We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a-posteriori error estimation for this new method are also proved. Key words: Finite element method, meshless finite element method, finite element methods for highly oscillatory solutions TICAM, The University of Texas at Austin, Austin, TX 78712. Research was partially supported by US Office of Naval Research under grant N00014--90--J1030 y Seminar for Applied Mathematics, ETH Zurich, CH--8092 Zurich, Switzerland....

. Motivated by [5] we describe a method related to scattered Hermite interpolation for which the solution of elliptic partial differential equations by collocation is well-posed. We compare the method of [5] with our method. x1. Introduction In this paper we discuss the numerical solution of elliptic partial differential equations using a collocation approach based on radial basis functions. To make the discussion transparent we will focus on the case of a time independent linear elliptic partial differential equation in IR 2 . In the following we assume we are given a set of nodes \Xi = f ~ ¸ 1 ; : : : ; ~ ¸ N g ae IR d , along with a continuous function ' : [0; 1) ! IR. We then refer to ~x 7! '(k~x\Gamma ~ ¸ k k 2 ), ~x 2 IR d , k 2 f1; : : : ; Ng, as radial basis functions centered at ~ ¸ k . Some of the most commonly used radial basis functions are the (reciprocal) multiquadrics '(r) = (r 2 + c 2 ) \Sigma1=2 , the Gaussians '(r) = e \Gammacr 2 , and the thin pla...

After a series of application papers have proven the approach to be numerically effective, this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with (possibly radial) basis functions. 0 Introduction We consider a general class of boundary or initial value problems for partial differential equations: Lu = f in\Omega ae IR d L : W\Omega ! L\Omega Bu = g in @\Omega B : W\Omega !W @\Omega : (0.1) Here, the operator L is a linear partial differential operator, and B is a "boundary " operator that prescribes values on (possibly only part of) the boundary @\Omega of the underlying bounded domain\Omega 2 IR d . The domain and range spaces can be viewed as certain instances of Sobolev or L 2 spaces such that appropriate trace theorems hold. We can also allow multiple differential, integral, or boundary operators, but we do not want to introduce too much notation at this stage. The goal of this paper is to prove the...

: This paper applies the global radial basis functions as a spatial collocation scheme for solving the Options Pricing model. Different numerical time integration schemes are employed for the time derivative of the model. In the case of the European options, it is shown that the major numerical error is from the time integration instead of the spatial approximation by comparing with the analytical solution. The numerical results for the American options indicate that this proposed scheme offers a highly accurate approximation compared with existing numerical methods. Since the basis functions are infinitely differentiable, the numerical approximation of the derivatives of the options price can be computed directly without using extra interpolation techniques. The numerical approximation of the optimal exercise boundary in the case of American options can also be obtained effectively by using the Newton's iterative scheme. Key words: Black-Scholes; American Options; Radial ...

During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ill-conditioning, which has previously limited the range that could be used for this parameter. We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBF-based methods against two standard techniques (a second-order finite di#erence method and a pseudospectral method), the former gave a much superior accuracy.

. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using so-called meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and co-worker's results [3] using the so-called element-free Galerkin method, Duarte and Oden's work [11] using h-p clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...

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In recent years approximation theory has found interesting applications in the elds of Arti cial Intelligence and Computer Science. For instance, a problem that ts very naturally in the framework of approximation theory is the problem of learning to perform a particular task from a set of examples. The examples are sparse data points in a

In this paper, we present a meshless discretization technique for instationary convection-diffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h- or p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection and elliptic problems.