A preconditioning technique for a class of PDE-constrained optimization problems
A preconditioning technique for the linear equations resulting in the application of the inexact Newton method to the solution of discretized PDE constrained optimization problems is presented. The block triangular preconditioner \(P\) is based on a perturbed version of the Gauß-Newton Hessian. \(P\) is used as preconditioner for FGMRES. The Schur complement systems arising in the inversion of \(P\) are solved by a PCG method with ML smoothed aggregation AMG as preconditioner. For numerical experiments, a simplified version of the Monge-Kantorovich mass transfer problem involving a hyperbolic PDE constraint is investigated. The inexact Newton method converges rapidly, the number of FGMRES iterations is low, even in the case where the solutions of the Schur complement systems are only approximated with low relative accuracy. The computational effort does not scale well with the problem size, due to some problems going back to the ML preconditioner. Nevertheless, the new approach seems to be competitive in comparison to reduced-Hessian based methods.
scientific article; zbMATH DE number 5988495
scientific article