% this file is called up by thesis.tex
% content in this file will be fed into the main document

%: ----------------------- name of chapter  -------------------------
\chapter{Conclusion}\label{C:conclusion} 
Recent advances in measurement tools for the cardiovascular system and the establishment of numerical techniques for cardiovascular mathematics motivated in this work the design of a reliable and robust technique for the inclusion of velocity data into hemodynamics simulations. Our goal is the estimation of blood flow filtering the noise affecting the data and the quantification of the uncertainty and credibility of numerical results. 

Current clinical methods for the estimation of such variable are based on simple interpolation techniques or manipulation of data \cite{brummer} and they do not rely on any constitutive law for the phenomena of interest. Thus, results are not accurate and they might not even be feasible; for this reason they may fail in guiding clinical intervention. The added value of our technique is to force the numerical solution to satisfy the equations governing the system; in fact, our estimator is based on the solution of a control problem where the misfit between velocity data and numerical solution is minimized  under the constraint of the fluid dynamics equations (the \gls{NSE}).

Another important aspect that makes our technique valuable is the inclusion of the additional knowledge regarding the measurement process and the statistical features of the measurement noise. When data are characterized by high correlation (which is very likely in case of complex systems as the cardiovascular one) this factor cannot be neglected as it significantly determines the statistical properties of the flow.

Moreover, on the basis of the analysis of mathematical properties of the optimization problem we are able to establish conditions on the number and location of data that ensure existence and uniqueness of the estimate. This result is the groundwork of a ``regularization'' technique to treat the case in which available data do not satisfy such conditions; based on data interpolation, this procedure proves to be competitive with current regularization techniques and it is parameter-free. 

Our assimilation method is robust and consistent in case of noise-free data; its performance is not affected by the nature of the flow (low or high Reynolds number) and it works nicely combined with stabilization techniques for advection dominated phenomena.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{The noise filtering properties}
Throughout this work we pointed out the filtering properties of this method. In this paragraph we would like to summarize this aspect in the case of 2-dimensional non-trivial geometries approximating vessels of interest. 

In Chapter \ref{C:da_nse}, Section \ref{S:da_nse_real_geometries}, we tested the deterministic estimator on a 2-dimensional approximation of the carotid. Here the assimilated solution is compared to the one obtained from a forward simulation prescribing noisy velocity on the inflow boundary as Dirichlet condition and disregarding internal data. Our results are 50\% more accurate than the non-assimilated ones. The same solutions are compared in the computation of the \gls{WSS}; for decreasing values of \gls{SNR} out estimator is up to the 50\% more accurate than the forward simulation with respect to a reference solution.

For the same test case in Chapter \ref{C:da_bayesian}, Section \ref{S:bayesian_real_geom}, comparison results show that statistical estimators are more accurate than deterministic ones in case of correlated data. In fact, the Bayesian estimation exploits the knowledge of the noise stochastic properties and yields a gain in accuracy of the 30\% with respect to the deterministic estimates. Furthermore, the analysis of the standard deviation of the velocity reveals that, except for the area close to the inflow boundary, the deviation from the mean of the assimilated solution is at most half the one of the given data. This fact implies that the uncertainty of the WSS computed using the assimilated solution is lower than the one obtained using given data only.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Future developments}
Although promising and potentially of great impact, \gls{DA} in computational hemodynamics is still at its beginning stage and has some weaknesses. For the \gls{DA} estimator to be applicable to 3-dimensional test cases with real geometries and data several computational issues have to be considered in order to improve the efficiency of the method.

