Optimization and geophysical inverse problems Page: 24 of 37
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examples are simulated annealing and genetic algorithms. Both of these approaches retain some
aspects of a random statistical search of the model space but use the gradually accumulating
information about acceptable models to direct the search into those parts of the model space
where good models are most likely to be found. These approaches appear to be feasible for
moderately sized problems where a full Monte Carlo approach would be prohibitive (Scales et
Simulated annealing is based upon an analogy with a natural optimization process in ther-
modynamics and uses a directed stochastic search of the model space. It requires no derivative
information. Its use in numerical optimization problems began with Kirkpatrick et al. (1983)
and its first use in geophysical problems appears to be Rothman (1985, 1986). A review of the
method and its application to geophysical problems can be found in Sen and Stoffa (1995) and
examples of its use in Sen and Stoffa (1991), Mosegaard and Vestergaard (1991), and Varela et
Another class of directed search methods are the evolutionary methods that make use of
analogies with the natural optimization processes found in the evolution of biological systems
(Holland, 1975; Goldberg, 1989). One class of such methods, genetic algorithms, applies the
operators of coding, selection, crossover, and mutation to a finite population of models and
allows the principle of "survival of the fittest" to guide the population toward a composition
that contains the optimum model. This approach has been applied to a number of geophysical
problems (see for instance Stoffa and Sen, 1991; Sen and Stoffa, 1992, 1995; Sambridge and
Drijkoningen, 1992; Kennett and Sambridge, 1992; Everett and Schultz, 1993; Sambridge and
Gallagher, 1993; Nolte and Frazer, 1994; Boschetti et al., 1996; Parker, 1999). Another class of
evolutionary methods, evolutionary programming (Fogel, 1962; Fogel, 1995; Back, 1996), uses
only the operators of selection and mutation and has only recently been applied to geophysical
problems (Minster et al., 1995; de Groot-Hedlin and Vernon, 1998).
Approaches that attempt some combination of stochastic and deterministic search methods
would appear to hold considerable promise. The general idea is to combine the global search
property of the stochastic methods with the efficiency of the deterministic methods. Of course,
one must always keep in mind that global optimization of a general function is computationally
intractable and so no method is sure to work. This is understood for most general nonlinear
problems, but global optimization has a further difficulty. Specifically, even if one has found a
global optimizer, it is impossible to recognize it for a general problem (Stephens and Baritompa,
1998). This does diminish the importance of work on global optimization methods, but, on the
contrary, it just shows how difficult the problem is and what we can hope to accomplish.
4.9 Very large problems
There exist some geophysical inverse problems that are so large that special methods of solution
have to be used. Tomography problems such as canonical problem 4 often fall into this class,
where it is not unusual to have on the order of 107 data and 105 model parameters. Linearization
about a reference model m0 is almost always performed in these problems, and the reference
model is usually held fixed. The choice of a method of solving such a linear system is-restricted
by the fact that it is not possible to fit the entire coefficient matrix into primary storage of most
computer systems, even though this matrix is usually sparse. This restriction eliminates many
standard solution methods, but there do exist approaches that require access to only one row
of the coefficient matrix at a time. These are iterative methods, but in this case the iterations
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Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al. Optimization and geophysical inverse problems, report, October 1, 2000; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901685/m1/24/: accessed February 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.