Iterative Looping Methods for Navier--Stokes Inverse Problems

Navier--Stokes inverse problems have applications ranging from supernovae to numerical weather prediction. Direct Adjoint Loooping (DAL) is the defacto approach for solving retrospective (backward in time) inverse problems, but it has not been applied to deterministic Navier--Stokes inverse problems in 2D or 3D. We demonstrate that DAL is ill-suited for solving retrospective 2D Navier--Stokes inverse problems. Alongside DAL, we study two other iterative looping methods: Simple Backward Integration (SBI) and the Quasi-Reversible Method (QRM). Our iterative SBI approach is novel while iterative QRM has been used previously for linear inverse problems. Both methods approximate the target state with more accuracy than DAL by an astonishing margin. We attribute this performance gap to additional terms present in SBI and QRM's respective backward integrations which are absent in DAL.