Development of an Efficient Response Surface Method for Highly Nonlinear Systems from Sparse Sampling Data Using Bayesian Compressive Sensing

Abstract

A main challenge for risk assessment on geotechnical systems is the computational effort required when stochastic sampling methods are used. Because the deterministic models used for geotechnical systems are often complicated and highly nonlinear, it is time-consuming to perform the deterministic analysis for each stochastic sample. The computational effort would become quite demanding, and even unrealistic, if direct Monte Carlo simulation (MCS) is used. To tackle this challenge, this study develops an efficient response surface method (RSM) that significantly improves computational efficiency and achieves the accuracy simultaneously. The proposed method is based on a novel sampling strategy called Bayesian compressive sensing (BCS). The proposed method is able to accurately reconstruct a highly nonlinear response surface from a small number of sampling points. Equations for the proposed RSM method are derived, and the attention is paid to extending the existing BCS method that deals only with low-dimensional data [e.g., one, two, or three-dimensional (1D, 2D, or 3D)] to high-dimensional data in RSM. The proposed method is illustrated using a highly nonlinear analytical function and a slope reliability analysis problem with consideration of spatial variability in soil properties. The results show that the proposed response surface method performs well and outperforms other response surface methods (e.g., response surface methods based on the kriging method or polynomial chaos expansion), particularly when sampling data are sparse.