Goal-oriented error control of stochastic system approximations using metric-based anisotropic adaptations

Highlights: • We address shock-dominated problems where we control approximation errors (deterministic and stochastic) committed on a stochastic quantity of interest through anisotropic mesh adaptivity in both spaces. • A new error estimate of the stochastic error committed in approximating a stochastic quantity of interested is proposed, based on the interpolation error in the parameters space weighted by the probability density function. • The continuous framework of Riemannian metric space is extended to stochastic space. • An optimal metric is computed as a solution of the stochastic optimization problem where we seek the optimal simplex tessellation in the parameters space that minimizes the L1 norm of the interpolation error. • We propose an adaptive strategy to control the total error, where, given a computational budget, we quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces. The simulation of complex nonlinear engineering systems such as compressible fluid flows may be targeted to make more efficient and accurate the approximation of a specific (scalar) quantity of interest of the system. Putting aside modeling error and parametric uncertainty, this may be achieved by combining goal-oriented error estimates and adaptive anisotropic spatial mesh refinements. To this end, an elegant and efficient framework is the one of (Riemannian) metric-based adaptation where a goal-based a priori error estimation is used as indicator for adaptivity. This work proposes a novel extension of this approach to the case of aforementioned system approximations bearing a stochastic component. In this case, an optimization problem leading to the best control of the distinct sources of errors is formulated in the continuous framework of the Riemannian metric space. Algorithmic developments are also presented in order to quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces. The capability of the proposed method is tested on various problems including a supersonic scramjet inlet subject to geometrical and operational parametric uncertainties. It is demonstrated to accurately capture discontinuous features of stochastic compressible flows impacting pressure-related quantities of interest, while balancing computational budget and refinements in both spaces.