Numerical Methods for Optimal Control Problems by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9783030019594
Publisher: Springer International Publishing

This is indeed the discretization of the Jacobian for the problem

that can be further approximated by means of the preconditioner proposed in [31]. The application of the preconditioner can be done either in a direct way or in the form of a multi-iterative scheme in which the application of P (2) is approximated iteratively with a Krylov preconditioned method as in [31].

Remark 1

Observe that we could have used the spectral approximations from Proposition 2 also for building preconditioners for the Box-constrained case, i.e., to solve the sequences of linear systems generated by the state and adjoint equations. This would have implied going toward a multi-iterative scheme in which we need to use another iterative method to apply these spectrally approximated preconditioners which are not anymore explicit preconditioners; see [17, 25] for some solutions in this direction. In here we preferred a more straightforward approach. We computed directly, from the matrices of the problems, sparse preconditioners in explicit form such that their application is performed via low-cost matrix-vector product; see again [10, 14] for these issues.

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