As discussed in Chapter \ref{C:preliminary} and \ref{C:da_nse} the bottleneck of the optimization process is the solution of the state equation and its adjoint. Besides the \gls{PCD} preconditioner, other effective preconditioners for the steady and unsteady problem should be considered. Among the others we mention the works by Benzi et al. \cite{benzi2005numerical,benzi_prec_2010} and Elman et al. \cite{elman_finite_2005} for the steady case and the ones by Veneziani et al. \cite{gauthier2004fast, quarteroni1999analysis, saleri2006pressure, veneziani2003block} and Cahouet-Chabard \cite{cachab} for the unsteady one.  
For the solution of the nonlinear optimization problem we propose a nonlinear procedure that stems from the combination of the \gls{DA} method for linear problems and fixed point iterative methods for the NSE. In literature there is plenty of methods for the solution of problems of this sort. For reasons related to the \gls{FE} library, {\tt lifeV}, that we utilize we plan to adopt the {\tt NOX} package\footnote{http://trilinos.sandia.gov/packages/nox/}, which is part of Sandia's Trilinos project. {\tt NOX} is short for Nonlinear Object-Oriented Solutions, and its objective is to enable the robust and efficient solution of nonlinear equations using globalized Newton methods such as line search and trust region methods. It can be easily customized and it is designed to work with any linear algebra package.

Another important issue is to be competitive with other assimilation approaches, such as the Kalman filter methods \cite{bertoglio} which do not require the solution of the adjoint equation. A way to speed up the computation is to combine efficient optimization strategies with model reduction. In doing this it is important to find a good compromise between accuracy and efficiency; in medicine this is a delicate issue. On one hand fast simulations allow fast clinical intervention, on the other hand, inaccurate results may fail in guiding correctly medical doctors' decisions. Several methods are available, among these we mention the reduced basis (RB) method for parameter-based reduction and the proper orthogonal decomposition \cite{podNewlook,podWillcox}. A very good review of the RB method can be found in \cite{reducedBasis} by Patera and Rozza focused on formulation, analysis, and computational procedures for reduced basis approximation and a posteriori error estimation for parametrized \gls{PDE}s. Se \cite{Peterson} and \cite{Rozza} for applications of the RB method to Stokes and Navier-Stokes equations. Concerning the POD method we refer to \cite{podFluid1,podFluid2} for applications to fluid dynamics and to \cite{podOpt} for applications to \gls{PDE} constrained optimization problems.

The author is currently working in collaboration with Prof. Haber (University of British Columbia, Vancouver, CA) and Dr. Horesh (IBM, T. J. Watson Research Center, Yorktown, NY) on the design of optimal reduced basis \cite{md_eh_lh} obtained as the solution of an inverse problem. More specifically, the optimal basis is the control variable of a \gls{PDE} constrained optimization problem where the misfit between reduced and non-reduced solution is minimized subject to the state and reduced-state equations. Our plan is to combine the \gls{DA} method with this technique, now designed for elliptic \gls{PDE}s only.

Concerning the unsteady formulation presented in Chapter \ref{C:da_unsteady} there are still many open issues which deserve further and deeper investigation. For the nonlinear problem we developed a \gls{DO} iterative procedure based on the assimilation, at each time step, via solution of an optimization problem. This is limiting for two reasons: optimality in time is only local and the assumption of data availability at each time step is not feasible. 

The first issue can be assessed exploiting different assimilation methods which are based on ``global'' optimization in time. In doing this we have to deal with the solution of the unsteady adjoint problem which may be highly computationally expensive as we already discussed in the introduction to Chapter \ref{C:da_unsteady}. Additionally, we might also consider an \gls{OD} approach to be formulated in a way similar to the one presented in \cite{MPAVCV}. 

For the second issue, the generalization to sparse data in time can be managed by either interpolating the data in time or performing advancing steps between assimilation instants.

Furthermore, as we did for space discretization in the steady case, it is important to test the consistency and the sensitivity of the method with respect to the time step and for different time discretization schemes. In this work the choice of the \gls{BE} method is dictated by its unconditional stability; an analysis of stability properties of the resulting assimilation scheme might allow the employment of more accurate and/or less expensive methods.




% ---------------------------------------------------------------------------
%: ----------------------- end of thesis sub-document ------------------------
% ---------------------------------------------------------------------------

