Showing papers on "Sparse grid published in 1997"

Face authentication with sparse grid Gabor information

Abstract: This paper investigates the application of statistical pattern recognition methods in the framework of the dynamic link matching approach. This method describes objects by means of local frequency information on nodes of a sparse grid. Matching of an input image with a reference is achieved by displacement and deformation of the grid. This method is applied here to the authentication of human faces in a cooperative scenario where candidates claim an identity that is to be checked. The matching error is not powerful enough to provide satisfying results in this case. We introduce an automatic weighting of the nodes according to their significance. Results show that for regular grids, this weighting leads to a significant improvement of the performance.

A multigrid algorithm for higher order finite elements on sparse grids.

Abstract: For most types of problems in numerical mathematics, efficient discretization techniques are of crucial importance. This holds for tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as accurate as possible as well as for the numerical solution of differential equations. Introduced by Zenger in 1990 and based on hierarchical tensor product approximation spaces, sparse grids have turned out to be a very efficient approach in order to improve the ratio of invested storage and computing time to the achieved accuracy for many problems in the areas mentioned above. Concerning the sparse grid finite element discretization of elliptic partial differential equations, recently, the class of problems that can be tackled has been enlarged significantly. First, the tensor product approach led to the formulation of unidirectional algorithms which are essentially independent of the number d of dimensions. Second, techniques for the treatment of the general linear elliptic differential operator of second order have been developed, which, with the help of domain transformation, enable us to deal with more complicated geometries, too. Finally, the development of hierarchical polynomial bases of piecewise arbitrary degree p has opened the way to a further improvement of the order of approximation. In this paper, we discuss the construction and the main properties of a class of hierarchical polynomial bases and present a symmetric and an asymmetric finite element method on sparse grids, using the hierarchical polynomial bases for both the approximation and the test spaces or for the approximation space only, resp., with standard piecewise multilinear hierarchical test functions. In both cases, the storage requirement at a grid point does not depend on the local polynomial degree p, and p and the resulting representations of the basis functions can be handled in an efficient and adaptive way. An advantage of the latter approach, however, is the fact that it allows the straightforward implementation of a multigrid solver for the resulting system which is discussed, too.

Scientific Visualization on Sparse Grids

Abstract: The ever growing size of data sets resulting from industrial and scientific simulations and measurements have created an enormous need for analysis tools allowing interactive visualization. A promising hierarchical approach in the area of numerical simulation is called sparse grids. We present two major visualization algorithms working directly on the sparse grid representation of the data set. One of them is interactive particle tracing, which continues to be an important utility for evaluating CFD simulations. The other one is volume ray casting, which is of interest in many areas dealing with three-dimensional scalar data. Additionally we have been able to employ texture hardware support for the necessary function interpolation. Hence, we are able to perform volume visualization methods on compressed data sets at interactive frame rates, which is not possible with other methods like wavelets or fractal compression. In particular, we are able to handle sparse grids of level 13, which correspond to regular volumes of 8193^3 voxels.

Adaptive Sparse Grid Multilevel Methods for Elliptic PDEs Based on Finite Differences.

Abstract: We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a one-dimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments. — Author's Abstract

Convergence results for 3D sparse grid approaches

Abstract: The convergence behaviour is investigated of solution algorithms for the anisotropic Poisson problem on partially ordered, sparse families of regular grids in 3D. In order to study multilevel techniques on sparse families of grids, first we consider the convergence of a two-level algorithm that applies semi-coarsening successively in each of the coordinate directions. This algorithm shows good convergence, but recursive application of the successive semi-coarsening is not sufficiently efficient. Therefore we introduce another algorithm, which uses collective 3D semi-coarsened coarse grid corrections. The convergence behaviour of this collective version is worse, due to the lack of correspondence between the solutions on the different grids. By solving for the trivial solution we demonstrate that a good convergence behaviour of the collective version of the algorithm can be retained when the different solutions are sufficiently coherent. In order to solve also non-trivial problems, we develop a defect correction process. This algorithm makes use of hierarchical smoothing in order to deal with the problems related to the lack of coherence between the solutions on the different grids. Now good convergence rates are obtained also for non-trivial solutions. All convergence results are obtained for two-level processes. The results show convergence rates which are bounded, independent of the discretisation level and of the anisotropy in the problem.