structures, improving measurements in complex vibratory systems, and developing new technologies and materials.

Improving offset detection algorithm of GNSS position time-series using spline function theory

SUMMARYIt is well known that unmodelled offsets in Global Navigation Satellite System (GNSS) position time-series can introduce biases into the station velocities. Although large offsets are usually reported or can be visually detected, automated offset detection algorithms require further investigation. This problem is still challenging as (small) geophysical offsets are usually covered by coloured noise and remain undetected. An offset detection algorithm has recently been proposed, which can detect and estimate offsets in both univariate and multivariate analyses. Although efficient in truly detecting offsets, this method still suffers from a high rate of detected fake offsets. To improve the offset detection performance, we attempt to stabilize the offset power spectrum to reduce the number of false detections. The spline function theory is adopted in the smoothness process of the power spectrum. The algorithm modified using the spline functions, referred to as As-mode, is compared with its original counterpart, called A-mode. The GNSS position time-series consisting of a linear trend, seasonal signals, offsets, and white plus coloured noise are simulated for the numerical comparison. The overall performance of the algorithm is significantly improved using the As-mode algorithm. The multivariate analysis shows that the truly detected offsets' percentage (true positive) increases from 52.9 per cent for A-mode to 61.1 per cent for As-mode. Further, the falsely detected offsets' percentage (false positive) is reduced from 40.6 per cent to 29.8 per cent. The algorithm was also tested on the DOGEx data set. The results indicate that the proposed method outperforms the existing solutions, with TP, FP and FN being 33.3 per cent, 32.3 per cent and 34.4 per cent, respectively. Also, in 90 per cent of the station, velocities are estimated at a 0.8 mm yr−1 distance from the simulated values.

Michael J. D. Powell. 29 July 1936—19 April 2015

Michael James David Powell was a British numerical analyst who was among the pioneers of computational mathematics. During a long and distinguished career, first at the Atomic Energy Research Establishment (AERE) Harwell and subsequently as the John Humphrey Plummer Professor of Applied Numerical Analysis in Cambridge, he contributed decisively towards establishing optimization theory as an effective tool of scientific enquiry, replete with highly effective methods and mathematical sophistication. He also made crucial contributions to approximation theory, in particular to the theory of spline functions and of radial basis functions. In a subject that roughly divides into practical designers of algorithms and theoreticians who seek to underpin algorithms with solid mathematical foundations, Mike Powell refused to follow this dichotomy. His achievements span the entire range from difficult and intricate convergence proofs to the design of algorithms and production of software. He was among the leaders of a subject area that is at the nexus of mathematical enquiry and applications throughout science and engineering.

The Application of Spline Functions in Wavelet Construction

On the basis of the spline function theory and the study of spline functions, the article discusses the refinement shift-invariant space of the functions in detail. The important quality and characteristics of the space are acquired to offer the methods for wavelet construction.

Tri-cubic polynomial natural spline interpolation for scattered data

In this paper an interpolation problem for 3D scattered data defined on a rectangular parallelepiped with natural boundary conditions is considered. By using spline function theory in Hilbert space...

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.

On A Generalization of the Dirichlet Integral

Abstract Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral where Ω is a bounded domain of an 𝑛-dimensional Euclidean space 𝑅𝑛, λ ≥ 0 is a fixed number, and 𝑢𝑥𝒊 is a generalized according to Sobolev with respect to 𝑥𝒊 derivative of the function 𝑢 defined on Ω. Minimization is realized with respect to the functions 𝑢 whose boundary values on Γ form a preassigned function, and for them 𝐹λ(𝑢) is finite.

On the estimation of smooth forward rate curves from a finite number of observations: A comment

The problem of estimating the forward rate curve from observed values of the yield-to-maturity is considered in the context of the theory of spline functions in Hilbert spaces and the solution is given in the interpolatory and smoothing case.

Generalized divided differences, with applications to generalized B-splines

Within the theory of spline functions, there has always been great interest in the study of B-splines, i.e. splines with a finite support. In addition to the classical polynomial case, there also exist various approaches to the definition of B-splines from other function classes, such as trigonometric or hyperbolic ones (cf. section 0). In the present paper we define B-splines from a rather general function space, with covers almost all existing approaches as special cases. The only condition that the spaces under consideration must satisfy is that of being translation invariant, a fundamental property which will be defined in the paper.Our definition of generalized B-splines is based on generalized divided differences, which go back to Popoviciu [14] and were further investigated by Mühlbach [10, 11]. In the first section of the paper, we therefore study these operators and prove new results, such as a contour integral representation and a multistep-formula; the latter expresses—in closed form—a generalized divided difference of order m+j using those of order m, for arbitrary j ∈ ℕ. This makes it possible to compute generalized divided differences recursively, even if the underlying function space is spanned by anon-complete Chebyshev system.

Interpolation by L-spline functions of many variables

The choice of function space allows us to make conclusions in the multidimensional case that are analogous to results in the theory of spline functions of one variable. We establish the minimum norm property, the existence and uniqueness of a solution of the interpolation problem, the property of best approximation, and the convergence of interpolation processes.

Hybrid generation of smooth functions by spline interpolation

The use of sparse data in hybrid computer simula tion has been provided with a formal procedure for generating smooth approximation functions with excellent reproduction quality. The method is use ful with data records over the independent variable or where a variable diode function generator might be used. The method depends on the theory of spline functions, which provides the specified quality of "smoothness" of the reconstructed function. An analog polynomial circuit receives parameters sequentially from the digital to generate the function. Besides illustrating an efficient method for hybrid function generation, the subject of spline approximation has been introduced to simulators in a simple language.