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

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.

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“…Fig 5 shows that for this mesh, for a given value of λ 2 , the influence of λ 1 on quality is not systematically monotonous, but, as could be expected, the accuracy tends to decrease when the size of the grid increases. However, it becomes monotonous when λ 2 is larger than 800 and λ 1 remains within the interval [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] mm. In this region of the plane (λ 1 , λ 2 ) we observed also a satisfactory level of distortion as captured by the Jacobian based measure J 10% .…”

“…The adaptive stochastic collocation method on simplex elements proposed by Van Langenhove et al [ 19 ] is used here for the following reasons: (a) it is non-intrusive, meaning the image registration tools are seen as a black box and no modification of the tools is required; (b) it is an adaptive approach with a constraint on the computational cost, meaning we should get the best possible response using a computational budget; (c) the method has been built to capture singularities such as rare events or localized high sensitivity of the model response. The main idea is to estimate the stochastic error on the quantity of interest by measuring the difference between the true value of the model output and the value estimated with the metamodel using an interpolation error on the parameter space (see below for more details).…”