Biomedical Engineering Reference
In-Depth Information
Linear interpolation of experimental data points in the interval [u
exp
h
, u
exp
Fig. 3.34
h
þ
1
] on the
linear interpolant of the experimental curve in that interval at arguments u
sim
i
The term in parenthesis in (
3.385
) denotes the Euclidian norm (L
2
-norm) of the
residual vector
jj
r
jj
E
which is to be minimized
fjj
r
jj
E
¼j
X
n
½
h
i
f
i
T
½
h
i
f
i
ð
h
i
; p
Þ
T
jjg!
min
:
ð
3
:
387
Þ
i
¼
1
The square of the Euclidian norm of the residual vector thus equals the sum of
squared residuals, equivalent to the least-square definition given in (
3.385
)
k
r
k
E
¼
X
n
½
f
i
f
i
ð
v
i
; p
Þ
2
:
ð
3
:
388
Þ
i
¼
1
The least-square solution of (
3.385
) is thus defined as a vector p which
minimizes the Euclidian norm of the residual vector r, i.e. minimizing the sum of
squares of the moduli of the residual r
i
. As outlined previously, in tissue and foam
material parameter optimization, function values are provided by the finite element
solver. This represents a non-linear least-square problem where, in addition, the
model function is not accessible. Therefore, an iterative numerical parameter
optimization algorithm, e.g. a direct or probabilistic method, must be employed to
find values of the parameters p
i
, which minimize the quality or objective function
U
k
. In general, however, in linear least-square fitting, adoption of iterative pro-
cedures is not necessary since the methods of differential calculus can be applied
to the model function. For this reason, the sum of the squares of the residual is
