Spatially Adaptive Refinement

摘要

While sparse grids allow one to tackle problems in higher dimensionalities than possible for standard grid-based discretizations, real-world applications often come along with requirements or restrictions which enforce problem-dependent adaptations of the standard sparse grid technique. Consider, for example, interpolations where the function values at grid points are obtained via time-consuming numerical simulations. Then, only very few grid points can be spent; classical convergence might be out of reach. Another hurdle is that real-world problems often do not meet the smoothness requirements of the sparse grid method. Thus, the standard approach has to be fine-tuned to the problem at hand, especially in higher-dimensional settings. Therefore, a suitable choice of basis functions can be required, as well as criteria for problem-adapted refinement. Fortunately, and in contrast to full grids, the hierarchical basis formulation of the direct sparse grid approach conveniently provides a reasonable criterion for spatially adaptive refinement practically for free. This can serve as a starting point to develop suitable modifications. We show several problems stemming from different fields of application and demonstrate modifications of the standard sparse grid approach. They enable one to cope with the properties and requirements of the corresponding problem and can serve as examples for similar challenges.