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

Christian Soize (1) , Charbel Farhat (2, 3, 4)

1 MSME - Laboratoire de Modélisation et Simulation Multi Echelle
2 ICME - Institute for Computational and Mathematical Engineering [Stanford]
3 Department of Mechanical Engineering [Stanford]
4 AA Stanford - Department of Aeronautics and Astronautics [Stanford]

A nonparametric probabilistic approach for modeling uncertainties in projection-based, nonlinear, reduced-order models is presented. When experimental data is 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.

Mots clés: reduced-order model; model order reduction; uncertainty quantification; modeling errors; model uncertainties; nonparametric stochastic approach

Domaines: Sciences de l'ingénieur [physics]; Probabilités [math.PR]; Statistiques [math.ST]; Mécanique [physics.med-ph]

Christian Soize, Charbel Farhat. A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and high-dimensional nonlinear models. International Journal for Numerical Methods in Engineering, 2017, 109, pp.837-888. ⟨10.1002/nme.5312⟩. ⟨hal-01353194⟩