Abstract
Propagation of random variables through computer codes of many inputs is primarily limited by computational expense. The use of sparse grids mitigates these costs somewhat; here we show how Sobol indices can be used to perform dimension adaptivity to mitigate them further. The method is compared to conventional adaptation schemes on sparse grids (Gerstner and Griebel, Computing 71(1), 65–87, 2003), and seen to perform comparably, without requiring the expense associated with a look-ahead error estimate. It is demonstrated for an expensive computer model of contaminant flow over a barrier.
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Notes
- 1.
Incidentally, a much weaker condition on f than that required by (4).
- 2.
This can be seen by considering what space of polynomials can be reconstructed on the sparse grid, namely \(\phi (x,y) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}y + a_{4}y^{2}\), which include no interaction (xy) terms.
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Acknowledgements
Special thanks to Andreas Mack for providing modelling and CFD simulation for the heavy gas release test-case.
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Dwight, R.P., Desmedt, S.G.L., Omrani, P.S. (2016). Sobol Indices for Dimension Adaptivity in Sparse Grids. In: Koziel, S., Leifsson, L., Yang, XS. (eds) Simulation-Driven Modeling and Optimization. Springer Proceedings in Mathematics & Statistics, vol 153. Springer, Cham. https://doi.org/10.1007/978-3-319-27517-8_15
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