RGA - algorithm for black box global optimization

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RGA  in near-real style.

 Inverse Helmholtz.

"Multibanana function"

The function  is induced by the inverse Helmholtz,  simulating 2.5 D microopical tomography
(acquisition is of cylindrical symmetry, cylindrical objects are illluminated by polarized monochrome linear  sources).

• acquisition :  a linear source illuminates an object of 6 wave  lengths'- (wl-) radius ,  shifted from the center of  symmetry: x = 10 wl,  y = 7 wl , from  11 positions on an circle of 200 wl -radius
• forward-scattering data:  wave fields along the 11 arcs  (the 1-st one has y -axis  as an axis of symmetry) of radius 600 wl with overlapping in a half of arc-length

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Inverse problem:
   given:

• refractive index of reference medium is equal 1.
• data (amplitude and phase) are given along the arcs with 0.5  wl intervals

   find  4  parameters:

• location of the cylinder (2 parameters), its radius and refractive index.

Objective function:  L₂-norm of residuals (differences of measured and synthetic data).

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Fig. 8.   2D-slice  of multibanana function is displayed (two parameters of the object location are fixed) .  Global minimum corresponds to  6.00 -> radius of the object  and 1.25 -> refractive index value : marked with red rhombus
a)   refractive index range  is [  0.7 : 1.3 ]
b)   refractive index range  is [ 1.1  : 1.5 ].
a)
b)

 The results of RGA-runs.

 The result  of RGA-run in a following domain of search
 

x	y	radius	refractive index
[ 9.0 : 11.0 ]	[ 6.0 : 8.0 ]	[ 4.0 : 7.0 ]	[ 0.7 : 1.4 ]

is as follows - with true values respectively
 

10.0007	6.9908	6.0027	1.2495
10.	7.	6.	1.25

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Respective OF-value is 0.0001278.
The total number of OF-calls is 1486.

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The result  of the other RGA-run:  another domain of search/true parameters
 

x	y	radius	refractive index
[ -2.0 : 0.0 ]	[ 5.0 : 7.0 ]	[ 6.0 : 9.0 ]	[ 0.7 : 1.5 ]

 is as follows:
 

-0.9719	6.0740	8.0081	1.3995
-1.	6.	8.	1.40

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Respective OF-value is 0.0004004.
The total number of OF-calls is 1850.

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'Corrupted' multibanana

Fig. 9.  2D-slice  of  the 'corrupted' multibanana function:  induced by the same  problem as it
     is represented with  Fig.8,   except the input data are just amplitudes (no  info  on phase).

 The results of RGA-runs.

The result  of RGA-run:  another domain of search/true parameters
 

x	y	radius	refractive index
[ 9.0 : 11.0 ]	[ 6.0 : 8.0 ]	[ 4.0 : 7.0 ]	[ 0.7 : 1.3 ]

 is as follows:
 

9.6315	7.7332	5.9693	1.2538
10.	7.	6.	1.25

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Respective OF-value is 0.0004196.
The total number of OF-calls is 1375.
N.B !  The  revealed parameters differ from the "true" ones  not because of  RGA-algorithm  shortcomings,
but  because  of  nonuniqueness of the problem.  Again: the value of OF is  0.0004.
As to the RGA-precision, see  the next page

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