Biomedical Engineering Reference
In-Depth Information
We define squared L
2
norm of residuum at current iteration by
(
c
i
,
j
u
n
(
l
)
−
n
i
,
j
u
n
(
l
)
−
w
i
,
j
u
n
(
l
)
−
s
i
,
j
u
n
(
l
)
−
e
i
,
j
u
n
(
l
)
R
(
l
)
−
r
i
,
j
)
2
=
.
i
,
j
i
+
1
,
j
i
,
j
−
1
i
−
1
,
j
i
,
j
+
1
i
,
j
The iterative process is stopped if
R
(
l
)
<
TOL
R
(0)
. Since the computing of
residuum is time consuming itself, we check it, e.g., after every ten iterations.
The relaxation parameter
ω
is chosen by a user to improve convergence rate of
the method; we have very good experience with
ω
=
1
.
85 for this type of prob-
lems. Of course, the number of iterations depends on the chosen precision TOL,
length of time step
τ
, and a value of the regularization parameter
ε
also plays a
role. If one wants to weaken this dependence, more sophisticated approaches
can be recommended (see e.g. [25, 35, 46] and paragraph below) but their imple-
mentation needs more programming effort. The semi-implicit co-volume method
as presented above can be implemented in tens of lines.
We also outline shortly further approaches for solving the linear systems
given in every discrete time step by (11.23). The system matrix has known
(penta-diagonal) structure and moreover it is symmetric and diagonally domi-
nant M-matrix. One could apply direct methods as Gaussian elimination, but this
approach would lead to an immense storage requirements and computational
effort. On the contrary, iterative methods can be applied in a very efficient way.
In the previous paragraph we have already presented one of the most popular
iterative methods, namely SOR. This method does not need additional storage,
the matrix elements are used only to multiply the old solution values and conver-
gence can be guaranteed for our special structure and properties of the system
matrix . However, if the convergence is slow due to condition number of the sys-
tem matrix (which increases with number of unknowns and for increasing
τ
and
decreasing
ε
), faster iterative methods can be used. For example, the precondi-
tioned conjugate gradient methods allow fast convergence, although they need
more storage. If the storage requirements are reduced, then they can be very
efficient and robust [25, 35]. For details of implementation of the efficient pre-
conditioned iterative solvers for co-volume level set method, we refer to [25],
cf. also [51]. Also an alternative direct approach based on operating splitting
schemes can be recommended [57, 58].
In the next section, comparing CPU times, we will show that semi-implicit
scheme is much more efficient and robust than explicit scheme for this type
of problems. The explicit scheme combined with finite differences in space is
