Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
2.   Sparse Grids   &  Adaptivity
A simple approach for multivariate wavelets are tensor  products of univariate multiscale functions. There are two different methodologies. In the first case, the univariate functions for the different coordinate directions have the same level of refinement. This leads to essentially isotropic basis functions.  In the second case, there is no restriction on the refinement level.  The tensor product approach is used in a straight forward manner and the wavelets are defined by
Depending on how much the level indices for the various coordinate directions differ, one ends up with   isotropic  as well as strongly anisotropic basis functions. Some examples are shown below for the 2D case.
Levels (3,1),  (1,3),  (2,2)
Levels (2,1), (1,2)
Level (1,1)

The approximation properties of the isotropic/ansiotropic bases are quite different as thorough mathematical analyses have shown [1-4].  Simply said, the isotropic bases are slightly better for functions with  non-grid alligned   (quasi-)singularities.  However, the  above anisotropic basis leads to a dramatically more efficient approximation to functions with

  • boundary layers or other grid alligned (quasi-)singularities.
  • bounded mixed derivatives.
Of course, one can only achieve this high efficiency with finite dimensional approximation spaces adapted to the target function. For example, for functions with bounded mixed derivatives the (almost) optimal  approximation spaces contain all wavelets with indices
 l and t are the multi-indices for the level and the translation and L is a certain maximal level which controls the target accuracy. For the truely adaptive case more complicated index sets without closed form may be necessary for the best efficiency.  The question how to find good index  for an unknown solution of a PDE  is addressed later. For the visualization of the index sets one plots the centers of the supports of the active basis functions. In case of the above simple index sets we get the so-called regular sparse grids, see the  left Figure below.  An adaptive sparse grid is shown right. It stems for the numerical solution of a hyperbolic equation with two diagonal shock fronts in the solution.