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Digital filter stepsize control in DASPK and its effect on control optimization performance. (English) Zbl 1226.49028

Biegler, Lorenz T. (ed.) et al., Real-time PDE-constrained optimization. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-21-4/pbk; 978-0-89871-893-5/ebook). Computational Science & Engineering, 183-195 (2007).
From the introduction: The problem is given by
\[ \min_{\mathbf u}\Phi({\mathbf u})= \int_0^T \Psi({\mathbf y},{\mathbf u},t)\,dt \quad\text{subject to}\quad {\mathbf F}(t,{\mathbf y},{\mathbf y}',{\mathbf u})=0, \quad {\mathbf y}(0)={\mathbf y}_0({\mathbf u}), \qquad g({\mathbf u})\leq 0, \]
where \(\Psi\) is the objective function, \({\mathbf F}\) is the DAE system which is a function of state \({\mathbf y}\) and control \({\mathbf u}\), and \(g({\mathbf u})\) are constraints. In the formulation considered here, the problem is solved by a shooting-type method: whenever the objective function needs to be evaluated for a given \({\mathbf u}\), we first solve the DAEs for \({\mathbf y}\). We select a derivative-based optimization method for two reasons: (a) the optimizer is fast and reliable, and (b) the DAE solver DASPK3.1 is able to efficiently compute the required derivatives via sensitivity analysis.
We implement the new digital controller in the large-scale DAE solver DASPK3.1, which has a facility for sensitivity analysis. The result version of DASPK, denoted DASPKmod, is tested on several simulation and sensitivity problems. It is found that its speed is generally compared to that of DASPK3.1, but its performance is more predictable for small perturbations in the problem or code parameters.
To determine whether the new stepsize controller could lead to faster convergence in the control of dynamical systems, we test the new solver in the optimization of an interesting problem from systems biology. We employ DASPKmod together with the derivative-based optimizer KNITRO to solve this optimization problem. The numerical results demonstrate that the optimization process (combined with DASPKmod) took substantially fewer iterations compared with the case when KNOTRO was used in conjunction with DASPK3.1.
For the entire collection see [Zbl 1117.49004].

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
34H05 Control problems involving ordinary differential equations

Software:

KNITRO; DASSL; DASPK 3.0
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