A generalized multi-fidelity simulation method using sparse polynomial chaos expansion

Abstract

The polynomial chaos expansion (PCE) method has received considerable attention in uncertainty quantification (UQ). Nevertheless, it is well known that the computational cost of PCE becomes expensive, or even unaffordable for high-dimensional problems (i.e., curse of dimensionality). To alleviate the computational burden, many multi-fidelity PCE methods have been proposed, which work by approximating the high-fidelity (HF) model with the sum of a low-fidelity (LF) model and a correction function. Although the current multi-fidelity PCE methods have proven to be effective in the examples they tested, their accuracy has a strong dependence on the LF model selection. Aimed at this issue, we develop a generalized multi-fidelity PCE (GMF-PCE) using the control variate method. Specifically, an adjustable coefficient is assigned to the LF model and its optimal value is theoretically derived. Furthermore, a flexible yet still simple enough way is provided to implement the developed method, i.e., the sparse PCE based on the least angle regression (LAR) is employed to approximate the LF model, and then the subset of LF-PCE expansion is automatically detected using LAR and corrected via HF computations. Two classic algebraic examples for UQ, namely the borehole problem and the Ishigami function, as well as an unsaturated flow and heat transport problem are used to examine the performance of GMF-PCE. The results show that with the same computational cost, GMF-PCE can achieve much higher accuracy compared to the sparse LAR-based PCE and MF-PCE (non-generalized version).