Optimization and geophysical inverse problems Page: 18 of 37
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the essential features of the complete solution with a surrogate simulation that can be executed
much more efficiently. An example of this approach is the use of straight rays in travel time
tomography. Even though rays clearly bend in the Earth, the straight ray approximation may
nevertheless be very useful in low contrast or anisotropic media, and it is very quick to compute.
Software (C++) for incorporating surrogates into a filter pattern search method for generally
constrained problems can be found at www.caam.rice.edu/~dougm. It is assumed that the user
will prefer to furnish their own specific application-specific surrogates. The work of Booker, et al.
(1999), and other work by the same group, is based on the krigipg surrogates from geophysics.
Still another approach that has been shown to work in difficult highly non-linear problems
is to obtain solutions to an approximate problem that behaves like the real problem in an
asymptotic limit. Approaches of this type appear to be effective for various transport problems
in high contrast media (Borcea et al., 1996; Borcea and Papanicolaou, 1998; Borcea, 1999;
Borcea et al., 1999; Dorn, 1998; Dorn et al., 1999). Work should be done to ascertain whether
these approximate solutions can be used in the surrogate management framework of Booker et
An important step in many geophysical inverse problems is the part of the parameterization
process where a continuous model space is converted to a discrete set of parameters through
an expansion in a set of basis functions. A common form of this discretization process is to
divide the model space into a set of non-overlapping cells with the parameters being the mean
values for the cells. A critical question arises in regard to the best choice for the scale of the
discretization, which often reduces to a choice for the dimensions of the cells. Awareness of the
following two points can be useful in making this choice. First, it is important to understand that
discretization is really a form of regularization, as the smoothness of the model and its ability to
fit the data are directly related to the scale of the discretization. For instance, whether a problem
is under-determined or over-determined is directly related to the scale of the discretization in
most problems. Second, the scale of the discretization can be included in the inversion process
as a parameter to be optimized. This is an opportunity to remove one type of subjectivity from
the inversion process, and it also has benefits in the appraisal stage.
Multiple scales can also be used to increase the efficiency of the optimization. For example,
optimization on coarse scales can give good starting values for the optimization on fine scales,
coarse discretizations can be used to obtain less expensive second derivative approximations
for the problems on fine scales, and coarse scale information can be used to design precondi-
tioners for optimization subproblems on the fine grids. Finally, properties of the underlying
infinite dimensional problem and the choice of discretization provides important information
about the 'scaling' (as used in the optimization language, see, e.g., Dennis and Schnabel, 1996;
Heinkenschloss and Vicente, 1999a) of the optimization problem.
4.4 Feasibility constraints
For some inverse problems the theory constraint can be stated in the form of a variational
principle (Berryman, 1990; Berryman and Kohn, 1990; Berryman, 1993, 1997), which has several
attractive implications for the inverse problem. It may be possible to demonstrate thai the set of
feasible solutions is convex, and nonlinear programming methods are well suited for these types
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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/m1/18/: accessed January 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.