AMS MSC 2000: 49M37, 90C55, 90C06We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.

Abstract. We consider an implant shape design problem arising in the context of facial surgery. The aim is to find the shape of an implant that deforms the soft tissue of the skin in a desired way. Assuming sufficient regularity, we introduce a reformulation as an optimal control problem where the control acts as a boundary force. The solution of that problem can be used to recover the implant shape from the optimal state. For a simplified problem, in the case where the state can be modeled as a minimizer of a polyconvex hyperelastic energy functional, we show existence of optimal solutions and derive-on a formal level-first order optimality conditions. Finally, preliminary numerical results are presented for the original optimal control formulation.

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