Many complex problems in physics, chemistry and engineering can only be examined by modeling and simulation. Frequently, the complexity of the problems translates to a high dimensionality. Due to this curse of dimensionality, the evaluation of these models can become extremely time consuming and should therefore be avoided whenever possible. High-dimensional characteristic maps can come to the rescue, approximating new simulations based on a small set of previous simulations. Using additional evaluations of the original model, the accuracy of the characteristic map can be adaptively increased.
An established method for creating characteristic maps for higher-dimensional problems are sparse grids. They reduce the dependency of the problem on the number of dimensions. We consider both the sparse grid combination technique and hierarchical sparse grids. The former use the solutions of several coarser instances to approximate a finer instance of the problem. The latter enable spatial adaptivity and thereby an even more accurate characteristic map with less evaluations of the original model.
Typically global polynomials and hat ansatz functions are used for sparse grids. We propose to replace polynomial ansatz functions by B-splines. B-splines do not suffer from numerical instabilities, result in continuously differentiable approximations and can be implemented in an efficient manner.
The aim of this project is to simplify and accelerate the working with complex models. We examine the efficiency of sparse grid characteristic maps with B-splines based on several applications including uncertainty quantification and model reduction of multi scale models.