Optimization and geophysical inverse problems Page: 23 of 37
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results to be obtained from the generalized pattern search (GPS) class of algorithms defined
and analyzed by Torczon (1997). Torczon and Trosset (1997), Lewis and Torczon (1996, 1998a,
1998b, 1999), and Lewis et al. (1998) give useful and interesting extensions of the algorithms
and the supporting theory, especially their work on problems with a finite number of linear
constraints, which could be bounds on the parameters in geophysical inverse problems. They also
give an interesting approach related to GPS with a very satisfying convergence theory for general
constraints. Audet and Dennis (2000a, 2000b, 2000c) extend this class of algorithms to handle
mixed continuous and discrete variables and general constraints, and they give a new analysis
showing convergence for discontinuous problems with appropriate optimality conditions for limit
points at which the problem is locally smooth. Thus, the GPS algorithms are especially attractive
in that they are broadly applicable, simple to implement and supported by a strong proof of
convergence to local optima. When used in conjunction with the feasibility constraint methods
mentioned previously (Berryman, 1997), such methods can take advantage of the variational
structure of the fully non-linear inversion problem.
The problem of multiple solutions is particularly important for nonlinear inverse problems,
as most optimization methods only provide a local extremum and separate procedures must
be used to find a more global extremum. Methods are available that are designed to find
the global extremum. For example, one such method uses a combination of smoothing and
continuation to find global solutions (Mor6 and Wu, 1997). Another method, the terminal
repeller unconstrained subenergy tunneling (TRUST) algorithm, has been used to solve a fairly
difficult geophysical inverse problem, the estimation of residual seismic static corrections (Barhen
et al., 1997). Another approach deals directly with the nonlinear nature of the problem and
uses recent advances in computational algebra to handle the polynomial equations that must be
solved (Everett, 1996; Vasco, 1999, 2000).
Grid search and stochastic search methods are also designed to find global extrema. For
small problems it may be possible to perform a grid search in which all members of the model
space are examined and either accepted or rejected. For somewhat larger problems a Monte
Carlo search may be possible and it has the advantage of being simple to implement and easy
to check (Mosegaard and Tarantola, 1995; Mosegaard, 1998). However, for most geophysical
inverse problems the number of model parameters and the required accuracy are such that a
complete Monte-Carlo search is unfeasible simply because of the number of times tle forward
problem would have to be calculated to achieve sufficient sampling of the model space. The
search of the model space can also be guided by a statistical Bayesian approach in which a
combination of prior information and the information contained in the observational data are
used to construct some measure of relative probability for the model space (see for example
Tarantola, 1987, or Sen and Stoffa, 1995). While these methods that attempt a general search
of the model space are appealing because of their simplicity and completeness, they are not yet
practical for most geophysical inverse problems. This is because the size of the model space,
which is of order 10M where M is the number of model parameters, is generally much too
large to allow a general search in finite computational time. Nevertheless, there continues to be
considerable effort devoted to the task of improving the efficiency of stochastic search methods.
Bosch et al. (2000) obtain promising results using a combination of importance sampling and
While neither enumerative nor completely random searchs of the model space have proven
to be effective methods of solving most large geophysical inverse problems, there are some
directed search methods, also called pseudo-random search, that have been more successful. Two
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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/m1/23/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.