Balancing Neumann-Neumann Methods for Elliptic Optimal Control Problems

We present Neumann-Neumann domain decomposition preconditioners for the solution of elliptic linear quadratic optimal control problems. The preconditioner is applied to the optimality system. A Schur complement formulation is derived that reformulates the original optimality system as a system in the state and adjoint variables restricted to the subdomain boundaries. The application of the Schur complement matrix requires the solution of subdomain optimal control problems with Dirichlet boundary conditions on the subdomain interfaces. The application of the inverses of the subdomain Schur complement matrices require the solution of subdomain optimal control problems with Neumann boundary conditions on the subdomain interfaces. Numerical tests show that the dependence of this preconditioner on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only.

[1]  Jean-David Benamou,et al.  A Domain Decomposition Method with Coupled Transmission Conditions for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations , 1996 .

[2]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part I: The Krylov-Schur Solver , 2005, SIAM J. Sci. Comput..

[3]  J. Lions,et al.  Sur le contrôle parallèle des systèmes distribués , 1998 .

[4]  O. Widlund,et al.  Balancing Neumann‐Neumann methods for incompressible Stokes equations , 2001 .

[5]  James E. Dennis,et al.  A Comparison of Nonlinear Programming Approaches to an Elliptic Inverse Problem and a New Domain Dec , 1994 .

[6]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[7]  Hoang Nguyen,et al.  Neumann-Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems , 2006, SIAM J. Sci. Comput..

[8]  J. Mandel.  Balancing domain decomposition , 1993 .

[9]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[10]  Volker Schulz,et al.  Fast Solution of Discretized Optimization Problems , 2001 .

[11]  R. Freund,et al.  Software for simplified Lanczos and QMR algorithms , 1995 .

[12]  A. Bounaim,et al.  A Lagrangian Approach to a DDM for an Optimal Control Problem , 1998 .

[13]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[14]  R. Hoppe,et al.  Primal-Dual Newton-Type Interior-Point Method for Topology Optimization , 2002 .

[15]  E. Haber,et al.  A multigrid method for distributed parameter estimation problems. , 2003 .

[16]  O. Ghattas,et al.  Parallel Netwon-Krylov Methods for PDE-Constrained Optimization , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[17]  Ekkehard W. Sachs,et al.  Block Preconditioners for KKT Systems in PDE—Governed Optimal Control Problems , 2001 .