Reduced order methods for the simulation and control of viscous flows

We review the application of reduced basis method as a computational reduction strategy for the approximation of viscous flows in systems characterized by physical and geometrical parameters, by focusing both on computational performances and stability, accuracy and reliability of results. We introduce proper geometrical parametrization settings provided by the use of small deformations on geometrical control points, to allow also a geometrical complexity reduction in terms of number of parameters, used to represent the computational domains. Numerical results will show some examples of optimal flow control and inverse problems related with possible applications in the mathematical modeling and numerical simulation of the human cardiovascular system.