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 | Model fitting by least squares |  |
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When standard methods of physics
relate theoretical data
to model parameters
,
they often use a nonlinear relation,
say
.
The power-series approach then leads to
representing theoretical data as
 |
(71) |
where
is the matrix of partial derivatives
of data values by model parameters,
say
,
evaluated at
.
The theoretical data
minus
the observed data
is the residual we minimize.
It is worth noticing that the residual updating
(2.73)
in a nonlinear problem is the same
as that in a linear problem (2.47).
If you make a large step
, however,
the new residual
will be different from that expected by
(2.73).
Thus you should always re-evaluate the residual vector at the new location,
and if you are reasonably cautious,
you should be sure the residual norm has actually decreased
before you accept a large step.
The pathway of inversion with physical nonlinearity
is well developed in the academic literature
and Bill Symes at Rice University has a particularly active group.
 |
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 | Model fitting by least squares |  |
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Next: Statistical nonlinearity
Up: THE WORLD OF CONJUGATE
Previous: THE WORLD OF CONJUGATE
2008-11-06