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Uncertainty quantification for unsteady fluid flow using adjoint-based approaches

Uncertainty quantification of numerical simulations has raised significant interest in recent years. One of the main challenges remains the efficiency in propagating uncertainties from the sources to the quantities of interest, especially when there are many sources of uncertainties. The traditional Monte Carlo methods converge slowly and are undesirable when the required accuracy is high. Most modern uncertainty propagation methods such as polynomial chaos and collocation methods, although extremely efficient, suffer from the so called "curse of dimensionality". The computational resources required for these methods grow exponentially as the number of uncertainty sources increases. 
The aim of this work is to address the challenge of efficiently propagating uncertainties in numerical simulations with many sources of uncertainties. Because of the large amount of information that can be obtained from adjoint solutions, we focus on using adjoint equations to propagate uncertainties more efficiently. 
Unsteady fluid flow simulations are the main application of this work, although the uncertainty propagation methods we discuss are applicable to other numerical simulations. We first discuss how to solve the adjoint equations for time-dependent fluid flow equations. We specifically address the challenge associated with the backward time advance of the adjoint equation, requiring the solution of the primal equation in backward order. Two methods are proposed to address this challenge. The first method solves the adjoint equation forward in time, completely eliminating the need for storing the solution of the primal equation. The other method is a checkpointing algorithm specifically designed for dynamic time-stepping. The adjoint equation is still solved backward in time, but the present scheme retrieves the primal solution in reverse order. This checkpointing method is applied to an incompressible Navier-Stokes adjoint solver on unstructured mesh. 
With the adjoint equation solved, we obtain a linear approximation of the quantities of interest as functions of the random variables describing the uncertainty sources in a probabilistic setting. We use this linear approximation to accelerate the convergence of the Monte Carlo method in calculating tail probabilities for estimating margins and risk. In addition, we developed a multivariate interpolation scheme that uses multiple adjoint solutions to construct an interpolant of the quantities of interest as functions of the uncertainty sources. This interpolation scheme converge exponentially to the true function, thus providing very accurate and efficient means of propagating of uncertainties and remains accurate independently of the locations of the available data.