Download e-book for kindle: A Direct Method for Parabolic PDE Constrained Optimization by Andreas Potschka

By Andreas Potschka

ISBN-10: 3658044756

ISBN-13: 9783658044756

ISBN-10: 3658044764

ISBN-13: 9783658044763

Andreas Potschka discusses an immediate a number of capturing technique for dynamic optimization difficulties limited by means of nonlinear, probably time-periodic, parabolic partial differential equations. unlike oblique equipment, this process immediately computes adjoint derivatives with no requiring the consumer to formulate adjoint equations, which might be time-consuming and error-prone. the writer describes and analyzes intimately a globalized inexact Sequential Quadratic Programming process that exploits the mathematical constructions 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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Extra info for A Direct Method for Parabolic PDE Constrained Optimization Problems

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3d) i = 0, ) ≥ 0, re (vnMS ) ≥ 0. 3e) i = 1, . . 3g) Throughout we assume that all discretized functions are sufficiently smooth to apply the numerical methods of Part II. This includes that all functions need to be at least twice continuously differentiable. In the case of the functions f PDE(l) and f ODE(l) we might even need higher regularity to allow for efficient adaptive error control in the numerical integrator. 3) we have developed an inexact Sequential Quadratic Programming (SQP) method which we describe in Part II.

1: The two main routes to approximate an infinite dimensional optimization problem (upper left box) with necessary finite dimensional optimality conditions (lower right box) are direct approaches (discretize-then-optimize, lower left path) versus indirect approaches (optimize-then-discretize, upper right path). property is paramount for successful deployment of the method in collaboration with practitioners. We have chosen a Direct Multiple Shooting approach for the following numerical advantages: First, it has been shown in Albersmeyer [2] and Albersmeyer and Diehl [4] that Multiple Shooting can be interpreted as a Lifted Newton method which might reduce the nonlinearity of a problem and enlarge thus the domain of fast local convergence (see Chapter 5).

22. Let an inexact Newton method with step sizes α = [α k ] be realized. , 0 ≤ hδk − [hδk ] < σ max(1/(1 + ρ), [hδk ]) for some σ < 1. Then the exact natural contraction factor satisfies Θk = Δzk+1 1 − σ (1 + 2ρ) α. < 1− 2 + ρ(1 − σ ) Δzk The inexact natural contraction factor is bounded by Θk = δ zk+1 1 (1 + ρ)(1 + σ ) < 1− 1− α. 18) 52 5 Inexact Sequential Quadratic Programming Proof. 7) in the form Θk ≤ 1 − 1 − αhδk δk − 1 − δk 2(1 − δk ) α. 19) We now find a bound for αhδk with optimal realizable step size α = [α k ] = min(1, 1/ (1 + ρ)[hδk ] ).

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A Direct Method for Parabolic PDE Constrained Optimization Problems by Andreas Potschka

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