Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

Fast evaluation of radial basis functions : moment based methods

In this paper we introduce a new algorithm for fast evaluation of univariate radial basis functions of the form s(x) = Σᶰn₌₁ dn(⃒x - xn⃒) to within accuracy . The algorithm has a setup cost of (N⃒log⃒log⃒log⃒) operations and an incremental cost per evaluation of s(x) of (⃒log⃒) operations. It is based on a hierarchical subdivision of the unit interval, the adaptive construction of a corresponding hierarchy of polynomial approximations, and the fast accumulation of moments. It can be applied in any case where the basic function smooth on (0, 1], and on any grid of centres ｛Xn｝. The algorithm does not require that be analytic at infinity, nor that the user specify new polynomial approximations or modify the data structures for each new , nor that the points Xn form any sort of regular array. Furthermore the algorithm can be extended to problems in higher dimensions

Some Basis Function Methods for Surface Approximation

This thesis considers issues in surface reconstruction such as identifying approximation methods that work well for certain applications and developing efficient methods to compute and manipulate these approximations. The first part of the thesis illustrates a new fast evaluation scheme to efficiently calculate thin-plate splines in two dimensions. In the fast multipole method scheme, exponential expansions/approximations are used as an intermediate step in converting far field series to local polynomial approximations. The contributions here are extending the scheme to the thin-plate spline and a new error analysis. The error analysis covers the practically important case where truncated series are used throughout, and through off line computation of error constants gives sharp error bounds. In the second part of this thesis, we investigates fitting a surface to an object using blobby models as a coarse level approximation. The aim is to achieve a given quality of approximation with relatively few parameters. This process involves an optimization procedure where a number of blobs (ellipses or ellipsoids) are separately fitted to a cloud of points. Then the optimized blobs are combined to yield an implicit surface approximating the cloud of points. The results for our test cases in 2 and 3 dimensions are very encouraging. For many applications, the coarse level blobby model itself will be sufficient. For example adding texture on top of the blobby surface can give a surprisingly realistic image. The last part of the thesis describes a method to reconstruct surfaces with known discontinuities. We fit a surface to the data points by performing a scattered data interpolation using compactly supported RBFs with respect to a geodesic distance. Techniques from computational geometry such as the visibility graph are used to compute the shortest Euclidean distance between two points, avoiding any obstacles. Results have shown that discontinuities on the surface were clearly reconstructed, and th

Iterative Techniques for Radial Basis Function Interpolation

The problem of interpolating functions comes up naturally in many areas of applied mathematics and natural sciences. Radial basis function methods provide an interpolant to values of a real function of $\textit{d}$ variables and are highly useful in many applications, especially if the function values are given at scattered data points. The need for iterative procedures arises when the number of interpolation conditions $\textit{n}$ is large, since hardly any sparsity occurs in the linear system of interpolation equations. Solving this system with direct methods would require O($\textit{n}$3) operations. This dissertation considers several iterative techniques. They were developed from an algorithm described by Beatson, Goodsell and Powell (1995), which is examined first. By gaining more and more theoretical insight into the original algorithm, new algorithms are developed and connections to known methods are made. We establish the important role a certain semi-inner product plays in the convergence analysis of the original algorithm, and the first proof of convergence is given. This leads to a new technique using line searches described later. Then it is shown that the original algorithm is equivalent to solving a certain symmetric and positive definite system of equations by Gauss-Seidel iterations. Thus iterative techniques like Jacobi iterations and conjugate gradient methods follow. This symmetric and positive definite system of equations can be derived from the original system of equations by preconditioning it with a certain matrix. The preconditioned conjugate gradient algorithm was first suggested for this problem by Dyn et al. (1983, 1986), motivated by the variational theory of thin plate splines. It is helpful to view the original algorithm as a linear operator working on a certain linear space equipped with the aforementioned semi-inner product. The original algorithm had the drawback that the residuals had to be updated at several stages during each iteration. Another algorithm defers the updates till the end of each iteration, which usually improves efficiency greatly, but divergence occurs in some cases. Therefore a line search method is developed that ensures convergence. The last technique described is a Krylov subspace method which proved to be very successful. It can be applied to any algorithm that fulfils certain criteria. If the underlying algorithm is convergent, the Krylov subspace technique speeds the convergence up. In cases of divergence, the Krylov subspace method enforces convergence. It is shown that the Krylov subspace method applied to the algorithm where updating the residuals is deferred till the end of each iteration is analogous to the conjugate gradient technique applied to the aforementioned symmetric and positive definite system of equations. All algorithms are related to each other and theoretical insight into some properties of one algorithm leads to an improved algorithm. These considerations provide a highly useful theory, linking different techniques for iterative radial basis function interpolation

Error Estimation and Adaptive Refinement of Finite Element Thin Plate Spline

The thin plate spline smoother is a data fitting and smoothing technique that captures important patterns of potentially noisy data. However, it is computationally expensive for large data sets. The finite element thin plate spline smoother (TPSFEM) combines the thin plate spline smoother and finite element surface fitting to efficiently interpolate large data sets. When the TPSFEM uses uniform finite element grids, it may require a fine grid to achieve the desired accuracy. Adaptive refinement uses error indicators to identify sensitive regions and adapts the precision of the solution dynamically, which reduces the computational cost to achieve the required accuracy. Traditional error indicators were developed for the finite element method to approximate partial differential equations and may not be applicable for the TPSFEM. We examined techniques that may indicate errors for the TPSFEM and adapted four traditional error indicators that use different information to produce efficient adaptive grids. The iterative adaptive refinement process has also been adjusted to handle additional complexities caused by the TPSFEM. The four error indicators presented in this thesis are the auxiliary problem error indicator, recovery-based error indicator, norm-based error indicator and residual-based error indicator. The auxiliary problem error indicator approximates the error by solving auxiliary problems to evaluate approximation quality. The recovery-based error indicator calculates the error by post-processing discontinuous gradients of the TPSFEM. The norm-based error indicator uses an error bound on the interpolation error to indicate large errors. The residual-based error indicator computes interior element residuals and jumps of gradients across elements to estimate the energy norm of the error. Numerical experiments were conducted to evaluate the error indicators' performance on producing efficient adaptive grids, which are measured by the error versus the number of nodes in the grid. A set of one and two-dimensional model problems with various features are chosen to examine the effectiveness of the error indicators. As opposed to the finite element method, error indicators of the TPSFEM may also be affected by noise, data distribution patterns, data sizes and boundary conditions, which are assessed in the experiments. It is found that adaptive grids are significantly more efficient than uniform grids for two-dimensional model problems with difficulties like peaks and singularities. While the TPSFEM may not recover the original solution in the presence of noise or scarce data, error indicators still produce more efficient grids. We also learned that the difference is less obvious when the data has mostly smooth or oscillatory surfaces. Some error indicators that use data may be affected by data distribution patterns and boundary conditions, but the others are robust and produce stable results. Our error indicators also successfully identify sensitive regions for one-dimensional data sets. Lastly, when errors of the TPSFEM cannot be further reduced due to factors like noise, new stopping criteria terminate the iterative process aptly

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells