A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver—FETI-DPH—for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising. Especially, in the full-space method with the new factorization, the number of iterations required for convergence is independent of regularization parameter values.

[1]  Philip E. Gill,et al.  Numerical methods for constrained optimization , 1974 .

[2]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[3]  P. Gill,et al.  Range-Space Methods for Convex Quadratic Programming. , 1982 .

[4]  Nicholas I. M. Gould,et al.  A weighted gram-schmidt method for convex quadratic programming , 1984, Math. Program..

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

[7]  Michael A. Saunders,et al.  Preconditioners for Indefinite Systems Arising in Optimization , 1992, SIAM J. Matrix Anal. Appl..

[8]  Roland W. Freund,et al.  Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices , 1992, SIAM J. Sci. Comput..

[9]  J. Mandel,et al.  Convergence of a substructuring method with Lagrange multipliers , 1994 .

[10]  R. Freund,et al.  A new Krylov-subspace method for symmetric indefinite linear systems , 1994 .

[11]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

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

[13]  Kay Hameyer,et al.  Algebraic multigrid for complex symmetric systems , 2000 .

[14]  David Day,et al.  Solving Complex-Valued Linear Systems via Equivalent Real Formulations , 2001, SIAM J. Sci. Comput..

[15]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

[16]  S. Reitzinger,et al.  Algebraic multigrid for complex symmetric matrices and applications , 2003 .

[17]  M. Heinkenschloss,et al.  Large-Scale PDE-Constrained Optimization , 2003 .

[18]  O. Widlund,et al.  A FETI-DP Corner Selection Algorithm for three-dimensional problems , 2003 .

[19]  Lorenz T. Biegler,et al.  SQP SAND Strategies that Link to Existing Modeling Systems , 2003 .

[20]  M. Heinkenschloss,et al.  Large-Scale PDE-Constrained Optimization: An Introduction , 2003 .

[21]  C. Farhat,et al.  FETI-DPH: A DUAL-PRIMAL DOMAIN DECOMPOSITION METHOD FOR ACOUSTIC SCATTERING , 2005 .

[22]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[23]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[24]  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..

[25]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange-Newton Solver and Its Application to Optimal Control of Steady Viscous Flows , 2005, SIAM J. Sci. Comput..

[26]  Andrew J. Wathen,et al.  Approximate Factorization Constraint Preconditioners for Saddle-Point Matrices , 2005, SIAM J. Sci. Comput..

[27]  Xiao-Chuan Cai,et al.  Parallel Full Space SQP Lagrange-Newton-Krylov-Schwarz Algorithms for PDE-Constrained Optimization Problems , 2005, SIAM J. Sci. Comput..

[28]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[29]  Anders Forsgren,et al.  Iterative Solution of Augmented Systems Arising in Interior Methods , 2007, SIAM J. Optim..

[30]  Martin Stoll,et al.  Combination Preconditioning and the Bramble-Pasciak+ Preconditioner , 2008, SIAM J. Matrix Anal. Appl..

[31]  H. Thorne.  Properties of linear systems in PDE-constrained optimization. Part I: Distributed control 1 , 2009 .

[32]  H. Thorne.  Properties of linear systems in PDE-constrained optimization. Part II : Neumann boundary control , 2009 .

[33]  C. Farhat,et al.  The FETI family of domain decomposition methods for inequality-constrained quadratic programming: Application to contact problems with conforming and nonconforming interfaces , 2009 .

[34]  Nicholas I. M. Gould,et al.  Preconditioning Saddle-Point Systems with Applications in Optimization , 2010, SIAM J. Sci. Comput..

[35]  Andrew J. Wathen,et al.  Optimal Solvers for PDE-Constrained Optimization , 2010, SIAM J. Sci. Comput..

[36]  Andrew J. Wathen,et al.  A new approximation of the Schur complement in preconditioners for PDE‐constrained optimization , 2012, Numer. Linear Algebra Appl..

[37]  Valeria Simoncini,et al.  Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems , 2012, Computational Optimization and Applications.

[38]  Martin Stoll,et al.  Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems , 2012, SIAM J. Matrix Anal. Appl..

[39]  S. ChoiY.  MINIMAL RESIDUAL METHODS FOR COMPLEX SYMMETRIC, SKEW SYMMETRIC, AND SKEW HERMITIAN SYSTEMS , 2013 .

[40]  Andrei Draganescu,et al.  Multigrid solution of a distributed optimal control problem constrained by the Stokes equations , 2011, Appl. Math. Comput..

[41]  Sou-Cheng T. Choi,et al.  Minimal Residual Methods for Complex Symmetric, Skew Symmetric, and Skew Hermitian Systems , 2013, ArXiv.

[42]  Youngsoo Choi,et al.  Simultaneous Analysis and Design in Pde-constrained Optimization a Dissertation Submitted to the Institute for Computational and Mathematical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy , 2013 .