Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

Abstract

ABSTRACT We consider the conjugate gradient method for the normal equations in the solution of discrete ill-posed problems arising from seismic tomography. We use a linear approach of traveltime tomography that is characterized by an ill-conditioned linear system whose unknowns are the slownesses in each block of the computational domain. The algorithms considered in this work regularize the linear system by stopping the conjugate gradient method in an early iteration. They do not depend on the singular-value decomposition and represent an attractive and economic alternative for large-scale problems. We review two recently proposed stopping criteria and propose a modified stopping criterion that takes into account the oscillations in the approximate solution.