  In simulation technology, computationally expensive objective functions are
often replaced by cheap surrogates, which can be obtained by interpolation.
Full grid interpolation methods suffer from the so-called curse of
dimensionality, rendering them infeasible if the parameter domain of the
function is higher-dimensional (four or more parameters). Sparse grids
constitute a discretization method that drastically eases the curse, while the
approximation quality deteriorates only insignificantly. However, conventional
basis functions such as piecewise linear functions are not smooth (continuously
differentiable). Hence, these basis functions are unsuitable for applications
in which gradients are required. One example for such an application is
gradient-based optimization, in which the availability of gradients greatly
improves the speed of convergence and the accuracy of the results.
  This thesis demonstrates that hierarchical B-splines on sparse grids are
well-suited for obtaining smooth interpolants for higher dimensionalities. The
thesis is organized in two main parts: In the first part, we derive new
B-spline bases on sparse grids and study their implications on theory and
algorithms. In the second part, we consider three real-world applications in
optimization: topology optimization, biomechanical continuum-mechanics, and
dynamic portfolio choice models in finance. The results reveal that the
optimization problems of these applications can be solved accurately and
efficiently with hierarchical B-splines on sparse grids.
