This work studies the effect of using Monte Carlo based methods to estimate high-dimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induces considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article we quantify this variability with mean squared error measures for two Monte-Carlo based Kalman filter variants, the ensemble Kalman filter and the square-root filter. Under weak assumptions, we derive exact expressions of the error measures. In other cases, we rely on matrix expansions and provide approximations. We show that covariance-shrinking (tapering) based on the Schur product of the prior sample covariance matrix and a positive definite function is a simple, computationally feasible, and very effective technique to reduce sample variability and to address rank-deficient sample covariances. We propose practical rules for obtaining optimally tapered sample covariance matrices. The theoretical results are verified and illustrated with extensive simulations.

. 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, ...

. We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a "loss of derivatives", and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of NashMoser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step. 1. Introduction It has been only very recently that the idea of multistep (or multilevel) interpolation ...

Abstract. While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline interpolation we also give certain saturation theorems. 1.

Abstract. We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions. 1.

Introduction We will treat systems of linear equations, each of the form Lu = f on\Omega ; (1.1) where\Omega is a domain in IR d and L is a linear operator, acting on complex valued functions on \Omega\Gamma The function f is given ("data") and each equation should have a nonempty space of solutions. 2 1. Introduction 1.1 Collocation Method for Single Equations Collocation is a well-known method to approximate the solution of equations of the form (1.1). The idea is to approximate the requested solution u by a function s u = n X j=1 \Phi j ff j 2 span f\Phi j g j=1;:::;n ; (1.1.1) where the \Phi j :\Omega ! C are cer

. In a recent paper with Jerome we have suggested the use of a smoothing operation at each step of the basic multilevel approximation algorithm to improve the convergence rate of the algorithm. In our original paper the smoothing was defined via convolution, and its actual implementation was done via numerical quadrature. In this paper we suggest a different approach to smoothing, namely the use of a precomputed hierarchy of smooth functions. This essentially reduces the cost of the smoothing to zero. x1. Background and Motivation Multilevel approximation with radial basis functions (RBFs) was first suggested in [4], and since then also investigated theoretically in [2,6]. The basic idea is to work with locally supported basis functions at different levels of resolution. This ensures stability and accuracy, something which is difficult to achieve with globally supported RBFs (cf. the well-known trade-off principle [7]). One starts with a coarse-level approximation, and then approximat...

. We generalize techniques dating back to Duchon [4] for error estimates for interpolation by thin plate splines to basis functions with positive and algebraically decaying Fourier transform. We include Lp-estimates for 1 p ! 2 that can also be applied to thin plate spline approximation. x1. Introduction Radial basis functions are a well-established tool for multivariate approximation problems. A radial basis interpolant to a continuous function f : IR d ! IR on a set X = fx 1 ; : : : ; xN g is formed by s f (x) = N X j=1 ff j \Phi(x \Gamma x j ): Here \Phi : IR d ! IR is a fixed, positive definite and symmetric function, and the coefficients ff j are determined by the interpolation conditions s f (x j ) = f(x j ), 1 j N . A more general setting adds certain polynomials to s f to form the interpolant and allows \Phi to be a more general function. For details we refer the reader to the overview articles [3, 5, 6, 8]. In many cases, the function \Phi is radial in the sense \...

In this paper we provide a framework for studying the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation and approximation on a compact Riemannian manifold. We apply this framework to obtain explicit estimates in cases of the circle and 2-sphere. In addition, we provide a technique for constructing strictly positive definite spherical functions out of radial basis functions, and we use it to make a spherical function that is locally supported.

| This paper presents the application of the compactly supported radial basis functions (CSRBFs) in solving a system of shallow water hydrodynamics equations. The proposed scheme is derived from the idea of piecewise polynomial interpolation using a function of Euclidean distance. The compactly supported basis functions consist of a polynomial which are non-zero on [0; 1) and vanish on [1; 1). This reduces the original resultant full matrix to a sparse matrix. The operation of the banded matrix system could reduce the ill-conditioning of the resultant coecient matrix due to the use of the global radial basis functions. To illustrate the computational eciency and accuracy of the method, the dierence between the globally and compactly supported radial basis function schemes is compared. The resulting banded matrix has shown improvement in both ill-conditioning and computational eciency. The numerical solutions are veried with the observed data. Excellent agreement is shown between the ...