AD-based Matrix-free Methods in PDE-Constrained Optimization
by
D. Keyes
NASA
Coauthors: P. Hovland, L. McInnes, W. Samyono

A strict implementation of the Reduced Sequential Quadratic Programming (RSQP) technique for equality-constrained optimization requires direct inversions of the reduced Hessian, the Jacobian of the state constraints, and the Jacobian-transpose. For PDE-constrained optimization, in which the state space is very high-dimensional (e.g., millions of degrees of freedom), strict RSQP is therefore infeasible. However, conventional approximations (quasi-Newton approximations to the reduced Hessian, inexact Jacobian solves, omission of Hessian terms on the RHS of the adjoint step) lead to suboptimal departures from Newton's method applied to the optimality (Karush-Kuhn-Tucker, or KKT) conditions. Newton-like convergence can be preserved if the entire KKT system is solved iteratively with a Krylov acceleration method. This requires only matrix-vector multiplications with Hessian and Jacobian blocks. Automatic differentiation packages have evolved to the point where they now routinely supply such Hessian-vector and Jacobian-vector products in a matrix-free mode, at a fraction of the operation and storage cost of dealing with the full-dimensional matrix objects.

The challenge in dealing directly with the KKT system is in its preconditioning. Due to problem scale, it must be parallelizable. To keep the Krylov iterations in high-dimensional vector spaces affordable, convergence must be rapid. A variety of parallel preconditioners can be constructed based on the preconditioning of the underlying PDE problem and various approximations to the Hessian blocks.

Newton-Krylov-Schwarz (NKS) is among the leading rootfinding techniques for large-scale nonlinear systems arising from the implicit discretization of PDEs on parallel computers. Additive Schwarz preconditionings (including multilevel forms) for systems involving the Jacobian provide linear robustness with good concurrency and locality properties.

The purpose of this presentation is to illustrate an effective family of preconditioners developed for large-scale PDE problems and to show how they may be used in conjunction with AD capabilities to solve optimization problems with large-scale PDE constraints. Several new issues arise that are not present in the PDE analysis problem alone, and some preliminary progress in sorting them out is described. The specific context is steady-state problems in fluid dynamics.

Date received: March 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caev-04.