The sparse grid combination technique for quantities of interest

Abstract

The curse of dimensionality is a major problem for large scale simulations. One way to tackle this problem is the sparse grid combination technique. While a full grid requires O{h_n^{-d}} grid points the sparse grid combination technique needs significantly less points. In contrast to the traditional combination technique, which combines solution functions themselves, this work puts its focus on the combination technique with quantities of interest and their surpluses. After introducing the concept of surpluses that describe how much the solution changes if the grids are refined, we defined the combination technique as a sum of these surpluses. We show how the concept of surpluses can be utilized to deduce error bounds for the quantity of interest and helps to adapt the combination technique to problems with different error models. To improve the error bound we introduce a new extrapolated version of the combination technique and see how the surpluses are affected. To evaluate our theoretical results we perform numerical experiments where we consider integration problems and the gyrokinetic plasma turbulence simulation GENE. The experimental results for the integration problems nicely confirm our derived theoretical results.