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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
| |
(87) |

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.
| |
(88) |

| (89) |

It is worth noticing that the residual updating
(89)
in a nonlinear problem is the same
as that in a linear problem (44).
If you make a large step , however,
the new residual
will be different from that expected by
(89).
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.

** Next:** Statistical nonlinearity
** Up:** THE WORLD OF CONJUGATE
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Stanford Exploration Project

4/27/2004