Efficient Visualization of Data on Sparse Grids

Abstract

Sparse grids are nowadays frequently used in numerical simulation. With their help the number of unknown in discrete PDE or approximation problems can drastically be decreased. On sparse grids in d space dimensions O(Nlog(N)^d−1) nodes are required to achieve nearly the same approximation quality for sufficiently smooth data as on a standard finite difference grid with N ^d nodes. This allows numerical methods with small error tolerances on a corresponding very fine mesh width. Mapping the complete sparse grid data on an N ^d standard grid in order to analyse them visually would burst a currently available work station storage for large O(N log(N)^d−1) numbers of nodal values provided by numerical code on a sparse grid. We present an efficient, procedural approach to sparse grids for the visual post-processing. A procedural interface addresses data on grid cells only temporarily, only if actually requested by the visualization methods. The cell access procedures extract local numerical data directly from the users sparse grid data base. No additional memory is needed. The procedural approach is combined with a multiresolution strategy based on a recursive traversal of the grid hierarchy. It includes hierarchical searching for features such as isosurfaces and a locally adaptive stopping on coarser grids in areas of sufficient smoothness. This data smoothness corresponds to a user prescribed error tolerance. Examples underline the benefits of the presented approach.