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 Vol. 27; no. 4; pp. 1305 - 1328
Main Authors: Prudencio, Ernesto E., Byrd, Richard, Cai, Xiao-Chuan
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
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
76D55
Krylov subspace method
Computing
Partial differential equation
Schwarz method
Non linear equation
49K20
65F10
Navier Stokes equation
Sequential method
Numerical solution
incom-pressible Navier-Stokes equations
65K10
90C90
93C20
Schwarz preconditioners
Newton method
65N55
parallel computing
Mathematical programming
Control flow
65K05
Reynolds number
Initial value problem
90C06
Quadratic programming
flow control
65Y05
Constrained optimization
Numerical analysis
Boundary value problem
Scientific computation
76N05
nonlinear partial differential equations constrained optimization
inexact Newton method
Two-dimensional calculations
Parallel computation
Domain decomposition
Preconditioning
65N22
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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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.
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ISSN:1064-8275
1095-7197
DOI:10.1137/040602997