ANOVA-based multi-fidelity probabilistic collocation method for uncertainty quantification

The probabilistic collocation method (PCM) has drawn wide attention in uncertainty quantification. Nevertheless, PCM may become prohibitively expensive for high-dimensional nonlinear problems. To alleviate the computational burden, we develop an ANOVA (analysis of variance)-based multi-fidelity probabilistic collocation method (AMF-PCM) in this study. Instead of directly approximating the computationally expensive system model (i.e., high-fidelity HF model) as in the traditional PCM, a computationally cheap while less accurate numerical model (i.e., low-fidelity LF model) is used in AMF-PCM. The central idea of AMF-PCM is to take advantage of both the accuracy of the HF model and the computational efficiency of the LF model. To address the high-dimensionality issue, we propose to respectively decompose the LF model and the discrepancy between the HF and LF models with functional ANOVA by approximating them with the summation of low-order ANOVA components using PCM. Then the final model approximation can be easily obtained from the combined results. The efficiency and accuracy of AMF-PCM are demonstrated by several numerical cases of coupled unsaturated flow and heat transport, where two ways of building a computationally cheap LF model (i.e., by simplifying the physics or using a coarser discretization) are employed. Compared to the traditional PCM that is solely based on the HF model, AMF-PCM achieves a better accuracy with a significantly lower computational cost.