By Andreas Potschka
Andreas Potschka discusses a right away a number of taking pictures strategy for dynamic optimization difficulties restricted by means of nonlinear, probably time-periodic, parabolic partial differential equations. not like oblique tools, this technique instantly computes adjoint derivatives with out requiring the consumer to formulate adjoint equations, that are time-consuming and error-prone. the writer describes and analyzes intimately a globalized inexact Sequential Quadratic Programming approach that exploits the mathematical constructions of this strategy and challenge type for speedy numerical functionality. The ebook gains purposes, together with effects for a real-world chemical engineering separation problem.
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Additional resources for A Direct Method for Parabolic PDE Constrained Optimization Problems
20 we have ≤ ρ i (1 + ρ i ) Δzk+1 − δ zk+1 . Δzk+1 − δ zk+1 i 0 Proof. 11) yield − (1 − α)δ zk ≤ Δzk+1 − (1 − α)δ zk + δ zk+1 − Δzk+1 δ zk+1 i i = (1 + ρ i ) Δzk+1 − (1 − α)δ zk . 13) on the left hand side then delivers the assertion. 21 is the inequality ρ i ≤ ρ i (1 + ρ i ). 14) to obtain ρ i = ρ i /(1 + ρ i ), or ρ i = ρ i /(1 − ρ i ) for ρ i < 1. 15), ρ i ≤ ρ max with ρ max ≤ 13 . 14) we can only conclude ρ i ≥ ρ i /(1 + ρ i ) (and not “≤”). 18 depends on the unknown hδk = ω δ zk which must be approximated.
The second assertion can be shown via k T k ˜ k )Δzk = −F(zk ) + rk − rk (Δz ) Δz = −F(zk ). , Nocedal and Wright ) we obtain rk (Δz ) k −1 J(zk )−1 (Δz k )T Δzk J(z ) k T k −1 ˜ ) M(z ) = J(z k k −1 = J(z ) + ) k −1 r 1 − (Δz(Δz k k )T Δzk J(z ) k T . The last assertion then follows from J(zk )−1 rk = Δzk − Δz∗ . 3) ˆ k ) ∈ RN×N . 2) is linear and based on a splitting ˆ k ) − ΔJ(zk ), J(zk ) = J(z ˆ k ) = J(z ˆ k )−1 . M(z ˆ k ) will be given by a Newton-Picard preconditioner (see ChapIn this thesis J(z ˆ k ) in this context, which include Jacobi, ter 6).
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. 24. The Newton-Picard preconditioners in Chapter 6 satisfy this relation at least partially which leads to scaling invariance of the Newton-Picard LISA-Newton method (see Chapter 6).