In the last 25 years, developments in mathematical models/methods together with the improvements in the data acquisition devices have made possible to use mathematics to study the behavior of the human cardiovascular system. Furthermore, cardiovascular mathematics has not been limited to be used as a descriptive qualitative tool, but instead, has started to be used for quantitative analysis of patients conditions and even treatment design. The robustness of this tool depends on the reliability of the results. Data Assimilation (DA) is a set of techniques that helps to improve the specificity of the model, by incorporating available data into the model and therefore leading to patient specific results. On the other hand, the numerical methods used in the simulations must be accurate enough to guarantee that the computed solution accurately describes the real behavior of the system.;This work is divided into two parts. In the first, we focus on the estimation of the compliance of a blood vessel using DA techniques. In particular, we use measurements of the displacement of the vessel wall to estimate its Young's modulus. We adopt the variational approach proposed Perego et al. (2011), and we focus on the issue of the computational costs associated with the solution of the inverse problem. The second part concerns the accurate simulation of flows at moderately large Reynolds numbers. In particular, we focus on the model proposed in Layton et al. (2012) for the discretization of the Leray system, and we propose a new interpretation of the method as an operator-splitting scheme, for a perturbed version of the Navier-Stokes equations, and we use heuristic arguments to calibrate one of the main parameters of the model.;For both these parts we will perform numerical experiments, on 3D geometries, to validate the approaches. In particular, for the first part, we will use synthetic measures to validate our approach, while for the second part, we will test the method on a benchmark proposed by the Food and Drug Administration, comparing out results with experimental data.
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