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

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 program and two examples illustrate the method. INTRODUCTION This paper describes the method of conjugate directions for solving linear operator equations in Hilbert space. This method is usually described in the numerous textbooks on unconstrained optimization as an introduction to the much more popular method of conjugate gradients. See, for example, Practical optimization by Gill et al. (1995) and its bibliography. The famous conjugate-gradient solver possesses specific properties, well-known from the original works of Hestenes and Stiefel (1952) and Fletcher and Reeves (1964). For linear operators and exact computations, it guarantees finding the solution after, at most, n iterative steps, where n is the number of dimensions in the solution space. The method of conjugate gradients doesn’t require explicit computation of the objective function and explicit inversion of the Hessian matrix. This makes it particularly attractive for large-scale inverse problems, such as those of seismic data processing and interpretation. However, it does require explicit computation of the adjoint operator. Claerbout (1992, 2003) shows dozens of successful examples of the conjugate gradient application with numerically precise adjoint operators. The motivation for this tutorial is to explore the possibility of using different types of preconditioning operators in the place of adjoints in iterative least-square inversion. For some linear or linearized operators, implementing the exact adjoint may pose a e-mail: [email protected]