B-Splines for Sparse Grids: Algorithms and Application to Higher-Dimensional Optimization

TL;DR

This paper introduces hierarchical B-splines on sparse grids for smooth interpolation in high-dimensional optimization, overcoming limitations of traditional basis functions and enabling efficient gradient-based methods.

Contribution

It develops new B-spline bases on sparse grids and demonstrates their effectiveness in real-world high-dimensional optimization applications.

Findings

Hierarchical B-splines provide smooth interpolants suitable for gradient-based optimization.

The approach efficiently solves complex optimization problems in engineering, biomechanics, and finance.

Results show high accuracy and computational efficiency in practical applications.

Abstract

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.