Bayesian uncertainty quantification and propagation using adjoint techniques

Abstract

This paper presents the Bayesian inference framework enhanced by analytical approximations for uncertainty quantification and propagation and parameter estimation. A Gaussian distribution is used to approximate the posterior distribution of the uncertain parameters. The most probable value of the parameters is obtained by minimizing the function defined as the minus of the logarithm of the posterior distribution and the covariance matrix of this posterior distribution is defined using asymptotic expansion as the inverse of the Hessian matrix of the aforementioned function, which is defined by the deviation of the computed quantities from corresponding experimental measurements. The gradient and the Hessian matrix of the objective function are computed using first and second-order adjoint approaches, respectively. The asymptotic approximation is also used to propagate the computed uncertainties of the model parameters to compute the uncertainty of the value of a quantity of interest. The presented approach is applied to the estimation of the uncertainties in the parameters of the Spalart-Allmaras turbulence model, based on experimental measurements that account for velocity and Reynolds stress distributions.