Parallel Multi-Level Restricted Schwarz with Pollution

Removing for PDE-Constrained Optimization.

Ernesto E. Prudencio and Xiao-Chuan Cai

Although Newton-Krylov-Schwarz (NKSz) methods have been

successfully applied to the numerical solution of many

simulation PDEs, little is known about their suitability to

the Karush-Kuhn-Tucker (KKT) systems arising from

PDE-constrained optimization problems.

                 In this talk we present multi-level

versions of the parallel fully coupled full space sequential

quadratic programming class of methods known as

Lagrange-Newton-Krylov-Schwarz (LNKSz).                   

                                                         In

LNKSz a Lagrangian functional is formed and differentiated

to obtain the KKT optimality system, which is then solved

with NKSz algorithms. The multi-level preconditioner for the

KKT Jacobian becomes a key component, combining restricted

Schwarz smoothers with a pollution removing coarse-to-fine

interpolation for the Lagrange multipliers.

Our tests involve the parallel numerical solution of some

boundary flow control problems with two-level LNKSz

and show the method providing both the robustness w.r.t.

Reynolds number and the scalability w.r.t. to number of

processors and mesh size.