# ITN SADCO "Large scale systems: sensitivity analysis and numerical methods" (Task 1.4)

**start of the project: **
2013
,
**end of the project: **
2014

**contract number: **
264735-SADCO (grant agreement number)

**funding institution: **
European Commission (EU)

## project members

### principal investigator

Task 1.4 a): Prof. Dr. Kurt Chudej and Prof. Dr. Hans Josef Pesch (both: University of Bayreuth)

Task 1.4 b): Prof. Dr. Christof Büskens (University of Bremen) and Astos Solutions (Stuttgart)

### project members

## aims of the project

Large scale systems arise in many engineering applications and provide a major challenge both for the analysis and for the development of efficient numerical methods, which we intend to address in this project. Those problems are typical for multi-physics problems as they arise, for instance, in minimum fuel hypersonic flight trajectory optimization, in the optimization of fast and secure load changes for fuel cell systems

[1] K. Chudej, H.-J. Pesch, und K. Sternberg,

Optimal control of load changes for molten carbonate fuel cell systems: A challenge in pde constrained optimization,

SIAM Journal on Applied Mathematics, 70 (2009), pp. 621–639.

or in modern laser welding techniques when the position, size and intensity of auxiliary laser beams are optimized in order to avoid hot cracking. Due to the complexity of those problems, we focus on the approach *first discretize, then optimize*, since adjoint equations can hardly be established here. Nevertheless, we will preserve some of their advantages.

### Task 1.4 a)

(Task 1.4-a) will focus on systems resulting from the discretization of systems of PDEs or of equations of different type such as partial differential algebraic equations, where parabolic equations play a dominant role. Especially, the particular structure of the huge scale finite dimensional nonlinear programming problems and the interplay between optimality conditions and discretization schemes shall be exploited. A promising numerical method in this direction is the primal-dual augmented Lagrangian method suggested in

[2] A. Forsgren and P. Gill, Primal-dual interior methods for nonconvex nonlinear programming,

SIAM J. Optimization, 8 (1998), pp. 1132–1152.[3] P. E. Gill and D. P. Robinson, A primal-dual augmented lagrangian, Numerical Analysis Report 08–2,

Department of Mathematics, University of California, San Diego, La Jolla, CA, 2008.

which couples augmented Lagrangian methods and primal-dual interior-point methods. In addition to the full discretization approach outlined above, domain decomposition methods in the spirit of multiple shooting techniques, see

[4] M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems,

J. Comput. Appl. Math., 173 (2005), pp. 169–198.[5] S. Ulbrich, Generalized sqp-methods with ‘parareal’ time-domain decomposition for time-dependent pde-constrained optimization,

in "Real-Time PDE-Constrained Optimization", L. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, eds., 2007, pp. 145–168.

shall be investigated for parabolic PDE optimal control problems. Furthermore, estimates of the discrete adjoint variables representing approximation of the continuous adjoint variables will be investigated. These topics will be studied in close collaboration with Tasks 1.2 and 1.3.

### Task 1.4 b)

(Task 1.4-b) concerns the efficient solution of the nonlinear program (NLP) resulting from a direct discretization of an optimal control problem. In the last years NLP solvers were developed which use the exact sparse Hessian matrix, calculated, e.g. by automatic differentiation (AD) or finite differences with special group strategies. From the exact Hessian and Jacobian the full KKT-matrix is available and hence a parametric sensitivity analysis of general perturbed solutions of NLP problems in the sense of Fiacco und Robinson can be performed. First numerical algorithms based on this approach have been shown by Christof Büskens to be suitable for real-time applications and have been successfully implemented in the SQP solver WORHP. Motivated by these results, we will investigate sensitivity based approaches for further increasing the efficiency of NLP solvers, like, e.g., regularisation of the Hessian for globalisation or appropriate stepsize control techniques. Furthermore, the relation to the techniques from Task 1.5 will be studied. The resulting routines will be made available for all interested users within the network.

Further information is available:

- description of the Post-doc position for task 1.4 a) (position no longer available)
- description of the PhD position for task 1.4 b) (position no longer available)
- homepage of the German SADCO team
- homepage of the ITN SADCO network
- International Training Network (ITN)
- Seventh Framework Programme (FP 7) of the European Commission