New algorithms for optimal online checkpointing.

*(English)*Zbl 1214.65038The computation of derivatives for optimizing time-dependent problems is frequently based on integration of the adjoint differential equation. For this, the knowledge of the complete forward solution may be required. Similar information is needed in the context of a priori error estimation with respect to a given functional. In the area of flow control, especially for three-dimensional problems, it is usually impossible to keep track of the full forward solution due to lack of storage capacity. Moreover, for many problems, adaptive time stepping procedures are needed toward integration schemes in time. Therefore, standard optimal offline checkpointing strategies are usually not well suited in that framework.

In the present paper, two algorithms for an online checkpointing procedure are presented that determine the checkpoint distribution on the fly. It is proven that, using these algorithms, a wider range of applications can be covered with optimal or almost optimal checkpointing. Throughout, it is assumed that the time step lengths determined for the forward integration are also used for the adjoint integration, as is often done in the partial differential equation constraint optimization literature. Numerical results underline the theoretical findings.

In the present paper, two algorithms for an online checkpointing procedure are presented that determine the checkpoint distribution on the fly. It is proven that, using these algorithms, a wider range of applications can be covered with optimal or almost optimal checkpointing. Throughout, it is assumed that the time step lengths determined for the forward integration are also used for the adjoint integration, as is often done in the partial differential equation constraint optimization literature. Numerical results underline the theoretical findings.

Reviewer: Ludwig Kohaupt (Berlin)

##### MSC:

65K10 | Numerical optimization and variational techniques |

65Y20 | Complexity and performance of numerical algorithms |

90C30 | Nonlinear programming |

49N90 | Applications of optimal control and differential games |

68W40 | Analysis of algorithms |