Technique has been developed for determining values of selected subsets of independent variables in mathematical formulations. Required computation time increases with first power of the number of variables. This is in contrast with classical minimization methods for which computational time increases with third power of the number of variables

Fymat, A. L.

English

NASA Technical Reports Server

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December 1975	 B75-10338 
•
	 NASA TECH BRIEF 
NASA Pasadena Office 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Minimization Search Method For Data Inversion 
In the physical, biological, or social sciences, an 
experimenter often makes a series of measurements of 
a dependent variable as correspond to values of a 
selected subset of the independent variables occurring 
in a known mathematical formulation. For instance, 
if X = f(a,b,c,d), measurements of X might be made 
at a selected value of a and at b = b 1 , b 2 , b 3, . . . etc. 
The remaining unknowns, independent variables c 
and d, may be of primary interest, but their values 
cannot generally be determined analytically from the 
known relationship. Instead, one must take successive 
guesses for c and d, until values of c and d are found 
that give a value of X sufficiently close to the 
measured X (at all the values of b for which 
measurements are taken). 
Several methods, such as least-squares techniques, 
are known for finding the values of such variables. 
However, the required computation time using a new 
technique, the minimization search method, increases 
with the 1st power of the number of variables. This is 
in contrast with classical minimization methods for 
which the computational time increases with the 3d 
power of the number of variables. 
Imaginary Part of Refractive Index, mj —.--Ø-
.	 Figure 1. Meaningful Values of mj and mr. 
The "Contour Lines' Represent Lines 
of Equal Value of the Function S.
The new method was developed to calculate 
refractive indices from measurements taken by a 
spectrometer described in NASA Tech Brief 
B75-10335. In that case it was desired to determine 
the values of the independent real and complex terms 
of the refractive index (mr and mj) which are related 
to a series of measurable quantities 
'q (intensities): 
I(sq; mr, mi) 
where Sq represents one or more independent 
variables with known values. The problem is to find 
for any and all q the particular couple (mr, mi) such 
that IIq 'qI £q, where 1q is the observed value, 1q is 
the calculated value, and Eq is the upper bound for 
accuracy. (The values Eq can be different for each q.) 
To find the values of mj and mr, a surface is defined 
[ij-Iq
S(sq; mr, mj) =	 L dqlq	 (1) 
where dq is a statistical weighting factor related to the 
manner in which the values of 5q (at which 
measurements are taken) are chosen and to the 
weights accorded the individual measurements Iq. 
Such a surface is shown in Figure 1. The minimum of 
the surface Smin, corresponds to the values of mi and 
mr within the required accuracy. The minimization 
search method can be used to find the point. 
The first step is to apply any, physical or theoretical 
information which limits the range of values for the 
minimum. In Figure 1 the rectangle ABCD represents 
the region of possible physically meaningful values of 
mi and mr. Then an initial guess, mr( 0) and mi(0), is 
made and defines point 0 that serves as an "origin" 
.
(continued overleaf) 
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Usually, only a few iterations are required to find 
Smjn . One way of testing for uniqueness is to repeat 
the search, starting at other widely differing points. A 
unique minimum requires that the same Smin be 
found each time. 
This method has also been applied successfuly to a 
true	 considerably larger number of unknowns. The Mr 
rectangle is then replaced by a multidimensional 
figure in the parameter-space of the dependent 
variables.
. 
rr1 
( mt, mi)	 (M+m	 in, S,d 
N. i, s, m+r	 ".2, S2 	
.-
__-5, ssL... 
1, 5, I
2,S3	
1--i 
r
Mt
	 Lt 
Smin 
'--1-- - - —. 
4'. S.	 3, S3 
4.	 fm	 3,	 + 
	
(mi, nij) !
	 (Mr, m) A-D	 C 
Rectangle ABCD From Figure 1

Figure 2. Minimization Search Method 
(see Figure 2). Next a rectangle is constructed parallel 
to ABCD and containing 0 (the rectangle need not be 
centered around 0). It is obtained after perturbing mj 
and mr both positively and negatively. (The 
magnitudes of these perturbations need not be equal 
- neither in the positive nor negative directions nor 
for each dependent variable). The value of S (equation 
1) is computed for the origin (S 0) and, when 
interrogated, the points at each corner of the 
rectangle and the additional four points on the 
rectangle sides corresponding to null perturbations in 
one of the two dependent variables. A random 
number generator (RNG) subroutine is attached to 
this set of eight possible improvements on the guess. 
Subsets of this set are then interrogated sequentially. 
The interrogation of any subset is carried out only if 
that of the subset previously generated by the RNG 
subroutine was unsuccessful in determining a better 
approximation to the solution. 
If none of the S values of the various subsets is 
smaller than S 0 , none of the eight points is closer to 
the minimum than S 0 ; and a new, smaller, rectangle 
should be drawn at 0. If a point belonging to any of 
these subsets or to a subset of the new rectangle (say 
2') corresponds to an S smaller than S 0, then it is 
closer to the minimum. 
The direction from 0 to 2' is taken as the direction 
along which S decreases. A new point (5 in Figure 2) is 
chosen by moving along the line (0, 2') according to 
the formula 5 = 2'+ c(2' -0) where c is a constant and 
the corresponding value of 5, that is S, is computed. 
If S 5 < S2', successive similar steps are taken along 
the line (0, 2') until point n at which 5n > Sn4 . If Sri 
> S 11, c is decreased in a limited number of steps 
until a point at which S < is found; and that 
point becomes the origin for a new rectangle. The 
entire process is then reiterated. 
B75-10338
Notes: 
1. A spectrophotometer which uses the method de-
scribed here to determine the refractive index of 
particles suspended in a gas or a liquid is described 
in NASA Tech Brief B75-10335: Developments in 
Spectrophotometry III: Multiple-Field-of-View 
Spectrometer To Determine Particle-Size Distri-
bution and Refractive Index 
2. Further discussion may be found in: 
a. "Satellite Determination of Nature and Micro-
structure of Atmospheric Aerosols," by 
A. L. Fymat in Satellites For Meteorology and 
Earth Resources, ed. Rassegna, Rome, Italy, 
vol. 14, March 1974, p. 327 
b. "Inverse Multiple Scattering Problems: II," by 
A. L. Fymat and R. E. Kalaba, Journal of 
Quantitative Spectroscopy and Radiative 
Transfer, vol. 14, March 1974, p. 919 
3. Requests for further information may be directed 
to:
Technology Utilization Officer 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Reference: TSP75-10338 
Patent status: 
This invention is owned by NASA, and a patent 
application has been filed. Inquiries concerning 
nonexclusive or exclusive license for its development 
should be addressed to: 
Patent Counsel 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Source: Alain L. Fymat of 
Caltech/JPL 
(NPO-99999) 
Category: 09 (Mathematics and 
Information Sciences) 
M(c)
.


Minimization search method for data inversion

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