Abstract
We present an overview of the Lagrange-Newton-Krylov-Schur (LNKS) method for solution of optimization problems that are governed by systems of partial differential equations. We discuss how to improve LNKS’s work efficiency by carrying out certain computations inexactly, without compromising convergence. LNKS solves the Karush-Kuhn-Tucker optimality conditions using a NewtonKrylov algorithm. Its key component is a preconditioner based on variants of quasiNewton reduced space Sequential Quadratic Programming (QN-RSQP) methods. LNKS combines the fast convergence properties of a Newton method with the ability of preconditioned Krylov methods to solve very large linear systems. Nevertheless, even with good preconditioners, solution of the optimization problem is several times more expensive than solution of the underlying PDE problem. To accelerate LNKS, its computational components are carried out inexactly: premature termination of iterative algorithms, inexact evaluation of gradients and Jacobians, and approximate line searches. We discuss several issues that arise with respect to these inexact computations, and the resulting trade-offs between speed and robustness.
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Biros, G., Ghattas, O. (2003). Inexactness Issues in the Lagrange-Newton-Krylov-Schur Method for PDE-constrained Optimization. In: Biegler, L.T., Heinkenschloss, M., Ghattas, O., van Bloemen Waanders, B. (eds) Large-Scale PDE-Constrained Optimization. Lecture Notes in Computational Science and Engineering, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55508-4_6
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DOI: https://doi.org/10.1007/978-3-642-55508-4_6
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