Michal Smolik

Bio: Michal Smolik is an academic researcher from University of West Bohemia. The author has contributed to research in topics: Vector field & Radial basis function. The author has an hindex of 6, co-authored 29 publications receiving 133 citations.

Large scattered data interpolation with radial basis functions and space subdivision

Abstract: We propose a new approach for the radial basis function (RBF) interpolation of large scattered data sets. It uses the space subdivision technique into independent cells allowing processing of large data sets with low memory requirements and offering high computation speed, together with the possibility of parallel processing as each cell can be processed independently. The proposed RBF interpolation was tested on both synthetic and real data sets. It proved its simplicity, robustness and the ability to handle large data sets together with significant speed-up. In the case of parallel processing, speed-up was experimentally proved when 2 and 4 threads were used.

A Comparative Study of LOWESS and RBF Approximations for Visualization

Abstract: Approximation methods are widely used in many fields and many techniques have been published already. This comparative study presents a comparison of LOWESS (Locally weighted scatterplot smoothing) and RBF (Radial Basis Functions) approximation methods on noisy data as they use different approaches. The RBF approach is generally convenient for high dimensional scattered data sets. The LOWESS method needs finding a subset of nearest points if data are scattered. The experiments proved that LOWESS approximation gives slightly better results than RBF in the case of lower dimension, while in the higher dimensional case with scattered data the RBF method has lower computational complexity.

Algorithm for placement of reference points and choice of an appropriate variable shape parameter for the RBF approximation

Abstract: Many Radial Basis Functions (RBFs) contain a shape parameter which has an important role to ensure good quality of the RBF approximation. Determination of the optimal shape parameter is a difficult problem. In the majority of papers dealing with the RBF approximation, the shape parameter is set up experimentally or using some ad-hoc method. Moreover, the constant shape parameter is almost always used for the RBF approximation, but the variable shape parameter produces more accurate results. Several variable shape parameter methods, which are based on random strategy or on an evolutionary algorithm, have been developed. Another aspect which has an influence on the quality of the RBF approximation is the placement of reference points. A novel algorithm for finding an appropriate set of reference points and a variable shape parameter selection for the RBF approximation of functions y = f(x) (i.e. the case when a one-dimensional dataset is given and each point from this dataset is associated with a scalar value) is presented. Our approach has two steps and is based on exploiting features of the given dataset, such as extreme points or inflection points, and on comparison of the first curvature of a curve. The proposed algorithm can be used for the approximation of data describing a curve parameterized by one variable in multidimensional space, e.g. a robot path planning, etc.

Fast Radial Basis Functions for Engineering Applications

Abstract: Radial basis functions (RBFs) are a mathematical tool mature enough for consistently handling engineering applications. As their theoretical foundation is very well established and the mathematical framework they are based on can be adapted in several ways, such a numerical means has proven to be effective in many technical fields. As a matter of fact, a candidate engineering application can be faced taking the advantage of the peculiar features of RBF such as the availability of many radial functions with global and compact support as well as the interpolation and regression they allow in a multidimensional space. Such a flexibility makes RBF really attractive for users but, actually, their great potential is just partially exploited. This is due to the difficulty in doing a first step towards the effective usage of RBF not only because they are not commonly part of the cultural background of an engineer, but also, and above all, for the numerical complexity of the RBF problems that scale-up very quickly with the number of the considered RBF centres. Fast RBF algorithms are available to alleviate this latter boundary and, additionally, high-performance computing (HPC) systems and solutions can give a further aid of course. Nevertheless, a consolidate tradition in using RBF for engineering applications is still missing and the beginners may be confused by open literature that, in many cases, is presented with language and symbolisms that are comfortable for mathematicians, but that sound rather cryptic and discouraging to engineers. The aim of this book is to put the interested readers in a better position with respect to RBF providing them with a clear vision of their potential, computational tools ready for practical use and precise guidelines to let them implement their own customised workflows in order to handle engineering applications employing RBF. To this end, the book is divided into two main parts. The first one covers the foundation of RBF, the tools available for their quick implementation and the guidelines for facing new challenges, whilst the second part is a collection of practical applications of RBF in several engineering sectors. Such applications deal with lots of technical and scientific topics including, among others, response surface interpolation in n-dimensional spaces, mapping of magnetic and pressure loads,