By Andreas Potschka
Andreas Potschka discusses a right away a number of taking pictures process for dynamic optimization difficulties limited by means of nonlinear, in all likelihood time-periodic, parabolic partial differential equations. not like oblique tools, this process immediately computes adjoint derivatives with out requiring the person to formulate adjoint equations, that are time-consuming and error-prone. the writer describes and analyzes intimately a globalized inexact Sequential Quadratic Programming procedure that exploits the mathematical buildings of this technique and challenge classification for quick numerical functionality. The ebook positive aspects 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
33]). The explicit algorithmic control of Jacobian approximations is usually enforced via an adaptively chosen termination criterion for an inner preconditioned Krylov solver for the solution of the linearized system. 3 Globalization of convergence 39 solve the linear system exactly (see Battermann and Heinkenschloss , Battermann and Sachs , Biros and Ghattas ). 5 is already possible. Our aim in this section is to present the theoretical tools to understand this undesirable effect and to introduce the idea of monotonicity tests.
The ﬁrst theorem which relies on ω characterizes step sizes αk which yield optimal values for a local descent estimate of generalized level functions T (z|A) in the Newton method. 16. Let D be convex, J(z) nonsingular for all z ∈ D. Let furthermore J satisfy the ω-condition in D, zk ∈ D, A ∈ GL(N), and G(zk |A) ⊂ D. Let Δzk denote the Newton direction and deﬁne the Kantorovich quantities hk := Δzk ω, hk := hk cond(AJ(zk )). Then we obtain for α ∈ [0, min(1, 2/hk (A))] that AF(zk + αΔzk ) ≤ tk (α|A) AF(zk ) , where tk (α|A) := 1 − α + (1/2)α 2 hk (A).
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. 8) to a computable estimate of the optimal step size [α k ] = min(1, (1 − 2δk )/[hδk ]) = min(1, 1/ (1 + ρ)[hδk ] ). 20 we obtain an a-posteriori Kantorovich estimate [hδk ]i = − δ zk+1 2 δ zk+1 i 0 (1 + ρ i )α 2 δ zk = − (1 − α)δ zk 2 δ zk+1 i (1 + ρ i )α 2 δ zk ≤ hδk .
A Direct Method for Parabolic PDE Constrained Optimization Problems by Andreas Potschka