Publications Details

Multilevel-multifidelity acceleration of PDE-constrained optimization

Monschke, Jason A.; Eldred, Michael S.

Many engineering design problems can be formulated in the framework of partial differential equation (PDE) constrained optimization. The discretization of a PDE leads to multiple levels of resolution with varying degrees of numerical solution accuracy. Coarse discretizations require less computational time at the expense of increased error. Often there are also reduced fidelity models available, with simplifications to the physics models that are computationally easier to solve. This research develops an up to second-order consistent multilevel-multifidelity (MLMF) optimization scheme that exploits the reduced cost resulting from coarse discretization and reduced fidelity to more efficiently converge to the optimum of a fine-grid high-fidelity problem. This scheme distinguishes multilevel approaches applied to discretizations from multifidelity approaches applied to model forms, and navigates both hierarchies to accelerate convergence. Additive, multiplicative, or a combination of both corrections can be applied to the sub-problems to enforce up to second-order consistency with the fine-grid high-fidelity results. The MLMF optimization algorithm is a wrapper around a subproblem optimization solver, and the MLMF scheme is provably convergent if the subproblem optimizer is provably convergent. Heuristics are developed for efficiently tuning optimization tolerances and iterations at each level and fidelity based on relative solution cost. Accelerated convergence is demonstrated for a simple one-dimensional problem and aerodynamic shape optimization of a transonic airfoil.