Superlinear Convergence of the GMRES for PDE-Constrained Optimization Problems

ABSTRACT Optimal control problems for PDEs arise in many important applications. A main step in the solution process is the solution of the arising linear system, where the crucial point is usually finding a proper preconditioner. We propose both proper block diagonal and more involved preconditioners, and derive mesh independent superlinear convergence of the preconditioned GMRES iterations based on a compact perturbation property of the underlying operators.

[1]  Thomas A. Manteuffel,et al.  On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations , 1990 .

[2]  M. A. Kaashoek,et al.  Classes of Linear Operators Vol. I , 1990 .

[3]  Martin Stoll,et al.  Preconditioners for state‐constrained optimal control problems with Moreau–Yosida penalty function , 2014, Numer. Linear Algebra Appl..

[4]  Owe Axelsson,et al.  Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators , 2004, Numerische Mathematik.

[5]  Owe Axelsson,et al.  Mesh Independent Superlinear PCG Rates Via Compact-Equivalent Operators , 2007, SIAM J. Numer. Anal..

[6]  Tuomo Rossi,et al.  A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension , 1999, SIAM J. Sci. Comput..

[7]  P. Philip.  Optimal Control of Partial Dierential Equations , 2013 .

[8]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[9]  Owe Axelsson,et al.  Equivalent operator preconditioning for linear elliptic problems , 2008 .

[10]  Barbara Kaltenbacher,et al.  Iterative Solution Methods , 2015, Handbook of Mathematical Methods in Imaging.

[11]  Owe Axelsson,et al.  A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control , 2017, J. Comput. Appl. Math..

[12]  Owe Axelsson,et al.  An efficient preconditioning method for state box-constrained optimal control problems , 2018, J. Num. Math..

[13]  Owe Axelsson,et al.  Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems , 2016, Numerical Algorithms.

[14]  M. Schultz,et al.  Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations , 1986 .

[15]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[16]  Thomas A. Manteuffel,et al.  Preconditioning and boundary conditions without H 2 estimates: L 2 condition numbers and the distribution of the singular values , 1993 .

[17]  T. Rees,et al.  A Fast Solver for anH1 Regularized PDE-Constrained Optimization Problem , 2015 .

[18]  Owe Axelsson,et al.  Equivalent operator preconditioning for elliptic problems , 2009, Numerical Algorithms.

[19]  Owe Axelsson,et al.  Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems , 2016, Numerical Algorithms.

[20]  Owe Axelsson,et al.  Superlinear convergence using block preconditioners for the real system formulation of complex Helmholtz equations , 2018, J. Comput. Appl. Math..

[21]  Martin Stoll,et al.  A fast solver for an H1 regularized PDE-constrained optimization problem , 2016 .

[22]  I. Moret.  A Note on the Superlinear Convergence of GMRES , 1997 .