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Trial and error on step length $ \alpha$

Because the method above can lead to over-shooting on the step length $ \alpha$ , it may lead to a blow-up problem. The Newton method requires a convex function, but for some field data sets that condition may not be met. To overcome this, we use trial and error to avoid step length $ \alpha$ being too large. If the hyperbolic penalty function on new $ r=r+\alpha\Delta r$ is greater than on original $ r$ , the step length $ \alpha$ is too large, and we overshoot; in that case, we reduce the step length $ \alpha$ by half until the stability condition is met.

\begin{flalign*}\begin{split}&\hspace{1cm}\alpha_j=0 \\ &\hspace{1cm}{\rm Iterat...
...ha_j \Delta r \\ &\hspace{2cm}u=u+\alpha_j \Delta u \\ \end{split}\end{flalign*}      




2011-09-13