Showing papers on "Sparse grid published in 2004"

Uncertainty modeling using fuzzy arithmetic based on sparse grids: applications to dynamic systems

Abstract: Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to compute expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In this paper, we illustrate the general applicability of this new method by computing two dynamic systems subjected to uncertain parameters as well as uncertain initial conditions. The first model consists of a system of delay differential equations simulating the periodic outbreak of a disease. In the second model, we consider a multibody mechanism described by an algebraic differential equation system.

Higher Order Quadrature on Sparse Grids

Abstract: Sparse grids have turned out to be a very efficient discretization scheme that, to some extent, breaks the curse of dimensionality and, therefore, is especially well-suited for higher dimensional scenarios. Besides the classical sparse grid application, the numerical solution of partial differential equations, sparse grids have been used for various topics such as Fourier transform, image compression, numerical quadrature, or data mining, so far. In this paper, we summarize and assess recent results concerning the application of sparse grids to integrate functions of higher dimensionality, the focus being on the explicit and adaptive use of higher order basis polynomials.

Multidimensional smoothing using hyperbolic interpolatory wavelets.

Abstract: We propose the application of hyperbolic interpolatory wavelets for large-scale -dimensional data fitting. In particular, we show how wavelets can be used as a highly efficient tool for multidimensional smoothing. The grid underlying these wavelets is a sparse grid. The hyperbolic interpolatory wavelet space of level uses basis functions and it is shown that under sufficient smoothness an approximation error of order ! can be achieved. The implementation uses the fast wavelet transform and an efficient indexing method to access the wavelet coefficients. A practical example demonstrates the efficiency of the approach.

Efficient fuzzy arithmetic for nonlinear functions of modest dimension using sparse grids

Abstract: Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy extension of any objective function. We consider the difficult case of the objective function being an expensive to compute multivariate function of modest dimension (say d up to 16) where only real-valued evaluations of f are permitted. This often poses a difficult problem due to non-applicability of common fuzzy arithmetic algorithms, severe overestimation, or very high computational complexity. Our approach is composed of two parts: First, we compute a surrogate function using sparse grid interpolation. Second, we perform the fuzzy-valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic. The new approach gives accurate results and requires only few function evaluations.

An Optimized; Grid Independent; Narrow Band Data Structure for High Resolution Level Sets

Abstract: Level sets have recently proven successful in many areas of computer graphics including water simulations[Enright et al. 2002] and geometric modeling[Museth et al. 2002]. However, current implementations of these level set methods are limited by factors such as computational efficiency, storage requirements and the restriction to a domain enforced by the convex boundaries of an underlying cartesian computational grid. Here we present a novel very memory efficient narrow band data structure, dubbed the Sparse Grid, that enables the representation of grid independent high resolution level sets. The key features our new data structure are