The range test and the no response test for Oseen problems: Theoretical foundation

摘要

The reconstruction of unknown shapes and inclusions is an important task for many applied sciences. In electromagnetics and acoustics waves are used to probe some unknown region. However, often different types of measurements are available in industrial or natural applications, such as measurements of fluid flow at some boundary of a region in space. Then, the task is to detect the location and shape of an unknown inclusion from these remote flow measurements.The goal of this work is to develop the theory to establish two new methods for finding unknown inclusions embedded in some fluid which is governed by the Oseen equation. We will introduce and analyze a range test based on ideas of Kusiak, Potthast and Sylvester (2003) and the no-response test as suggested by Luke and Potthast (2003) for acoustic and electromagnetic problems.In this work, we will provide a new approach to show convergence of the no-response test by showing its equivalence to a regularized version of the range test. So in addition to two novel methods in inverse fluid dynamics this also provides further basic steps of methodological development on sampling and probing methods.