Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 433

Sensitivities can be retrieved by solving this set of equations for i D 1;2;:::;KC1.
In particular, for our working example we have

LJ LJ LJ LJ Ǜ .j/
i

LJ LJ LJ LJ u .j/

Z

.j/
i

.j/

D J R

DǛ i

@ J R

@ u

@ J R

@Ǜ i

D C D . u d/ i C 1 Ǜ i ; iD1;2;:::;K

LJ LJ LJ LJ u .j/ LJ LJ LJ LJ Ǜ .j/

K

Z

.j/

K C 1

.j/

D J R

D

@ J R

@ u

@ J R

@Ǜ i

D C D . u d/. K C 1 / C 2 . ref /:

C 1

Notice that from the state equations ( 6.30 ), ( 6.31 ), we have for i D 1;2;:::;KC 1

LJ LJ LJ LJ u .j/ . i / D i C b r i C i C 3. u .j/ / 2 i ;

@ @ u

and

LJ LJ LJ LJ

.j/

F
@Ǜ i

@

Df i ;iD 1;2;:::;K

LJ LJ LJ LJ

.j/

@ @

D u .j/ :

Then, the sensitivities equations read
8
<

i C b r i C i C 3. u .j/ / 2 i D f i in

K C 1 C b r K C 1 C K C 1 C 3. u .j/ / 2 K C 1 D u .j/

(6.35)

in

:

i D 0; i D 1;2;:::;KC 1 on :

Notice that these equations are linear in the sensitivities. Finally, we have

LJ LJ LJ LJ LJ LJ LJ LJ

Z Z

.j/ .j/

D J R

DǛ i

D J R

D

. u .j/ d/ i C 1 Ǜ .j i ; . u .j/ d/ K C 1 C 2 .j/ : D D

Gradients of the functional with respect to the control variables following this
approach requires therefore the solution of the K C 1 sensitivity equations.

Gradient Computation Through Adjoint Equations

In the following, we omit the iteration index j for simplicity. In the previous section
we computed the operator @

LJ LJ LJ LJ u

@ u

applied to the sensitivities i . Let us consider the