By Andreas Potschka
Andreas Potschka discusses a right away a number of taking pictures strategy for dynamic optimization difficulties restricted by way of nonlinear, potentially time-periodic, parabolic partial differential equations. unlike oblique tools, this technique immediately computes adjoint derivatives with no requiring the person to formulate adjoint equations, which might be time-consuming and error-prone. the writer describes and analyzes intimately a globalized inexact Sequential Quadratic Programming procedure that exploits the mathematical constructions of this technique and challenge category for quick numerical functionality. The e-book gains functions, together with effects for a real-world chemical engineering separation problem.
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Additional info for A Direct Method for Parabolic PDE Constrained Optimization Problems
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). 26. Let ˆ κlin := σr (I − Mˆ J). ˆ ζ0 ∈ RN . 21) converges for all F, N ˆ Conversely, if LISA converges for all F, ζ0 ∈ R , then κlin < 1.
Otherwise we loose the possibility for structure exploitation which is important for numerical efﬁciency reasons as we discuss in Chapter 8. In this thesis we restrict ourselves to piecewise constant control discretizations in time for reasons of simplicity. We now introduce artiﬁcial initial values (sl,i , vi ), i = 0, . . , nMS , for the semidiscretized PDE states ul (t) and the ODE states v(t), respectively. 2a) v(t) ˙ = f ODE(l) (qi−1 , ul (t), v(t)), t ∈ Ii, v(t i−1 ) = vi−1 . 2b) has a unique solution, denoted by the pair (ul,i (t; qi−1 , sl,i−1 , vi−1 ), vl,i (t; qi−1 , sl,i−1 , vi−1 )).
There can be between 101 and 102 time steps per interval. Efﬁcient derivative generation. It is inevitable for the solution of large-scale optimization problems to use derivative-based methods. 3d). In Chapter 10 we describe such a method which efﬁciently computes consistent derivatives of ﬁrst and second order in an automated way. Structure exploitation. 3). 3). Furthermore we develop preconditioners in Chapter 6 which exploit special spectral properties of the shooting Wronksi matrices. These spectral properties arise due to ellipticity of the operator A.