Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 395

differences and the complementary nature of the methods. In particular, we will
consider two classes of methods,
1. stochastic approaches , when some probabilistic knowledge of the uncertainty
affecting the model and the noise affecting the measures are available; in
particular we refer to methods related to Kalman filtering and its extension to
nonlinear problems; these methods will be addressed in Sect. 6.2 ;
2. deterministic approaches , when no clue on the statistical features of uncertainty
is available; in particular, we will see variational methods based on the minimiza-
tion of the mismatch between the data and numerical results, constrained by the
background model; these methods are introduced in Sect. 6.3 .
The above distinction is not strict. In fact, available statistical information can be
included in variational models.
The FSI problem and more in general the problems involved in cardiovascular
mathematics—usually represented by a system of partial differential equations—are
complex and per se challenging. In the framework of DA, the issue of computational
costs becomes even more important, as DA typically involves the solution of inverse
problems . In practice, these problems can be solved by iterative approaches, where
the solution of the FSI system (or more in general of the “forward” problem)
needs to be performed at each iteration. It is promptly realized that this requires
specific techniques to make the computational costs affordable. We address this
issue in Sect. 6.3 , in particular referring to methods for reducing the costs of each
iteration by representing the solution on a “smart” low-dimensional basis functions
set that strongly reduces the number of degrees of freedom required by a traditional
numerical method (like finite element or spectral methods).
Detailed examples are provided in Sect. 6.4 . In particular, we consider the
assimilation of velocity measures with the numerical simulation of the Navier-
Stokes equations for improving the estimate of blood velocity on an artery. We
address two different deterministic approaches and how they can be reinterpreted
or improved by a stochastic Bayesian perspective. Finally, we present in detail the
problem of estimating vascular compliance by solving an inverse FSI problem.
Again, we present both a stochastic approach based on Kalman filtering and a
deterministic constrained minimization approach. In the latter case, we present a
technique for reducing the computational costs by representing the solution on a
low-dimensional basis obtained with a Proper Orthogonal Decomposition (POD)
approach.
As we have pointed out, the methodological picture in the field of DA is
pretty articulated, encompassing statistical as well as numerical issues for inverse
problems [ 17 , 71 ]. Here, we mention some references for the reader interested in
having more details on the topics covered only partially in this introduction. The
importance of uncertainty quantification in any field of scientific computing has
been recently underlined in [ 22 ]. An excellent introduction to statistical methods
for computational inverse problems is given in the topics [ 13 , 28 , 47 , 68 ]. A recent
collection of contributions in the numerical solution of inverse problems and
computational costs reduction is [ 7 ].