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Hierarchical Gradient-Based Optimization with B-Splines on Sparse Grids

  • Julian ValentinEmail author
  • Dirk Pflüger
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 109)

Abstract

Optimization algorithms typically perform a series of function evaluations to find an approximation of an optimal point of the objective function. Evaluations can be expensive, e.g., if they depend on the results of a complex simulation. When dealing with higher-dimensional functions, the curse of dimensionality increases the difficulty of the problem rapidly and prohibits a regular sampling. Instead of directly optimizing the objective function, we replace it with a sparse grid interpolant, saving valuable function evaluations. We generalize the standard piecewise linear basis to hierarchical B-splines, making the sparse grid surrogate smooth enough to enable gradient-based optimization methods. Also, we use an uncommon refinement criterion due to Novak and Ritter to generate an appropriate sparse grid adaptively. Finally, we evaluate the new method for various artificial and real-world examples.

Keywords

Grid Generation Sparse Grid Macro Cell Chebyshev Point Grid Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was financially supported by the Juniorprofessurenprogramm of the Landesstiftung Baden-Württemberg.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute for Parallel and Distributed Systems (IPVS)Universität StuttgartStuttgartGermany

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