Viscoelastic inverse problems from seismic to medical scale

The objective of this project is to develop efficient algorithms for the determination of the physical parameters of a medium. We are in particular interested in properties which are not directly measurable, such as the Earth`s crust (seismic imaging) or patients` interiors (medical imaging). To get access to the material properties inside a body, penetrating waves are used, as they convey information on the medium in which they propagate. As an example, consider the waves resulting from an earthquake; after having traveled through the Earth, they are recorded at the Earth`s surface. These measurements are then used to characterize the sub-surface (Earth`s crust and even deeper). An example in medicine is X-Ray imaging, where Röntgen radiation waves outside of the patient are measured to recompute the absorption coefficient inside the patient. In the future, we can think of observed gravitational waves, that will allow us to detect properties of the Universe. In order to ``see`` inside a medium, an appropriate wavelength of the induced waves is necessary. For instance, X-rays can penetrate humans but not massive material, such as stones, which can be much better examined with seismic (mechanical) waves. After penetrating waves have been measured, the reconstruction relies on numerical simulations which necessitate to realistically model the medium, in order to extract its properties. For instance, conventional medical ultrasound imaging assumes that the body is essentially water. Conventional seismic imaging assumes a linear elastic material. In this project, we consider that the materials (Earth`s or medical samples) are viscous. Such materials have less elasticity and tend to wrinkle. Mathematically and for simulations, they are much harder to deal with than elastic materials (which do not wrinkle) and therefore, it is also harder to ``see inside`` using computational methods. The overall numerical simulations for reconstruction require efficient and innovative numerical methodologies (optimization method, deployment on supercomputers) and appropriate mathematical analysis and modeling, which are the heart of our research. This project is carried out by Dr. Florian Faucher at the Faculty of Mathematics at the University of Vienna, mentored by Prof. Otmar Scherzer.