An inverse problem in Fluid Mechanics applied in Biomedicine

In this thesis, new advances are presented in inverse problems of Fluid Mechanics in steady state, with direct applications in the recovery of domain deformations and obstacles, and whose purpose is to contribute to the detection of aortic valve conditions (such as insufficiency or stenosis).

The first main result of this thesis is an asymptotic approximation result between the obstacle detection problems and the recovery of a non-negative permeability parameter that assumes significantly large values in the regions with obstacles or the value 0 in other parts. This result is supported by numerical tests that confirm the approximation result.

The second result of this thesis presents a logarithmic inequality for the identification problem of the permeability parameter on Navier-Stokes equations from local measurements of fluid velocity. Numerical tests on the recovery of smooth and non-smooth parameters by a minimization problem and adaptive refinement algorithms are also included.

Finally, a parameter identification problem for the Oseen and Navier-Stokes equations is studied in order to recover a permeability parameter from local or global measurements of the fluid velocity. Several numerical experiments using Navier-Stokes flow illustrate the applicability of the method, for the localization of a simulated 2D cardiac valve from synthetic MRI and also recovering of the permeability parameter from 3D synthetic MRI.