Optimization and geophysical inverse problems Page: 28 of 37
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* Better solutions to the forward problem. It is a common situation in geophysical inverse
problems that the solution of the forward problem is extremely difficult and very time con-
suming on a computer, which imposes a severe limitation upon the solution of the inverse
problem. Many problems, particularly those that involve three-dimensional distributions
of material properties within the earth, are simply not being done because of this limita-
tion. In some cases it is possible to use simplified and approximate solutions, but this can
introduce additional uncertainty into the inverse problem. Thus there is a continuing need
for more efficient and more accurate methods of solving geophysical forward problems.
* More computational resources. The types and sizes of geophysical problems that are being
solved today is limited by the available computational resources, with a latent list of addi-
tional problems that await improvements in those resources. In addition to improvements
in speed, the computational needs for geophysical inverse problems include:
- Large amounts of memory, cache, and disk space.
- Management policy that allows long residency of data.
- More access to massively parallel systems.
- Better visualization capabilities.
It is obvious that these two needs, better solutions to the forward problem and more compu-
tational resources, are closely related, as the solution of the forward problem is often the most
computationally intensive part of the inverse problem and it is here that additional computa-
tional resources would be most effective.
The traditional approach of dividing a geophysical inverse problem into the separate stages
of formulation, solution, and appraisal, with optimization included primarily in the solution
stage, should be re-thought. A more general and more effective paradigm may be:
" Optimization should be included in all stages of the inverse problem. This would allow a
number of improvements:
- The scale of the discretization, which is often part of the formulation stage, could be
chosen in an optimum manner.
- The role of regularization in the determination of the solution would become more
evident and could be selected in a more optimum manner.
- The trade-off between fitting the data and satisfying the regularization condition
could be made more objective.
Certain general tendencies have developed in the formulation and solution of geophysical
inverse problems that need to be re-examined. Other possibilities that are available and should
be considered include:
" Include the theory connecting the data and the model as a constraint rather than as part of
the objective function. In principle, this change does not really change the formulation of
the problem, but it can have a significant effect upon the ability of optimization algorithms
to effectively find a solution. The main effects of this approach are:
- Couplings between the variables that makes the problem appear more nonlinear to
the optimizer are avoided.
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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/m1/28/: accessed November 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.