Least squares surface approximation to scattered data using multiquadric functions

This report documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadric function with the criteria for the fit being the least mean squared resifual for the data points. The principal problem is the selection of knot points (or base points for the multiquadric basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and multiquadric parameter is explored, with some unexpected results in terms of 'near repeated' knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.http://archive.org/details/leastsquaressurf00franApproved for public release; distribution is unlimited

Stable PDE Solution Methods for Large Multiquadric Shape Parameters

We present a new method based upon the paper of Volokh and Vilney (2000) that produces highly accurate and stable solutions to very ill-conditioned multiquadric (MQ) radial basis function (RBF) asymmetric collocation methods for partial differential equations (PDEs). We demonstrate that the modified Volokh-Vilney algorithm that we name the improved truncated singular value decomposition (IT-SVD) produces highly accurate and stable numerical solutions for large values of a constant MQ shape parameter, c, that exceeds the critical value of c based upon Gaussian elimination

Splines and local approximation of the earth's gravity field

Bibliography: pages 214-220.The Hilbert space spline theory of Delvos and Schempp, and the reproducing kernel theory of L. Schwartz, provide the conceptual foundation and the construction procedure for rotation-invariant splines on Euclidean spaces, splines on the circle, and splines on the sphere and harmonic outside the sphere. Spherical splines and surface splines such as multi-conic functions, Hardy's multiquadric functions, pseudo-cubic splines, and thin-plate splines, are shown to be largely as effective as least squares collocation in representing geoid heights or gravity anomalies. A pseudo-cubic spline geoid for southern Africa is given, interpolating Doppler-derived geoid heights and astro-geodetic deflections of the vertical. Quadrature rules are derived for the thin-plate spline approximation (over a circular disk, and to a planar approximation) of Stokes's formula, the formulae of Vening Meinesz, and the L₁ vertical gradient operator in the analytical continuation series solution of Molodensky's problem

On the Selection of a Good Shape Parameter for RBF Approximation and Its Application for Solving PDEs

Meshless methods utilizing Radial Basis Functions~(RBFs) are a numerical method that require no mesh connections within the computational domain. They are useful for solving numerous real-world engineering problems. Over the past decades, after the 1970s, several RBFs have been developed and successfully applied to recover unknown functions and to solve Partial Differential Equations (PDEs).However, some RBFs, such as Multiquadratic (MQ), Gaussian (GA), and Matern functions, contain a free variable, the shape parameter, c. Because c exerts a strong influence on the accuracy of numerical solutions, much effort has been devoted to developing methods for determining shape parameters which provide accurate results. Most past strategies, which have utilized a trail-and-error approach or focused on mathematically proven values for c, remain cumbersome and impractical for real-world implementations.This dissertation presents a new method, Residue-Error Cross Validation (RECV), which can be used to select good shape parameters for RBFs in both interpolation and PDE problems. The RECV method maps the original optimization problem of defining a shape parameter into a root-finding problem, thus avoiding the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned.With minimal computational time, the RECV method provides shape parameter values which yield highly accurate interpolations. Additionally, when considering smaller data sets, accuracy and stability can be further increased by using the shape parameter provided by the RECV method as the upper bound of the c interval considered by the LOOCV method. The RECV method can also be combined with an adaptive method, knot insertion, to achieve accuracy up to two orders of magnitude higher than that achieved using Halton uniformly distributed points

Advances in radial and spherical basis function interpolation

The radial basis function method is a widely used technique for interpolation of scattered data. The method is meshfree, easy to implement independently of the number of dimensions, and for certain types of basis functions it provides spectral accuracy. All these properties also apply to the spherical basis function method, but the class of applicable basis functions, positive definite functions on the sphere, is not as well studied and understood as the radial basis functions for the Euclidean space. The aim of this thesis is mainly to introduce new techniques for construction of Euclidean basis functions and to establish new criteria for positive definiteness of functions on spheres. We study multiply and completely monotone functions, which are important for radial basis function interpolation because their monotonicity properties are in some cases necessary and in some cases sufficient for the positive definiteness of a function. We enhance many results which were originally stated for completely monotone functions to the bigger class of multiply monotone functions and use those to derive new radial basis functions. Further, we study the connection of monotonicity properties and positive definiteness of spherical basis functions. In the processes several new sufficient and some new necessary conditions for positive definiteness of spherical radial functions are proven. We also describe different techniques of constructing new radial and spherical basis functions, for example shifts. For the shifted versions in the Euclidean space we prove conditions for positive definiteness, compute their Fourier transform and give integral representations. Furthermore, we prove that the cosine transforms of multiply monotone functions are positive definite under some mild extra conditions. Additionally, a new class of radial basis functions which is derived as the Fourier transforms of the generalised Gaussian φ(t) = e−tβ is investigated. We conclude with a comparison of the spherical basis functions, which we derived in this thesis and those spherical basis functions well known. For this numerical test a set of test functions as well as recordings of electroencephalographic data are used to evaluate the performance of the different basis functions

Development of mesh-free methods and their applications in computational fluid dynamics

Ph.DDOCTOR OF PHILOSOPH

Interval simplex splines for scientific databases

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1995.Includes bibliographical references (p. 130-138).by Jingfang Zhou.Ph.D

Intelligent Control Strategies for an Autonomous Underwater Vehicle

The dynamic characteristics of autonomous underwater vehicles (AUVs) present a control problem that classical methods cannot often accommodate easily. Fundamentally, AUV dynamics are highly non-linear, and the relative similarity between the linear and angular velocities about each degree of freedom means that control schemes employed within other flight vehicles are not always applicable. In such instances, intelligent control strategies offer a more sophisticated approach to the design of the control algorithm. Neurofuzzy control is one such technique, which fuses the beneficial properties of neural networks and fuzzy logic in a hybrid control architecture. Such an approach is highly suited to development of an autopilot for an AUV. Specifically, the adaptive network-based fuzzy inference system (ANFIS) is discussed in Chapter 4 as an effective new approach for neurally tuning course-changing fuzzy autopilots. However, the limitation of this technique is that it cannot be used for developing multivariable fuzzy structures. Consequently, the co-active ANFIS (CANFIS) architecture is developed and employed as a novel multi variable AUV autopilot within Chapter 5, whereby simultaneous control of the AUV yaw and roll channels is achieved. Moreover, this structure is flexible in that it is extended in Chapter 6 to perform on-line control of the AUV leading to a novel autopilot design that can accommodate changing vehicle pay loads and environmental disturbances. Whilst the typical ANFIS and CANFIS structures prove effective for AUV control system design, the well known properties of radial basis function networks (RBFN) offer a more flexible controller architecture. Chapter 7 presents a new approach to fuzzy modelling and employs both ANFIS and CANFIS structures with non-linear consequent functions of composite Gaussian form. This merger of CANFIS and a RBFN lends itself naturally to tuning with an extended form of the hybrid learning rule, and provides a very effective approach to intelligent controller development.The Sea Systems and Platform Integration Sector, Defence Evaluation and Research Agency, Winfrit