Parallel Full Space SQP Lagrange-Newton-Krylov-Schwarz Algorithms for PDE-Constrained Optimization Problems

Optimization problems constrained by nonlinear partial differential equations have been the focus of intense research in scientific computing lately. Current methods for the parallel numerical solution of such problems involve sequential quadratic programming (SQP), with either reduced or full space approaches. In this paper we propose and investigate a class of parallel full space SQP Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithms. In LNKSz, a Lagrangian functional is formed and differentiated to obtain a Karush--Kuhn--Tucker (KKT) system of nonlinear equations. An inexact Newton method with line search is then applied. At each Newton iteration the linearized KKT system is solved with a Schwarz preconditioned Krylov subspace method. We apply LNKSz to the parallel numerical solution of some boundary control problems of two-dimensional incompressible Navier--Stokes equations. Numerical results are reported for different combinations of Reynolds number, mesh size, and number of parallel processors. We also compare the application of the LNKSz method to flow control problems against the application of the Newton--Krylov--Schwarz (NKSz) method to flow simulation problems.

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