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OPTIMIZATION
Introduction [1]:
The most general method for solving nonlinear inverse problems needs a complete
exploration of the model space. Due to the fact for any other purposes other than
academic, this method is usually too computer intensive, usual methods normally limit
their scope to achieving some “best” model, as in a model maximizing the probability
density σ M(m) or minimizing some error function S(m).
If the forward problem isn’t very nonlinear, the required functions σ M(m) and/or S(m)
are normally well behaved and usually have only one extreme point, which could be
obtained by using the gradient method, for example, i.e., methods that use the function’s
properties at a specific point mn to decide on the future search direction so as to acquire
the updated model mn+1. For very nonlinear problems, however, there is a great risk that
the problem won’t converge to the required extreme points but rather to local min/max
points that don’t form the required solution to the problem. It has been shown that for
model spaces with more than a few parameters, it is more economical to select random
points in the space model, rather than building an evenly spread grid that is dense enough
to guarantee that at least one point will fall in the required optimal area.
Any method that uses a random (or pseudo-random) generator at any point is named
Monte Carlo.
The interest of Monte Carlo methods for inversion is that they can solve problems of
relatively large size without any linearization.
For the search for the domain of permitted models we will assume some known
information of the parameter domain, which can be described in the simple form:

m
inf


 m  msup
(  I M )
(1)
Where:
M = the model space
M = a particular point in M
The Monte Carlo method of inversion involves using a pseudo-random number generator
to generate random models within the section defined by (1). Computation of these
models, m, results in the estimated data, dcal = f(m). A quantitative comparison between
dcal and dobs, the observed data serves as a criterion for the suitability of any specific
model, m. This computation is stopped once the number of accepted models is enough to
suggest that that the model space has been explored.
An interesting reference to the use of Monte Carlo methods of inversion in geophysics is
the Press (1968) method of searching for the region of admissible models. Press studied
the density of the Earth’s mantle, as well as the velocity of seismic waves, as a function
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of the radius r, using as data measured eigenperiods of the Earth’s vibration, seismic
wave travel time, the total Earth’s mass and the Earth’s moment of inertia.
The parameters he wanted to estimate were the density ρ(r), the velocityof the longitudal
waves α(r), and the velocity of the tranverse waves β(r). These functions were considered
at 23 values of of r (the rest of the values were achieved by interpolation), resulting in a
total of 69 parameters.
Approximately five million models were randomly geneerated and tested. Of these only
six gave estimated data close enough to the observed values. Figure 1 shows four of the
Earth’s models obtained this way. This figure gives quite a good idea of the domain of
admissible Earth models, the invers problem has basically been solved.
Fig1. Four of the accepted Earth’s models of β, transverse waves velocity and
ρ, the density found by Press (1968). The heavy curves define the expected
region of results.
The example above illustrates some of the difficulties of this method. If the region error
bars are chosen too large, the number of experiments may be very large; if they are
chosen too small, they will control the results. It is also difficult to determine that the
number of trials suffices.
The problem:
This project is based on a previous study [2] done on muscle tissue. There it was shown
that the tissue impedance characteristics comply with the following nonlinear
mathematical equations:
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V23( , l ,t )  K cos2   t l sin 2 
1/ 2
(2)
V89 ( ,  t ,  l )  K[cos 2 (   1 )   t  l sin 2 (   1 )] 1/ 2
(3)
V56 ( ,  t ,  l )  K[cos 2 (   2 )   t  l sin 2 (   2 )] 1/ 2
(4)
K = I 2a  l  t
1/ 2
(5)
Where:
σt = electic conductivity (parralel to the fibers)
σl = electic conductivity perpandicular to σt (transverse to the fibers)
α = angle between the direction transverse to the fibers and the straight line
through the four electrodes [2]
I = alternating current through the medium
V = voltage response from the medium to the current I
For both V and I the reference is to each channel saparetely
a = distance between two adjacent electrodes.
γ1,2 = represent the angles between the first row of electrodes and the second and
third rows respectively
The optimization concerns finding the σt, σl, α parameter values so that the above
function minimization is achieved.
Using the matlab function, we obtain a set of predicted parameters for different starting
points. These starting points are spread randomly within a priori region based on existing
literary values. By selecting the minimum of the different function’s minima achieved
one can assume that the function global minimum was obtained.
The tighter the ‘starting point’ lattice the higher the probability of falling on the global
minimum, however, the price for that is a very complex and time consuming process.
The matlab Optimization toolbox [3] offers different routines that deal with optimization
of non-linear functions. These routines vary in algorithms and line search strategies. Trial
and error brought us to use the fsolve matlab function.
The fsolve matlab function:
This function finds the roots of the non-linear equation. Values of X are found so that:
F(X) = 0
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(6)
The optimization problem in this case is a non-linear least square problem and uses the
Gauss-Newton or Levenberg-Marquardt methods.
The choice of the algorithm and other parameters can be set manually by the user.
The options used in the optimization being:
1.
Terminate for x = 1e-4 (default). This defines the worst case precision required of
the independent variable, x. The optimization will carry on until termination criteria is
reached.
Terminate for f = 1e-4 (default). This defines algorithm termination based on
precision required of f at solution (In theory f = 0).
Main Algorithm = Gauss Newton
Search Algorithm = The line search algorithm used is a safeguarded cubic
polynomial method. This method requires fewer function evaluations but rather more
gradient evaluations.
Max iterations = 300 (default). The default number of iterations is defined as
100*n where n is the number of independent variables.
2.
3.
4.
5.
The Optimization algorithm:


Choose estimation starting point
Activate fsolve optimization function:
 Set function options different to default
Sensitivity points in the algorithm:
1.
Unless the function is continuous and has one minimum only, there is no
guarantee that the minimum achieved is the global minimum. Starting the
optimization at different starting points helps locate the global minimum. The smaller
the distance between these starting points the higher the probability to arrive at the
optimal results and the accuracy level rises.
2.
The function to be solved should be continuous. When successful, fsolve
returns one root only. fsolve may converge to a non-zero point in which case another
starting point may help.
3.
The noise level and characteristics affect the optimization accuracy. Sharp
spikes in the function or a noise level higher than 10[dB] throw the optimization
algorithm off course
The Xo beginning points are limited to a defined boundary based on previous studies.
Assuming these boundaries are correct and the requested estimated parameters are
located within these limits this factor shouldn’t have any affect. However, if the
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parameter values aren’t included within this boundary the estimation will not converge to
any value.
REFERENCES
[1] Tarantola, A., ”Monte Carlo Methods”, Inverse Problem Theory Methods for Data
Fitting and Model Parameter Estimation, pp 167-170, Elsvier, Netherlands,1987
[2] F.L.H Gielen; K.L. Boon, ‘Electrical Conductivity & Histological Structure of
Skeletal Muscle’.
[3]“Optimization Toolbox – User’s Guide”, 1992.