Showing papers on "Sparse grid published in 2000"

Higher order impedance boundary conditions for sparse wire grids

Abstract: Higher order impedance boundary conditions designed for modeling wire grids of thin conducting wires are established. The derivation is based on the exact analytical summation of the individual wire fields. This allows one to write an approximate boundary condition on the grid surface, which connects the averaged electric field and the averaged current (or the electric field and the averaged magnetic fields on the two sides of the grid surface). The condition depends on the tangential derivatives of the averaged current (up to the sixth order). This approach provides an extension of the averaged boundary conditions method (well established for dense grids) to sparse grids. Numerical examples demonstrate very good accuracy of the solutions for the field reflected from grids with the wire separation as large as half of the wavelength.

Application of adaptive sparse-grid technique to a model singular perturbation problem

Abstract: In this paper we show how, under minimal conditions, a combination extrapolation can be introduced for an adaptive sparse grid. We apply this technique for the solution of a two-dimensional model singular perturbation problem, defined on the domain exterior of a circle.

A Class of Periodic Function Spaces and Interpolation on Sparse Grids

Abstract: We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces.

The exponent of discrepancy of sparse grids is at least 2:1933

Abstract: We study bounds on the exponents of sparse grids for L2‐discrepancy and average case d‐dimensional integration with respect to the Wiener sheet measure. Our main result is that the minimal exponent of sparse grids for these problems is bounded from below by 2.1933. This shows that sparse grids provide a rather poor exponent since, due to Wasilkowski and Woźniakowski [16], the minimal exponent of L2‐discrepancy of arbitrary point sets is at most 1.4778. The proof of the latter, however, is non‐constructive. The best known constructive upper bound is still obtained by a particular sparse grid and equal to 2.4526....

A proof of the consistency of the finite difference technique on sparse grids

Abstract: In this paper, we give a proof of the consistency of the finite difference technique on regular sparse grids [7, 18]. We introduce an extrapolation-type discretization of differential operators on sparse grids based on the idea of the combination technique and we show the consistency of this discretization. The equivalence of the new method with that of [7, 18] is established.

A Sparse Grid PDE Solver; Discretization, Adaptivity, Software Design and Parallelization

Abstract: Sparse grids are an efficient approximation method for functions, especially in higher dimensions d≥3 Compared to regular, uniform grids of a mesh parameter h, which contain h −d points in d dimensions, sparse grids require only h −1| log h| d−1 points due to a truncated, tensor-product multi-scale basis representation The purpose of this paper is to survey some activities for the solution of partial differential equations with method based sparse grids Furthermore some aspects of sparse grids are discussed such as adaptive grid refinement, parallel computing, a space-time discretization scheme and the structure of a code to implement these methods

Experience with the Solution of a Finite Difference Discretization on Sparse Grids

Abstract: In a recent paper [10], we described and analyzed a finite difference discretization on adaptive sparse grids in three space dimensions. In this paper, we show how the discrete equations can be efficiently solved in an iterative process. Several alternatives have been studied before in Sprengel [16], where multigrid algorithms were used. Here, we report on our experience with BiCGStab iteration. It appears that, applied to the hierarchical representation and combined with Nested Iteration in a cascadic algorithm, BiCGStab shows fast convergence, although the convergence rate is not truly independent of the meshsize.

Scalable Parallel Algorithms ForPredictive Modelling

Abstract: Data Mining applications have to deal with increasingly large data sets and complexity. Only algorithms which scale linearly with data size are feasible. We present parallel regression algorithms which after a few initial scans of the data compute predictive models for data mining and do not require further access to the data. In addition, we describe various ways of dealing with the complexity (high dimensionality) of the data. Three methods are presented for three different ranges of attribute numbers. They use ideas from the finite element method and are based on penalised least squares fits using sparse grids and additive models for intermediate and very high dimensional data. Computational experiments confirm scalability both with respect to data size and number of processors.

Volume Ray Casting on Sparse Grids

Abstract: These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of partial differential equations. The volume visualization algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse grids as a data compression method in order to visualize huge data sets even on workstations with low main memory. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets. Besides the standard sparse grid interpolation algorithm and the so called combination approach, we have developed a new sparse grid interpolation method, which harnesses the texture-mapping hardware of Silicon Graphics workstations for accelerating purposes. Therefore, hardware based volume rendering becomes possible on compressed data sets at interactive frame rates.

Parallel Adaptively Refined Sparse Grids

Abstract: A parallel version of a finite difference discretization of PDEs on sparse grids is proposed. Sparse grids or hyperbolic crosspoints can be used for the efficient representation of solutions of a boundary value problem, especially in high dimensions, because the number of grid points depends only weakly on the dimension. So far only the ‘combination’ technique for regular sparse grids was available on parallel computers. However, the new approach allows for arbitrary, adaptively refined sparse grids. The efficient parallelisation is based on a dynamic load-balancing approach with space-filling curves.