Identification of Robin Coefficient for Non-Stationary Stokes System Using the Levenberg–Marquardt Method

We consider the identification of a Robin coefficient $R$ defined on a nonaccessible part $\Gamma_{\text{out}}$ of the boundary, from noisy velocity measurements available on an accessible part $\Gamma_0$, for the nonstationary Stokes system with mixed Dirichlet, Neumann and Robin boundary conditions. This ill-posed inverse problem is formulated as a nonlinear least-squares problem, which we solve by the Levenberg--Marquardt method. The linearized subproblems are expressed by means of the sensitivity and adjoint problems. Under suitable assumptions, we prove that, for an initial guess close enough to the exact coefficient and a small noise level, the iterates satisfy a quadratic convergence estimate up to a term proportional to the noise level. For the discrete iteration, we further show that, under noise and under an explicit condition, the iterates started close enough to the least-squares solution converge linearly to it, at a rate bounded explicitly in terms of the smallest eigenvalue of the Gauss--Newton matrix, the noise level and the curvature of the forward map, and we derive an a posteriori stopping test. The discrete problem is verified with manufactured solutions and an exact discrete gradient, and the rates predicted from the eigenvalue and the noise level alone are reproduced by the experiments to two digits. On a bifurcation with two outlets, velocity measurements at the inlet determine the equivalent resistance of the outlets well, and the split of the coefficients between them poorly: a network model, in which the inlet velocity depends on the coefficients through the equivalent resistance only, explains the weak direction. When the data are generated by a finer model, the time discretization error of the model used for the inversion exceeds the effect of the coefficients on the data, and the iteration moves away from the exact coefficient; we quantify this limitation and propose a stopping test that accounts for the model error. The approach is motivated by applications to blood flow in cerebral aneurysms and to air flow in the lungs.