Parallel Preconditioners for KKT Systems Arising in Optimal Control of Viscous Incompressible Flows

摘要

Recently, interest has increased in model-based optimal flow control of viscous fluids, that is, the determination of optimal values of parameters for systems governed by the fluid dynamics equations. For example, the objective could be minimizing drag on a solid body, and the controls might consist of velocities or tractions on some part of the boundary or of the shape of the boundary itself. Such problems are among the most computationally challenging optimization problems. The complexity stems from their being constrained by numerical approximations of the fluid equations, commonly the Navier-Stokes or the Euler equations. These constraints are highly nonlinear and can number in the millions for typical systems of industrial interest. The current state-of-the-art for solving such flow-constrained optimization problems is reduced sequential quadratic programming (RSQP) methods. General mathematical analysis of these methods as well as CFD-related research have appeared. In addition, parallel implementations of RSQP methods exhibiting high parallel efficiency and good scalability have been developed. These methods essentially project the optimization problem onto the space of control variables (thereby eliminating the flow variables), and then solve the resulting reduced system using a quasi-Newton method. The advantage of such an approach is that only two linearized flow problems need to be solved at each iteration. However, the convergence of quasi-Newton based RSQP methods (QN-RSQP) deteriorates as the number of control variables increases, rendering large-scale problems intractable.