Solution of linear systems from an optimal control problem arising in wind simulation

Several solution strategies for a class of large, sparse linear systems with a block 2 × 2 structure arising from the finite element discretization of an optimal control problem in wind simulation are introduced and analyzed. Block preconditioners and a sparse direct solver on the original coupled system are compared with a preconditioned GMRES iteration applied to a reduced system (Schur complement). Theoretical and experimental results demonstrate the effectiveness of the reduced system approach. Copyright © 2009 John Wiley & Sons, Ltd.

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

[2]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[3]  J. Scott,et al.  HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D , 2010 .

[4]  Jacques Simon,et al.  High definition local adjustment model of 3D wind fields performing only 2D computations , 2011 .

[5]  Ronald B. Morgan,et al.  GMRES with Deflated Restarting , 2002, SIAM J. Sci. Comput..

[6]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[7]  Patrick R. Amestoy,et al.  An Approximate Minimum Degree Ordering Algorithm , 1996, SIAM J. Matrix Anal. Appl..

[8]  D. Whittaker,et al.  A Course in Functional Analysis , 1991, The Mathematical Gazette.

[9]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[10]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[11]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[12]  Gene H. Golub,et al.  A Preconditioner for Generalized Saddle Point Problems , 2004, SIAM J. Matrix Anal. Appl..

[13]  J. Boyle,et al.  HSL MI 20 : an efficient AMG preconditioner , 2007 .

[14]  M. Isabel Asensio,et al.  A convection model for fire spread simulation , 2005, Appl. Math. Lett..

[15]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[16]  J. Lions,et al.  Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles , 1968 .

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

[18]  Ilse C. F. Ipsen.  A Note on Preconditioning Nonsymmetric Matrices , 2001, SIAM J. Sci. Comput..