allow for an approximation to the true solution as accurate as posssible. Typically such index sets are computed by a solve-refine cycle, see sections 9.5 and 9.6 for some examples.

However, in some cases one wants to use special, simple trial spaces which require much less implementation effort: uniform and level-adaptive spaces. These correspond to uniform grids and regular sparse grids respectively. We call a trial space

uniform |
contains all indices for a rectangular block of levels with
where is the maximal level along the th coordinate |
very simple implementation |

level adaptive |
contains all indices for an arbitrary set of levels | simple implementation |

adaptive |
may contain arbitrary indices | complicated |

For algorithmical reasons, there are, however, a few further restrictions on index sets. E.g. so-called cone conditions must hold, see [1,5]. The AWFD user doesn't have to care about this, as all functions for the creation and manipulation of adaptive index sets automatically make sure that the constraints are meet.