High dimensional sensitivity analysis using surrogate modeling and high dimensional model representation

Kubicek, Martin and Minisci, Edmondo and Cisternino, Marco (2015) High dimensional sensitivity analysis using surrogate modeling and high dimensional model representation. International Journal for Uncertainty Quantification, 5 (5). pp. 393-414. ISSN 2152-5099 (https://doi.org/10.1615/Int.J.UncertaintyQuantific...)

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In this paper, a new non-intrusive method for the propagation of uncertainty and sensitivity analysis is presented. The method is based on the cut-HDMR approach, which is here derived in a different way and new conclusions are presented. The cut-HDMR approach decomposes the stochastic space into sub-domains, which are separately interpolated via a selected interpolation technique. This leads to a dramatic reduction of necessary samples for high dimensional spaces and decreases the influence of the Curse of Dimensionality. The proposed non-intrusive method is based on the coupling of an interpolation technique with the cut-HDMR (high dimension model representation) approach. The new conclusions obtained from the new derivation of the cut-HDMR approach allow one to interpolate each stochastic domain separately, including all stochastic variables and interactions between variables. Moreover, the same conclusions allow one to neglect non-important stochastic domains and therefore, drastically reduce the number of samples to detect and interpolate the higher order interactions. A new sampling strategy is introduced, which is based on a tensor product, but it uses the idea of Smoylak sparse grid for higher domains. For this work, the multi-dimensional Lagrange interpolation technique is selected and is applied for all parts of the cut-HDMR approach. However, the nature of the method allows one to use a combination of various interpolation techniques. The sensitivity analysis is performed on the surrogate model using the Monte Carlo sampling. In this work, the Sobol's approach is followed and sensitivity indices are established for each variable and interaction. Moreover, due to the obtained conclusions, the separate surrogate models allow one to visualize the uncertainty in the high dimensional space via histograms. The usage of a histogram for each stochastic domain allows one to establish full statistical properties of a given stochastic domain. This helps the user to better understand the stochastic propagation for the model of interest. The proposed interpolation technique and sensitivity analysis approach are tested on a simple example and applied on the well-known Borehole problem. Results of the proposed method are compared to the Monte Carlo sampling using the mean value and the standard deviation. Results of the sensitivity analysis of the Borehole case are compared to the literature results and the statistical visualization of each variable is provided.