Flow and shape reconstructions from remote measurements

We develop a point source method (PSM) to obtain flow field reconstructions from remote measurements. The PSM belongs to the class of decomposition methods in inverse scattering because it solves the nonlinear and ill-posed inverse shape reconstruction problem by a decomposition into a linear ill-posed problem and a nonlinear well-posed problem. As a model problem, we investigate the reconstruction of the flow field of two-dimensional stationary Oseen equation, display math which is obtained by linearizing the Navier–Stokes equation with kinematic viscosity μ > 0 around the constant velocity u0. In contrast to acoustics or electromagnetics, the use of the PSM in fluid dynamics leads to a number of challenges in terms of the analysis and the proper setup of the scheme, in particular, because the null-spaces of the integral operators under consideration are no longer trivial and the fundamental solution is not symmetric in its spatial coordinate. We provide a suitable formulation of the method and prove convergence of flow reconstructions by the PSM. For the realization of the reconstruction when the inclusions are not known, we employ domain sampling. We will demonstrate the feasibility of the method for reconstructing one or several objects by numerical examples