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Physical nonlinearity

Methods of physics may relate modeled data $\bold d_{\rm theor}$ to model parameters $\bold m$, with a nonlinear relation, say $\bold d_{\rm theor} =\bold f(\bold m)$. The power-series approach then leads to representing modeled data as:
\begin{displaymath}
\bold d_{\rm theor} \eq
\bold f(\bold m_0 + \Delta \bold m)...
...\approx\quad
\bold f\bold (\bold m_0) + \bold F\Delta \bold m
\end{displaymath} (107)

where $\bold F$ is the matrix of partial derivatives of data values by model parameters, say $\partial d_i /\partial m_j$, evaluated at $\bold m_0$. The modeled data $\bold d_{\rm theor}$ minus the observed data $\bold d_{\rm obs}$ is the residual we minimize.
$\displaystyle \bold 0 \quad\approx\quad
\bold d_{\rm theor} - \bold d_{\rm obs}$ $\textstyle =$ $\displaystyle \bold F\bold \Delta\bold m +[\bold f(\bold m_0) - \bold d_{\rm obs}]$ (108)
$\displaystyle \bold r_{\rm new}$ $\textstyle =$ $\displaystyle \bold F\bold \Delta\bold m + \bold r_{\rm old}$ (109)

It is worth noticing that the residual updating (109) in a nonlinear application is the same as that in a linear application (55). If you make a large step $\Delta \bold m$, however, the new residual is different from that expected by (109). 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.

There are occasions to change the weighting function during model fitting. Then, one simply restarts the calculation from the current model. In the code, you would flag a restart with the expression forget=true.


next up previous [pdf]

Next: Coding nonlinear fitting problems Up: THE WORLD OF CONJUGATE Previous: THE WORLD OF CONJUGATE

2014-12-01