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10 10
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10 70

60

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function value

50
2
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30
4
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Figure 7.3: Third Weber Example

differencing. This, of course, is consistent with the theory, which does not claim that implicit
п¬Ѓltering is a global optimizer.

7.6.2 Parameter ID
We consider the parameter ID example from В§1.6.2 using the data from В§2.6.1. Recall that in this
example we use as data the values of the exact solution for c = k = 1 at the points ti = i/100
for 1 в‰¤ i в‰¤ 100. The initial iterate was (5, 5)T ; the sequence of scales was {2в€’k }12 . Implicit
k=1
п¬Ѓltering, like the globally convergent algorithms in the п¬Ѓrst part of the book, is fairly insensitive
to the choice of initial iterate, as we will see when we revisit this example in В§8.5.2.
We report on both low (rtol = atol = 10в€’3 , Figure 7.4) and high (rtol = atol = 10в€’6 ,
Figure 7.5) accuracy computations. Note that after 200 function evaluations the function re-
duction from the central difference BFGS form of implicit п¬Ѓltering п¬‚attens out in both plots at
roughly the expected level of O(tol) while the other methods have not. This effect, which is
not uncommon, is one reason for our preference for the BFGS central difference form of the
algorithm.

The performance of the central difference BFGS form of implicit п¬Ѓltering should be very good,
since (see exercises 7.7.1 and 7.7.2) the difference approximation of the gradient is exact. We
would expect that good performance to persist in the perturbed case. We illustrate this with results
on two problems, both given by (6.12). One is an unperturbed problem (aj = bj = cj = 0
for all j) where H is a diagonal matrix with (H)ii = 1/(2i) for 1 в‰¤ i в‰¤ N . The other is a
perturbed problem with
Оѕ0 = (sin(1), sin(2), . . . , sin(N ))T , Оѕ1 = 0, Оѕ2 = (1, . . . , 1)T ,

a1 = a2 = .01, a3 = 0, b1 = (1, . . . , 1)T , b2 = 0, and c1 = c2 = 10ПЂ.

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130 ITERATIVE METHODS FOR OPTIMIZATION

Central Differences Central Differences
4 2
10 10

2
10
0

function value
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10
2
10

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0 0
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Figure 7.4: Parameter ID, tol = 10в€’3

Central Differences Central Differences
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0
10
2
function value

10
2
10
0
10
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10

2 6
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0 0
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10 10

5 2
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Figure 7.5: Parameter ID, tol = 10в€’6

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IMPLICIT FILTERING 131

Central Differences Central Differences
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Figure 7.6: Unperturbed Quadratic, N = 4

Central Differences Central Differences
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function evaluations function evaluations

Figure 7.7: Unperturbed Quadratic, N = 32

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132 ITERATIVE METHODS FOR OPTIMIZATION

Central Differences Central Differences
2 2
10 10

0 0

function value
10 10

2 2
10 10

4 4
10 10
0 1000 2000 3000 4000 0 1000 2000 3000 4000
function evaluations function evaluations

Forward Differences Forward Differences
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10 10

0
10
0
function value
10
2
10
2
10
4
10

6 4
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function evaluations function evaluations

Figure 7.8: Perturbed Quadratic, N = 4

Central Differences Central Differences
2 4
10 10