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Extremal regularization appears to be practical for large scale
problems, as the Moré-Hebden algorithm with conjugate gradient
inner solves either converges in a
reasonable number of steps or doesn't converge when the constraint
(target noise level) forces too many small singular values into the
act. All of these terms are relative - small, doesn't converge, etc.
Modulo floating point arithmetic, the algorithm will *always* work
if enough effort is expended. The issue of course is reasonable level
of effort, and that is in some sense a translation of the concept of
``noise level'' - it's the misfit between the data and what you
can achieve with an easily computable model, no more.
Thus extremal regularization as implemented in this report appears to
give a reasonable approach to relative weighting in model and data
space when an independent estimate of noise level is somehow
available. This is the case for example in the examples mentioned
in the introduction. Maybe quiet parts of seismic traces furnish pure
noise series which might give a usable estimate of noise level -
provided that the modeling operator is sophisticated enough to fit the
rest!

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Stanford Exploration Project

4/20/1999