A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

Summary A nonparametric probabilistic approach for modeling uncertainties in projection‐based, nonlinear, reduced‐order models is presented. When experimental data are available, this approach can also quantify uncertainties in the associated high‐dimensional models. The main underlying idea is twofold. First, to substitute the deterministic reduced‐order basis (ROB) with a stochastic counterpart. Second, to construct the probability measure of the stochastic reduced‐order basis (SROB) on a subset of a compact Stiefel manifold in order to preserve some important properties of a ROB. The stochastic modeling is performed so that the probability distribution of the constructed SROB depends on a small number of hyperparameters. These are determined by solving a reduced‐order statistical inverse problem. The mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations. Its potential is demonstrated through several example problems from computational structural dynamics.