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The adjoint operator
projects the data space back to the
model space and is defined by the dot product test
| ![\begin{displaymath}
\left({\bf d},\,{\bf A\,m}\right) \equiv
\left({\bf A}^T\,{\bf d},\,{\bf m}\right)\end{displaymath}](img50.gif) |
(27) |
for any
and
. The method of conjugate gradients is
a particular case of the method of conjugate directions, where the
initial search direction
is
| ![\begin{displaymath}
{\bf c}_n = {\bf A}^T\,{\bf r}_{n-1}\;.\end{displaymath}](img51.gif) |
(28) |
This direction is often called the gradient, because it
corresponds to the local gradient of the squared residual norm with
respect to the current model
. Aligning the initial
search direction along the gradient leads to the following remarkable
simplifications in the method of conjugate directions.
Next: Orthogonality of the gradients
Up: Fomel: Conjugate directions
Previous: ALGORITHM
Stanford Exploration Project
9/11/2000