Sparse Grid Methods for Higher Dimensional Approximation: Theory and Experiments on Benefits and Limitations of Sparse Grid Approximation Methods and Generalizations to Low Effective Dimensionality

Approximation of functions depending on more than three variables is relevant for the simulation of dynamic models. Especially stochastic models lead to a wide range of applications like simulation of polymeric liquids, kinetic equations for forces acting on offshore wind parks, or even the pricing of financial products. Sparse Grids have been designed to significantly reduce the cost to approximate high-dimensional functions under certain smoothness conditions. This book presents tools to design and work with Sparse Grids, including construction properties, algorithms, data structures and error estimates. It also discusses inherent limitations and their benefits when it comes to the simulation of stochastic models, namely to the representation of probability density functions. Furthermore, the book reveals new space- and dimension-adaptive Sparse Grids which are suitable for functions that effectively depend on only a few of their input variables. Read more