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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| Published in: | SIAM journal on scientific computing Vol. 27; no. 4; pp. 1305 - 1328 |
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| Language: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2006
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| ISSN: | 1064-8275, 1095-7197 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Byrd, Richard Cai, Xiao-Chuan Prudencio, Ernesto E. |
| Author_xml | – sequence: 1 givenname: Ernesto E. surname: Prudencio fullname: Prudencio, Ernesto E. – sequence: 2 givenname: Richard surname: Byrd fullname: Byrd, Richard – sequence: 3 givenname: Xiao-Chuan surname: Cai fullname: Cai, Xiao-Chuan |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Parallel Full Space SQP Lagrange--Newton--Krylov--Schwarz Algorithms for PDE-Constrained Optimization Problems |
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