Field and Shape Reconstruction in Fluid Dynamics

Abstract

Inverse problems are concerned with the reconstruction of quantities from remote measurements. Inverse fluid flow problems are important for many applications, for example for determining the state of the atmosphere from measurements on the planets surface and further remote sensing techniques. Here, we investigate the reconstruction of some fluid flow and shape reconstruction for inclusions within the flow from boundary measurements. As a model problem we consider the Oseen equation, which is obtained by linearizing the Navier-Stokes equations.In a first step we develop a point source method for the reconstruction of flow field from remote measurements. In contrast to field reconstructions in acoustics or electromagnetics, here we need a proper setup of the scheme as the fundamental solution of the Oseen equation is not symmetric or anti-symmetric in its arguments; moreover the null-spaces of the integral operators under consideration are no longer trivial, such that the corresponding convergence analysis of the point source method is particularly difficult.Further we extend our study to develop methods to test for analytic extensibility in fluid dynamics for the inverse fluid flow problems. We study and analyze three different approaches for the analytical continuation, the range test, the no-response test and a convergence test. We prove the convergence of these methods when applied to the Oseen equation. In particular, we exhibit a new approach to show convergence of the no-response test. A strong relationship between the convergence test and the no response test is shown.A numerical demonstration of the point source method and the convergence test is presented to exhibit the feasibility of these methods. To carry out the reconstructions we employ either domain sampling or the LASSO scheme is used for the reconstruction of flow field and the shape of unknown obstacles.