WHAT ARE ADJOINTS FOR? THE METHOD OF CONJUGATE GRADIENTS

The adjoint operator ${\bf A}^T$ projects the data space back to the model space and is defined by the dot product test  

$$\left({\bf d},\,{\bf A\,m}\right) \equiv \left({\bf A}^T\,{\bf d},\,{\bf m}\right)$$ (27)

for any ${\bf m}$ and ${\bf d}$. The method of conjugate gradients is a particular case of the method of conjugate directions, where the initial search direction ${\bf c}_n$ is  

$${\bf c}_n = {\bf A}^T\,{\bf r}_{n-1}\;.$$ (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 ${\bf m}_{n-1}$. Aligning the initial search direction along the gradient leads to the following remarkable simplifications in the method of conjugate directions.