Surface generating on the basis of experimental data

The problem of surface generating on the basis of experimental data is presented in this lecture. Special attention is given to the implementation of moving ordinary least squares and moving total least squares. Some results done in the Institute for Applied Mathematics in Osijek are mentioned which were published in the last several years

Variable Shape Parameter Strategies in Radial Basis Function Methods

The Radial Basis Function (RBF) method is an important tool in the interpolation of multidimensional scattered data. The method has several important properties. One is the ability to handle sparse and scattered data points. Another property is its ability to interpolate in more than one dimension. Furthermore, the Radial Basis Function method provides phenomenal accuracy which has made it very popular in many fields. Some examples of applications using the RBF method are numerical solutions to partial differential equations, image processing, and cartography. This thesis involves researching Radial Basis Functions using different shape parameter strategies. First, we introduce the Radial Basis Function method by stating its history and development in Chapter 1. Second, we explain how Radial Basis Functions work in Chapter 2. Chapter 3 compares RBF interpolation to polynomial interpolation. Chapters 4 and 5 introduce the idea of variable shape parameters. In these chapters we compare and analyze the variable shape parameters in one and two dimensions. In Chapter 6, we introduce the challenges in interpolations due to errors in boundary regions. Here, we try to reduce the error using different shape parameter strategies. Chapter 7 lists the conclusions resulting from the research

Assessments On Surface Interpolation Methods For Local Geoid Modelling

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2016Thesis (M.Sc.) -- İstanbul Technical University, Instıtute of Science and Technology, 2016Farklı interpolasyon yaklaşımlarıyla oluşturulan yerel geoit modeli ile yüksek doğruluklu yükseklikler elde edilir.High precision heights are obtined with local geoid modelling what are determined by using different interpolation algorithms.Yüksek LisansM.Sc

A radial basis function approach to reconstructing the local volatility surface of European options

A key problem in financial mathematics is modelling the volatility skew observed in options markets. Local volatility methods, which is one approach to modelling skew, requires the construction of a volatility surface to reconcile discretely observed market data and dynamics. In this thesis we propose a new method to construct this surface using radial basis functions. Our results show that this approach is tractable and yields good results. When used in a local volatility context these results replicate the observed market prices. Testing against a skew model with known analytical solution shows that both prices and hedging parameters are acurately reconstructed, with best case average relative errors in pricing of 0.0012. While the accuracy of these results exceeds those reported by spline interpolation methods, the solution is critically dependent upon the quality of the numerical solution of the resultant local volatility PDE’s, heuristic parameter choices and data filtering

An Improved Method using RBF Neural Networks to Speed up Optimization Algorithms

The paper presents a method using Radial Basis Function (RBF) neural networks to speed up deterministic search algorithms used for the optimization of superconducting magnets for the LHC accelerator project at CERN. The optimization of the iron yoke of the main LHC dipoles requires a number of numerical field computations per trial solution as the field quality depends on the excitation and local iron saturation in the yoke. This results in computation times of about 30 minutes for each objective function evaluation (on DEC-Alpha 600/333). In this paper we present a method for constructing an RBF neural network for a local approximation of the objective function. The computational time required for such a construction is negligible compared to the deterministic function evaluation, and thus yields a speed-up of the overall search process. The effectiveness of this method is demonstrated by means of two- and three-parametric optimization examples. The achieved speed-up of the search routine is up to 30 %

Mesh Free Methods for Differential Models In Financial Mathematics

Philosophiae Doctor - PhDMany problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided