Civil Engineering Reference
In-Depth Information
In a vector machine we can collect the calculations of all
x
ij
with
i
+
j
=
const
. The calculation then proceeds in 2
m
groups. It is well known that
we can save time in a vector machine by overlapping about eight arithmetic oper-
ations. The time saved is proportional to
m
2
, and is worthwhile if it exceeds the
time required for the initialization of the 2
m
groups. Normally, this is the case
when
m>
approx. 40.
A different approach can be used on a parallel machine. We sketch the case
of two processors [Wittum 1989a]. First we divide the domain into two parts.
The first processor takes care of the nodes
(i, j)
with
j
≤
n/
2, and the second
one takes care of those with
j>n/
2. Once the first processor has dealt with
(
1
,
1
), (
1
,
2
), ..., (
1
,n/
2
)
, the second one is signaled. While the second proces-
sor works on its assigned values in the row
i
=
1, the first can do row
i
+
1
=
2.
The two processors continue to work in parallel on succeeding rows.
We have to provide memory with access by two processors only at the bound-
aries, i.e. for
j
=
n/
2 and
j
=
n/
2
+
1. It is clear how memory can be freed for
access by the other processor.
Without considering the memory restriction, there is also another approach
which we could take. The first processor works on the entire first row. With the
delay of one node, it signals the second processor to begin work on the second
row. The remaining rows are then dealt with alternately by the two processors. In
particular, with several processors, we get a complete parallelization after a short
initialization time.
For more on parallelization, see, e.g., Hughes, Ferencz and Hallquist [1987],
Meier and Sameh [1988], Ortega and Voigt [1985] and Ortega [1988].
Nonlinear Problems
The CG method can be carried over to nonlinear problems for which the function
f
to be minimized is not necessarily a quadratic function. This avoids iterating
with the Newton method, where the solution of the corresponding linear system
of equations would again require an iterative method.
As in §2, let
f
be a
C
1
n
. Very often
function defined on an open set
M
⊂ R
f
has the form
n
1
2
x
Ax
b
x
f(x)
=
−
d
i
φ(x
i
)
−
i
=
1
with
φ
∈
C
1
(
R
)
. The first term has a more significant effect on poor condition-
ing than the second [Glowinski 1984]. Suppose we have a matrix
C
which is
appropriate for preconditioning
A
(otherwise we choose
C
=
I
).
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