Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 404

In fact, we have

Z Z Z

E . w MV / D w MV p Z d z D w p W j Z d w p z d z

n n n

R R R

Z Z

D w p. w ; z /d w d z D E . w /:

n n

R R

It is also possible to verify that @ 2

J . w MV /

@ w 2

D 2E >0,so w MV is indeed a

minimum [ 68 ].

Maximum A Posteriori Estimator

Other estimators may be computed with a similar approach but referring to a
different risk

J

. w /,

Z Z Z

J . w / D E .J. e // D J. e /p W d w D J. e /p W;Z d w d z

R

n

R

n

R

n

for different choices for J./. For instance, another possible choice is the “uniform”
(thresholding) cost:

8
<

0 for k e k 1 <"
1

J. e / D

2" for k e k 1 " ;

:

where kk 1 is the maximum norm. With this cost function, we obtain

0 1

Z Z Z Z

1
2"

@ A

J

. w / D

J. e /p W j Z d w p Z d z D p W j Z d w p Z d z :

R

n

R

n

R

n n

nf w C Π";" n

R g

By definition

Z Z

p W j Z d w D 1 p W j Z d w

R

n

nf w C Π";" n g w C Π";" n

so that

Z Z Z

1
2"

1
2"

J

. w / D p Z d z p W j Z d w p Z d z :

R

n

R

n

w C Π";" n