Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing
<|reference_start|>Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing: First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.<|reference_end|>
@article{cesarano2024homotopy,
    title={Homotopy methods for higher order shape optimization: A globalized
  shape-Newton method and Pareto-front tracing},
    author={A. Cesarano and B. Endtmayer and P. Gangl},
    journal={arXiv preprint arXiv:2405.03421},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.03421},
    primaryClass={math.NA cs.NA}
}