Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 431

constant square matrix of compatible dimensions, reads

. u ; v/ D u T Av C v T A u D u T .A C A T /v; D G

while its Fréchet derivative reads

LJ LJ LJ LJ u D u T .A C A T /;

D D u

LJ LJ LJ LJ u

with the understanding, in this case, that the application of the operator D

D u

to

LJ LJ LJ LJ u

v is the usual matrix vector product of the one-row matrix D

D u

and the vector v.

LJ LJ LJ LJ u

G
D u

. u ;v/Dv and D

As another example, consider G . u / D u ,thenD G D

. In general, the derivative of a linear operator (the Laplacian operator in this
case) is the linear operator itself.
The usual chain rule holds for the differentiation of composite functions

D. G ı F /. u ;v/D D G . F . u /; D F . u ;v//;

or

LJ LJ LJ LJ u D LJ LJ LJ LJ F . u / D LJ LJ LJ LJ u :

G ı F G ı F/

D

D u

D. / D.

F

D u

Gradient-Based Optimization Approaches

A common and effective approach to deal with optimization constrained by partial
differential equations is to include directly the constraint in the functional to be
minimized. In this way, the minimization procedure is recast in an unconstrained
case and the solution is obtained with classical arguments. In particular, the first
order necessary conditions are obtained by setting to 0 the gradient of the functional.
In our case, this means that the solution u is computed as a function of the
control variables ˛ and and the total derivative of

J R , regarded as function of
these variables, is set to 0. This procedure admits an iterative implementation. Let
us denote the state problem ( 6.30 ), ( 6.31 ) with the abstract notation

F

. u ;˛;/D 0.
Assume that an initial guess ˛ .0/ and .0/ is given. Typically, we take .0/

D ref .

Then, we perform the following steps for j D 0;1;2;::::
- find the state variable u .j/ solution to F . u ;˛ .j/ ; .j/ / D 0;

J R . u .j/ ;˛ .j/ ; .j/ /=D˛ LJ LJ ˛ .j/ and D J R u .j/ ;˛ .j/ ; .j/ =D LJ LJ .j/ ;

- compute D

- if kD J R . u .j/ ;˛ .j/ ; .j/ /=DŒ˛;k is sufficiently small, solution is reached;