Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial

This tutorial describes the classic method of conjugate directions: the generalization of the conjugate-gradient method in iterative least-square inversion. I derive the algebraic equations of the conjugate-direction method from general optimization principles. The derivation explains the “magic” properties of conjugate gradients. It also justifies the use of conjugate directions in cases when these properties are distorted either by computational errors or by inexact adjoint operators. The extra cost comes from storing a larger number of previous search directions in the computer memory. A simple ratfor program and three examples illustrate the method.