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

TL;DR

This paper extends metric-based anisotropic adaptive methods for goal-oriented error control to stochastic system approximations, enabling efficient and accurate simulation of complex systems with uncertainties.

Contribution

It introduces a novel framework for goal-oriented error estimation and adaptive refinement in stochastic systems using Riemannian metrics, with algorithmic developments for error component adjustment.

Findings

Accurately captures discontinuous stochastic flow features

Balances computational cost with refinement accuracy

Demonstrates effectiveness on a supersonic scramjet inlet problem

Abstract

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 optimisation 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.