A Direct Method for Parabolic PDE Constrained Optimization by Andreas Potschka

By Andreas Potschka

Andreas Potschka discusses an immediate a number of capturing approach for dynamic optimization difficulties limited through nonlinear, very likely time-periodic, parabolic partial differential equations. unlike oblique equipment, this technique immediately computes adjoint derivatives with out requiring the person to formulate adjoint equations, which are time-consuming and error-prone. the writer describes and analyzes intimately a globalized inexact Sequential Quadratic Programming process that exploits the mathematical buildings of this process and challenge type for quick numerical functionality. The ebook good points functions, together with effects for a real-world chemical engineering separation problem.

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Numerical experience seems to suggest that more than one back projection step does not improve convergence considerably and should thus be avoided in all known cases. However, repeated back projection steps provide the theoretical benefit of making a proof of global convergence of the RMT possible. In particular, the RMT does not lead to iteration cycles on the notorious example by Ascher and Osborne [7] in contrast to the NMT. 6 Natural Monotonicity for LISA-Newton methods In this section we give a detailed description of an affine covariant globalization strategy for a Newton-type method based on iterative linear algebra.

27. 24. Let A, B ∈ GL(N) yield transformations of F, ˆ F = AF, ˆ J = AJB, ˆ −1 . M = B−1 MA Then LISA is affine invariant under A and B. Proof. Assume ζ i = B−1 ζi . Then we have ˆ −1 AJB ˆ −1 AFˆ ˆ B−1 ζi − B−1 MA ζ i+1 = (I − M J)ζ i − M F = I − B−1 MA = B−1 I − Mˆ Jˆ ζi − Mˆ Fˆ = B−1 ζi+1 . Induction yields the assertion. 25. A full-step LISA-Newton method is affine invariant under transformations A, B ∈ GL(N) with ˆ F(z) = AF(Bz) ˆ if the matrix function M(z) satisfies −1 ˆ . 6 Natural Monotonicity for LISA-Newton methods 55 Proof.

Then we denoted by rk+1 i can define an i-dependent inexact Simplified Newton step via = (−F(zk + αδ zk ) + rk ) + rk+1 . J(zk )δ zk+1 i i on α k in mind As in the above formula, we need to keep the dependence of δ zk+1 i but drop it in the notation for the sake of brevity. It is now paramount for the efficiency of the Newton-type method to balance the accuracies of the inner iterations with the nonlinearity of the problem. 11) which predict the solution to first order in α. We substitute the nonrealizable contraction factor Θk now by Θk = δ zk+1 / δ zk , which can be computed efficiently.

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