Geoscience Reference
In-Depth Information
In concrete problems, the choice of zeroth approximation
X
0
is usually accom-
plished from the physical reasons. This choice is bound up with the “guessing”
ofthesolution.Indeed,thecloserthezerothapproximationistoafinalsolution
the less number of the iterations is necessary and the better their convergence
is. Usually, certain a priori mean values are taken as a zeroth approximation.
It could be mean-climatic data for the problems of atmospheric optics. Some-
times there is a possibility to obtain anywise the approximate solution rough
as it is, and this solution is to take as
X
0
. Usually such a choice of zeroth ap-
proximation essentially increases the effectiveness of iteration process (4.23).
Mentionthatowingtotheproblemofthenonlinearity,thesolutioncouldbe
not unique, i. e. to depend on the concrete choice of
X
0
.Thesequestionswe
willdiscussinSect.4.4indetail.
Standard condition
ρ
ε
ρ
(
X
n
+1
,
X
n
)
<
,where
(. . .) is a certain metric, and
ε
is a parameter describing the solution accuracy, is used theoretically as a cri-
teria for breaking off the iteration. Usually the Euclid metric is used as
ρ
(...)
(4.10), because it is coordinated with the metric of the observations. Never-
theless, the other variants are possible, for example, the rigorous condition:
max
k
ε
(Box and Jenkins 1970). However, everything is
much more complicated during the practical calculations. The accumulation
of the errors of the computer calculation together with possible special fea-
tures of the discrepancy behavior around the minimum point leading to value
ρ
|
x
n
+1,
k
−
x
n
,
k
|
<
=
1,...,
K
(
X
n
+1
,
X
n
) is finishing to diminish with
n
increasing, hence, the condition
for the breaking of the iterations could be not valid for too small
ε
.Thus,to
ε
provide the solution independency of the concrete choice of value
, the other
conditionsareoftenusedforbreakingofftheiterations.Thus,theeffectiveway
is analyzing value
ρ
(
X
n
+1
,
X
n
)asafunctionof
n
and breaking off the iteration
when its stable decreasing changes to the oscillations around a certain mag-
nitude (Vasilyev O and Vasilyev A 1994). In the simplest variant, the decision
about the breaking off is assumed in the interactive regime. Another simple
way is a choice of the solution corresponding to the minimum of the discrep-
ancy for the fixed iteration number. Note that the peculiarities of the iteration
convergence are caused by the conditions of the concrete problem and need
the special study within the range of the preliminary numerical experiments
(Vasilyev O and Vasilyev A 1994).
It is easy to understand the reason for the appearance of the strong spread of
values
ρ
(
X
n
+1
,
X
n
) (i. e. the large difference of the desired values of two neigh-
bor iterations) from the physical meaning. Indeed, the matrix of the partial
derivatives
G
n
depends on current magnitude
X
n
,andvector
X
n
is quite pos-
sibly falling within the area where the measured values will extremely weakly
depend on some component of vector
x
n
,
j
. However, it means that magnitude
x
n
,
j
could vary strongly without essential influence to the measured values.
Algorithm (4.23) operates in this very way. There are diverse approaches to
remove this difficulty. They all are based on the paradoxical idea of
convergence
retarding
,whichdoesnotallowthevector
X
n
values to distinguish too strong
at the neighbor iterations (the more haste the less speed). We will return
to this question repeatedly, and now consider one of the simplest possibili-
ties.
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