Optimal Interpolation Formula Research Articles

On the problem of optimal interpolation of functions

In this work, the problem of constructing optimal interpolation formulas is discussed. Here, first, an exact upper bound for the error of the interpolation formula in the Sobolev space is calculated. The existence and uniqueness of the optimal interpolation formula, which gives the smallest error, are proved. An algorithm for finding the coefficients of the optimal interpolation formula is given. By implementing this algorithm, the optimal coefficients are found.

Construction of optimal interpolation formula exact for trigonometric functions by Sobolev’s method

The paper is devoted to derivation of the optimal interpolation formula in W2(0,2)(0,1) Hilbert space by Sobolev’s method. Here the interpolation formula consists of a linear combination ΣNβ=0Cβφ(xβ) of the given values of a function φ from the space W2(0,2)(0,1). The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients Cβ(z) of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions sinx and cosx. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula. Работа посвящена построению оптимальной интерполяционной формулы методом Соболева в гильбертовом пространстве W2(0,2)(0,1). Здесь интерполяционная формула состоит из линейной комбинации ΣNβ=0Cβφ(xβ) заданных значений функции φ из пространство W2(0,2)(0,1). Отличие функций от интерполяционной формулы рассматривается как линейный функционал, называемый функционалом погрешности. Погрешность интерполяционной формулы оценивается нормой функционала погрешности. Мы получаем оптимальной интерполяционной формулы путем минимизации нормы функционала погрешности на коэффициенты Cβ(z) интерполяционной формулы. Полученная оптимальная интерполяция формула точна для тригонометрических функций sinx и cosx. В конце статьи мы приводим некоторые численные результаты, которые подтверждают наши теоретические результаты.

Optimal Interpolation Model for Synthetic Aperture Radar Wind Retrieval

The variational model inversion (VAR) method for synthetic aperture radar (SAR) wind retrieval based on Bayesian theory can overcome the limitation of the traditional wind streak algorithm by introducing background wind and considering all sources of error, but its optimal solution is unstable and the time latency is long. In this paper, we propose a new wind retrieval method by applying optimal interpolation (OI) theory to construct a formula that considers the SAR information, background information coming from the numerical prediction model, and their associated wellcharacterized errors. The retrieved wind vector can be acquired by the analytic solution of the OI formula. Experimental results of the simulation data and Sentinel-1 SAR data show that the OI wind retrieval method can effectively reduce the background wind error and is more sensitive to wind speed than wind direction. Compared with other methods, the accuracy of the OI method is similar to that of the VAR method, but significantly higher than that of the direct wind retrieval (DIRECT) method. The time latency of the OI method is the shortest, and the calculation efficiency is much higher than that of the VAR method. The OI method can be effectively applied to SAR wind retrieval and has unique advantages.Sea surface wind is a crucial parameter for studying the physical quantity of the sea surface and plays an important role in many fields such as weather forecasts [1, 2], wind energy resource management [3,4], wave numerical simulation [5,6], and oil spill monitoring [7,8]. Because of the limited spatial and temporal coverage, high-precision sea surface wind acquired from buoys, ships and offshore platforms are not meeting the growing demand [9].In recent decades, with the development of satellite remote sensing, the technology of acquiring sea surface wind using satellite sensor detection data has gradually matured and improved. Among various satellite sensors, microwave radiometers and scatterometers play an important role in providing global sea surface wind. However, microwave radiometers and scatterometers can acquire only low spatial resolution (12.5-50 km) sea surface wind. This relatively low spatial resolution is

Integration by table look-up for p-version finite elements on curved tetrahedra

We develop a method of evaluating element level matrices and vectors associated with p-version finite element computations on tetrahedral meshes with curved boundaries. The procedure uses optimal interpolation formula [Q. Chen, I. Babuška, Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Methods Appl. Mech. Engrg. 128 (1995) 405–417; Q. Chen, I. Babuška, Approximate optimal polynomial interpolation points in the tetrahedron, Comput. Methods Appl. Mech. Engrg. 137 (1997) 89–94] for non-polynomial portions of integrands followed by exact integration. Exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look-up.We present error-analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric-modelling system. Numerical results are presented for problems involving the Helmholtz equation.

Interpolation Operators on the Space of Holomorphic Functions on the Unit Circle

The aim of the paper is to get an estimation of the error of the general interpolation rule for functions which are real valued on the interval [—a, a],a ∈ (0,1), have a holomorphic extension on the unit circle and are quadratic integrable on the boundary of it. The obtained estimate does not depend on the derivatives of the function to be interpolated. The optimal interpolation formula with mutually different nodes is constructed and an error estimate as well as the rate of convergence are obtained. The general extremal problem with free weights and knots is solved.

Optimal interpolation of analytic functions

We construct an optimal interpolation formula for a particular class of analytic functions, optimization being over a set of interpolation methods which are not necessarily linear. Optimal nodes and the norm of the error are found for the optimal interpolation formula.