Fundamental Splines on Sparse Grids and Their Application to Gradient-Based Optimization
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Most types of hierarchical basis functions for sparse grids are not continuously differentiable. This can lead to problems, for example, when using gradient-based optimization methods on sparse grid functions. B-splines represent an interesting alternative to conventional basis types since they have displayed promising results for regression and optimization problems. However, their overlapping support impedes the task of hierarchization (computing the interpolant), as, in general, the solution of a linear system is required. To cope with this problem, we propose three general basis transformations. They leave the spanned function space on dimensionally adaptive sparse grids or full grids unchanged, but result in triangular linear systems. One of the transformations, when applied to the B-spline basis, yields the well-known fundamental splines. We suggest a modification of the resulting sparse grid basis to enable nearly linear extrapolation towards the domain’s boundary without the need to spend boundary points. Finally, we apply the hierarchical modified fundamental spline basis to gradient-based optimization with sparse grid surrogates.
This work was financially supported by the Ministry of Science, Research and the Arts of the State of Baden-Württemberg. We thank the referees for their valuable comments.
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