Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 443

the convergence is not necessarily guaranteed. Again, the final goal is to keep the
size N of the finite dimensional approximation of the solution as small as possible,
thanks to the information contained in the basis.
From the computational standpoint, this snapshot-based approach relies on the
off-line/on-line paradigm, namely

1. computation of the basis is “off-line,” and it is intended to be an accurate
(and therefore expensive) numerical approximation of the solution for different
configurations that are considered to be relevant to the basis;
2. solution of the optimization problem, and in particular the computation of the
coefficients U N;i along the iterations of the minimization procedure is “on-line,”
and contributes to the actual cost of the control procedure.

In this way, the computational costs are factorized, the major contribution being
carried out in a step preliminary to the optimization. This paradigm clearly makes
sense whenever the “off-line” part can be recycled for the solution of several
optimization problems. 9
Among the different snapshot-based strategies, we mention the reduced basis
method and the POD . In the former, the snapshots are computed for values of
the parameters that are evaluated to perform the best control of the error on the
basis of rigorous error estimates (see [ 65 , 70 ]). In particular, we mention [ 52 ]for
an application of the reduced basis method to FSI problems. The latter is known
also as Karhunen-Loève decomposition or principal component analysis and it is
illustrated more in detail in the next paragraph.

POD Basis Selection

We start assuming that a set of size M of solutions is available, for instance, by
computing snapshots for M different values of the parameter of interest after a
uniform sampling of an appropriate range. We assume that M is still large for the
purpose of reducing the computational costs and that a reduction of the size of the
basis is required, by properly filtering redundancy in the snapshots set. Denote by
j 2 R

N the M snapshots of the (approximate) solution, with j D 1;:::;M. Then,
we perform the following steps.

M

X

1
M

1. Sample average : D j .

j D 1

M M , whose elements are defined as c ij WD

2. Sample Covariance : Compute C 2 R

1
M . i / T . j /. Matrix C is positive semidefinite and symmetric so the

9 This can be problematic in a clinical context, where patient-specific geometries differ one from
the other and the snapshot computation is not trivially recycled. Anatomical atlas mapping ideal to
real geometries are required.