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The use of conjugate gradients helps to speed convergence by choosing a direction that is a linear combination of the past and current steepest descent vectors Luenberger (1984). Following Mora (1987), I use a conjugate gradient approach given by Polak and Ribiére (1969)
| ![\begin{displaymath}
c_n = g_n + g^{*}_n \frac{ \left( g_n - g_{n-1} \right)}{g^{*}_{n-1}g_{n-1} },\end{displaymath}](img35.gif) |
(10) |
where cn is the conjugate gradient update. Note that this is equivalent to the formulation in Mora (1987) where data and model space covariances are represented by identity operators. Equation 5 thus modifies to
| ![\begin{eqnarray}
m_{n+1} ({\bf x}) = & m_n ({\bf x}) + \gamma_n ({\bf x}) c_n ({...
... 2, \nonumber \ \; = & \gamma_n ({\bf x}) g_n ({\bf x}) , & n=1.\end{eqnarray}](img36.gif) |
|
| (11) |
The computation of conjugate gradient direction in equation 11 comes essentially at no cost because the previous gradient vector, gn-1, easily can be stored in memory.
Next: Step-length Definition
Up: Review of Frequency-domain waveform
Previous: Gradient Vector Definition
Stanford Exploration Project
1/16/2007