Optimization and geophysical inverse problems Page: 21 of 37
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4.7 Linearized problems
Given the considerable machinery that exists for solving linear inverse problems, there is a
tendency to formulate problems so that linear methods can be used whenever possible. For
problems that are not too strongly non-linear, this can be done by a process of linearization.
Consider the reduced form of the theory constraint (equation 10) and write
d =a(mo + bm) a(mo) + m, (22)
where it has been assumed that mo is a reference model and that higher order derivative terms
are small enough to be ignored. Then, defining bd = d - a(mo), we have the linearized problem
3d = a(m ) 3m . (23)
So long as the the solution does not stray too far from the reference model m, this problem
can be solved with standard linear methods, which also includes the standard linear estimates
of resolution and uncertainty. The situation where the reference model is unknown is handled
by an iterated linearization procedure in which a new reference model is taken to be mo + bm
and the entire linearization and solution process repeated. This type of linearized approach to
the solution of an inverse problem is commonly used in the location of earthquakes where it is
known as Geiger's method (Lee and Stewart, 1981).
The process of solving a non-linear inverse problem by solving a series of linearized problems
is in principle no different from some of the standard iterative methods developed for solving
non-linear problems, such as the line search and trust region methods (Dennis and Schnabel,
1996). An advantage of using these established non-linear methods for problems of this type
is that convergence proofs exist and well-tested algorithms are available. Thus, in the case of
many geophysical problems, it is difficult to justify the linearization of a problem when efficient
methods of solving the non-linear problem are available.
4.8 Nonlinear problems
Many geophysical inverse problems fall into the category where both the objective function and
the constraints are significantly nonlinear in the model parameters. The choice of whether to
use methods that do or do not require derivatives is especially important in this case because
both first and second derivatives may be required by the methods that do use derivatives.
There is a variety of methods for this type of problem that use derivatives and global and
local convergence proofs are available for most of these methods. For a discussion of these
methods it is important to distinguish between the two related formulations (18) and (19).
For problems (18) without constraints one could apply Newton's method. This requires
both first and second derivatives, i.e. the Hessian matrix. Obtaining the second derivatives
may be difficult, either because the mathematical expressions are difficult to derive or because
they are computationally expensive. There is a class of geophysical inverse problems, such as
global seismic tomography, where the problem is so large that second derivatives are out of
the question, as are any matrix inversions. The nonlinear conjugate gradient (CG) methods
and variants such as LSQR (Paige and Saunders, 1982) have turned out to be the methods of
choice for these types of problems and their performance seems to be satisfactory (Nolet, 1984,
1985; Newman and Alumbaugh, 1997). Alternatively, inexact CG-Newton methods or limited
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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. (https://digital.library.unt.edu/ark:/67531/metadc901685/m1/21/: accessed March 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.