Speaker
Description
Modern 4D-MRI imaging techniques are able to capture fully space-time resolved blood flow velocity data. This data facilitates a diagnosis of the severity of cardiovascular diseases, provided that, in a first step, additional information such as pressure, wall-shear stresses, etc., is inferred from the measured data. In this talk, we treat the inverse problem of deducing pressure values from MRI velocity measurements by an optimal control formulation. The flow inside a blood vessel is modeled by the transient Navier-Stokes equations. The equations are controlled by inhomogeneous Do-Nothing conditions imposed on the in- and outflow boundaries, which result from a truncation of the computational domain. While these conditions are widely used in numerics and are stable for pure outflow, instabilities arise when backflow occurs. On the continuous level, no global existence theory is available. Despite these difficulties associated with the Do-Nothing conditions, we present a formulation of a continuous optimal control problem, which is well posed. Our results thus bridge a gap in the existing literature, which so far has focused on numerical implementations of related optimal control problems. We show numerical examples that highlight the potential of our approach.