Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 434

adjoint of this operator, which is the operator @

LJ LJ LJ LJ u

@ u

such that

< @

LJ LJ LJ LJ u LJ LJ LJ LJ u

@ u @ u

./; v > D <; @

.v/ >; (6.36)

for any v belonging to an appropriate functional space. Here < ; > indicates
a duality pairing. In particular, in a finite dimensional setting, < ; > typically
denotes the usual Euclidean dot product, while in the continuous setting, it denotes
one of the integrals
8
<

Z

< u ;v> u v; for scalar functions;

Z

< u ; v > u v ; for vector functions;

:

Z

<U;V> U W V for tensor functions:

In our example, we have

LJ LJ LJ LJ u

Z

v C b rv C vC 3 u 2 v :

@ u

<; @ .v/ >D

Integrating by parts, and choosing to vanish on ,weget 7

LJ LJ LJ LJ u

Z

b r C C 3 u 2 v:

<; @

@ u

.v/ >D

Therefore, the adjoint operator reads
@ LJ LJ LJ LJ u D b r C C 3 u 2 :

@ u

We consider the following adjoint problem, whose solution, as we will see later, is
crucial to find the derivatives of

J

with respect to the parameters.

LJ LJ LJ LJ u

LJ LJ LJ LJ u

< @

@ u @ J R

@ u

./; v >D .v/; (6.37)

7 We remind that we assumed b to be divergence free.