Trends in PDE Constrained Optimization pp 67-84 | Cite as

# Optimal Design with Bounded Retardation for Problems with Non-separable Adjoints

## Abstract

In the natural and enginiering sciences numerous sophisticated simulation models involving PDEs have been developed. In our research we focus on the transition from such simulation codes to optimization, where the design parameters are chosen in such a way that the underlying model is optimal with respect to some performance measure. In contrast to general non-linear programming we assume that the models are too large for the direct evaluation and factorization of the constraint Jacobian but that only a slowly convergent fixed-point iteration is available to compute a solution of the model for fixed parameters. Therefore, we pursue the so-called One-shot approach, where the forward simulation is complemented with an adjoint iteration, which can be obtained by handcoding, the use of Automatic Differentiation techniques, or a combination thereof. The resulting adjoint solver is then coupled with the primal fixed-point iteration and an optimization step for the design parameters to obtain an optimal solution of the problem. To guarantee the convergence of the method an appropriate sequencing of these three steps, which can be applied either in a parallel (Jacobi) or in a sequential (Seidel) way, and a suitable choice of the preconditioner for the design step are necessary. We present theoretical and experimental results for two choices, one based on the reduced Hessian and one on the Hessian of an augmented Lagrangian. Furthermore, we consider the extension of the One-shot approach to the infinite dimensional case and problems with unsteady PDE constraints.

## Keywords

Simulation Optimization PDE Automatic differentiation Fixed-point solver Retardation factor One-shot Piggyback Numerics## Notes

### Acknowledgements

The work was funded by the DFG (Deutsche Forschungsgesellschaft) as part of the *DFG Schwerpunktprogramm 1253 – Optimization with partial differential equations*. Our sincere thanks are due to several other groups of the SPP 1253 (Bock et al., Schulz et al.) for their stimulating comments and discussions. We are especially grateful for the many helpful suggestions and for the encouraging interest shown by other research groups outside of the SPP 1253: Alonso et al., Farrell et al., Koziel et al., Oschlies et al., De los Reyes et al., Thiele et al., and Wang et al.

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