An Introduction and Summary of Use of Optimal Control Methods for PDE’s
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In optimal control formulations of partial differential equations the aim is to find a control function that steers the solution to a desired form. A Lagrange multiplier, i.e. an adjoint variable is introduced to handle the PDE constraint. One can reduce the problem to a two-by-two block matrix form with square blocks for which a very efficient preconditioner, PRESB can be applied. This method gives sharp and tight eigenvalue bounds, which hold uniformly with respect to regularization, mesh size and problem parameters, and enable use of the second order inner product free Chebyshev iteration method, which latter enables implementation on parallel computers without any need to use global data communications. Furthermore this method is insensitive to round-off errors. It outperforms other earlier published methods. Implementational and spectral properties of the method, and a short survey of applications, are given.
KeywordsOptimal control Preconditioner Inner product free
- 7.Axelsson, O., Liang, Z.-Z.: An optimal control framework for the iterative solution of a missing boundary data problem. Work in ProgressGoogle Scholar