We present an algorithm to generate a smooth curve interpolating a set of data on an n-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in \cite{fatima-knut-rolling} for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a curve. This would allow to project data from the ellipsoid to a space where interpolation problems can be easily solved. However, even if one chooses to roll along a geodesic, the fact that explicit forms for Euclidean geodesics on the ellipsoid are not known, would be a major obstacle to implement the rolling part of the algorithm. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space in \Rn+1 equipped with an appropriate Riemannian metric, so that geodesics are given in explicit form and, consequently, the kinematics of the rolling motion are easy to solve. By doing so, we can rewrite the algorithm to generate a smooth interpolating curve on the ellipsoid which is given in closed form.
The main objective of this paper is to explain how the application of various interpolation methods influence the determination of vertical crustal movements at any given point. The paper compares several methods of interpolation and verifies their suitability, including kriging, minimum curvature, nearest neighbor, natural neighbor, polynomial regression, inverse distance to a power, and triangulation with linear interpolation. The calculations show that the chosen interpolation method has significant influence on the final result of the study. Nearest neighbor method was chosen to be the best., Kamil Kowalczyk, Jacek Rapinski and Marek Mroz., and Obsahuje bibliografii
The article is focused on the choice of interpolation types for toolpaths in NC programs. Interpolation type affects the interpretation of an NC program by the control system used in a CNC machine tool. It is important for a production engineer to know about the consequences of the use of various interpolation types in a specific CAM system for toolpath creation. For testing purposes, a general profile of a blade typically utilized in energy devices was used, with toolpaths for contour milling of its profile based on three interpolation types available in the CATIA CAM system (linear int., linear + circular int. and spline interpolation). The comparison of toolpaths is based on the number of blocks of the NC program, feed-rate profile measurement and measurement of machining time. For the machining and measurements, a real three-axis machine tool with the Sinumerik 840D control system has been used. and Obsahuje seznam literatury
The Global Navigation Satellite System (GNSS) can provide the daily position time series for the geodesy and geophysical studies. However, due to various unpredictable factors, such as receiver failure or bad observation conditions, missing data inevitably exist in GNSS position time series. Most traditional time series analysis methods require the time series should be completed. Therefore, filling the missing data is a valuable step before analyzing the GNSS time series. In this study, a new method named Iteration Empirical Mode Decomposition (Iteration EMD) is proposed to fill the missing data in GNSS position time series. The simulation experiments are performed by randomly removing different missing percentages of the synthetic time series, with the added different types noise. The results show that Iteration EMD approach performs well regardless of high or low missing percentage. When the missing percentage increases from 5 % to 30 % with a step of 5 %, all the Root Mean Square Errors (RMSE) and Mean Absolute Errors (MAE) of Iteration EMD are smaller than Interpolation EMD. The relative improvements at different percentages of Iteration EMD relative to Interpolation EMD are significant, especially for the high missing percentage. The real GNSS position time series of eight stations were selected to further evaluate the performance of Iteration EMD with an average missing percentage 8.15 %. Principal Component Analysis (PCA) was performed on the filled time series, which is used to assess the interpolation performance of Iteration EMD and Interpolation EMD. The results show that Iteration EMD can preserve variance 75.9 % with the first three Principal Components (PC), more than 66.5% of interpolation EMD. Therefore, we can conclude that Iteration EMD is an efficient interpolation method for GNSS position time series, which can make full use of available information in existing time series to fill the missing data.
The article focuses on a list of options for using interpolations, often referred to as interpolation of higher types. The article pays attention to several representatives of control systems. The issue is also conceived in relation to the preparation of the NC program by CAD/CAM, where the data is prepared for control systems. The available functions of CAD/CAM systems are very diverse. The preparation of the NC program is also related to the issue of postprocessors and therefore they are mentioned in the article as well. Let this article be a basis for those who are interested in the creation of NC programs using non-standard interpolations and serve as an introduction to this issue. and Obsahuje seznam literatury
Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal Lp-regularity is shown. By means of this purely operator theoretic approach, classical results on Lp-regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface diffusion for the diffusion equation is included.
A bicubic model for local smoothing of surfaces is constructed on the base of pivot points. Such an approach allows reducing the dimension of matrix of normal equations more than twice. The model enables to increase essentially the speed and stability of calculations. The algorithms, constructed by the aid of the offered model, can be used both in applications and the development of global methods for smoothing and approximation of surfaces.
Direct interpolation of daily runoff observations to ungauged sites is an alternative to hydrological model regionalisation. Such estimation is particularly important in small headwater basins characterized by sparse hydrological and climate observations, but often large spatial variability. The main objective of this study is to evaluate predictive accuracy of top-kriging interpolation driven by different number of stations (i.e. station densities) in an input dataset. The idea is to interpolate daily runoff for different station densities in Austria and to evaluate the minimum number of stations needed for accurate runoff predictions. Top-kriging efficiency is tested for ten different random samples in ten different stations densities. The predictive accuracy is evaluated by ordinary cross-validation and full-sample crossvalidations. The methodology is tested by using 555 gauges with daily observations in the period 1987-1997. The results of the cross-validation indicate that, in Austria, top-kriging interpolation is superior to hydrological model regionalisation if station density exceeds approximately 2 stations per 1000 km2 (175 stations in Austria). The average median of Nash-Sutcliffe cross-validation efficiency is larger than 0.7 for densities above 2.4 stations/1000 km2 . For such densities, the variability of runoff efficiency is very small over ten random samples. Lower runoff efficiency is found for low station densities (less than 1 station/1000 km2 ) and in some smaller headwater basins.
Including the previously untreated borderline cases, the trace spaces (in the distributional sense) of the Besov-Lizorkin-Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the traces are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the cases previously untreated. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J. Franke and B. Jawerth to the anisotropic scales.