Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver
Large-scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. Reduced quasi-Newton sequential quadratic programming (SQP) methods are state-of-the-art approaches for such problems. These methods take full advantage of existing...
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| Veröffentlicht in: | SIAM journal on scientific computing Jg. 27; H. 2; S. 687 - 713 |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2005
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| ISSN: | 1064-8275, 1095-7197 |
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| Abstract | Large-scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. Reduced quasi-Newton sequential quadratic programming (SQP) methods are state-of-the-art approaches for such problems. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of using a full space Newton solver combined with an approximate reduced space quasi-Newton SQP preconditioner. The basic components of the method are Newton solution of the first-order optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the Karush--Kuhn--Tucker (KKT) linear systems arising at each Newton iteration using a symmetric quasi-minimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation initialized by a two-step stationary method. Accordingly, we term the new method {\it Lagrange--Newton--Krylov--Schur} (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In part I of this two-part article, we investigate the effectiveness of the KKT linear system solver. We test our method on two optimal control problems in which the state constraints are described by the steady-state Stokes equations. The objective is to minimize dissipation or the deviation from a given velocity field; the control variables are the boundary velocities. Numerical experiments on up to 256 Cray T3E processors and on an SGI Origin 2000 include scalability and performance assessment of the LNKS algorithm and comparisons with reduced SQP for up to $1,000,000$ state and50,000 decision variables. In part II of the article, we address globalization and inexactness issues, and apply LNKS to the optimal control of the steady incompressible Navier--Stokes equations. |
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| AbstractList | Large-scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. Reduced quasi-Newton sequential quadratic programming (SQP) methods are state-of-the-art approaches for such problems. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of using a full space Newton solver combined with an approximate reduced space quasi-Newton SQP preconditioner. The basic components of the method are Newton solution of the first-order optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the Karush--Kuhn--Tucker (KKT) linear systems arising at each Newton iteration using a symmetric quasi-minimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation initialized by a two-step stationary method. Accordingly, we term the new method {\it Lagrange--Newton--Krylov--Schur} (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In part I of this two-part article, we investigate the effectiveness of the KKT linear system solver. We test our method on two optimal control problems in which the state constraints are described by the steady-state Stokes equations. The objective is to minimize dissipation or the deviation from a given velocity field; the control variables are the boundary velocities. Numerical experiments on up to 256 Cray T3E processors and on an SGI Origin 2000 include scalability and performance assessment of the LNKS algorithm and comparisons with reduced SQP for up to $1,000,000$ state and50,000 decision variables. In part II of the article, we address globalization and inexactness issues, and apply LNKS to the optimal control of the steady incompressible Navier--Stokes equations. |
| Author | Biros, George Ghattas, Omar |
| Author_xml | – sequence: 1 givenname: George surname: Biros fullname: Biros, George – sequence: 2 givenname: Omar surname: Ghattas fullname: Ghattas, Omar |
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| Cites_doi | 10.1016/S0045-7930(98)00050-4 10.1016/0956-0521(92)90116-Z 10.1002/nme.1620380209 10.1007/BF01588794 10.1088/0266-5611/17/6/319 10.1137/1.9781611970937 10.1137/0913013 10.1137/0908020 10.1137/0805017 10.1137/S1064827595281575 10.1002/(SICI)1097-0207(19970815)40:15<2759::AID-NME188>3.0.CO;2-T 10.1137/0729100 10.1137/S1064827502415661 10.1137/S1052623497327854 10.1006/jcph.1997.5744 10.1007/s007910050008 10.1017/CBO9780511624100 10.1137/0723011 10.1137/0915022 10.1007/b98874 10.1137/S0895479893245371 |
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| Keywords | 65W10 76D55 Krylov subspace method Partial differential equation Non linear equation 90C52 Sequential method 90C55 65K10 90C90 Navier-Stokes Schur complement Optimal control (mathematics) Lagrangian method State constraint Mathematical programming Parallel algorithm Numerical linear algebra Initial value problem Adjoint method 65Y20 76D05 76D07 Lagrange-Newton-Krylov-Schur methods nonlinear equations Numerical analysis Boundary value problem Optimality condition Lagrangian function Preconditioning preconditioners Direct method 65F10 Navier Stokes equation 93C20 adjoint methods Newton method 65N55 indefinite systems 65K05 PDE-constrained optimization parallel algorithms 49K20 Step method optimal control Quadratic programming 65Y05 Constrained optimization Two step method Linear system Scientific computation Matrix inversion 65J22 sequential quadratic programming finite elements Finite element |
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| Title | Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver |
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