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...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2006, Vol.27 (4), p.1305-1328
Main Authors: PRUDENCIO, Ernesto E, BYRD, Richard, CAI, Xiao-Chuan
Format: Article
Language:English
Subjects:
Algorithms
Boundary conditions
Calculus of variations and optimal control
Design optimization
Exact sciences and technology
Lagrange multiplier
Mathematical analysis
Mathematics
Methods
Microelectromechanical systems
Navier-Stokes equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Numerical methods in mathematical programming, optimization and calculus of variations
Partial differential equations, boundary value problems
Quadratic programming
Reynolds number
Sciences and techniques of general use
Variables
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Summary: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.
ISSN:1064-8275
1095-7197
DOI:10.1137/040602997