Goal-oriented uncertainty propagation using stochastic adjoints

Abstract

We propose a framework based on the use of adjoint equations to formulate an adaptive sampling strategy for uncertainty quantification for problems governed by algebraic or differential equations involving random parameters. The approach is non-intrusive and makes use of discrete sampling based on collocation on simplex elements in stochastic space. Adjoint or dual equations are introduced to estimate errors in statistical moments of random functionals resulting from the inexact reconstruction of the solution within the simplex elements. The approach is demonstrated to be accurate in estimating errors in statistical moments of interest and shown to exhibit super-convergence, in accordance with the underlying theoretical rates. Goal-oriented error indicators are then built using the adjoint solution and exploited to identify regions for adaptive sampling. The error-estimation and adaptive refinement strategy is applied to a range of problems including those governed by algebraic equations as well as scalar and systems of ordinary and partial differential equations. The strategy holds promise as a reliable method to set and achieve error tolerances for efficient aleatory uncertainty quantification in complex problems. Furthermore, the procedure can be combined with numerical error estimates in physical space so as to effectively manage a computational budget to achieve the best possible overall accuracy in the results.