<
p id="
p-1">Linear systems of equations arise in traveltime tomogra
phy, deconvolution, and many other geo
physical a
pplications. Nonlinear
problems are often solved by successive linearization, leading to a sequence of linear systems. Overdetermined linear systems are solved by minimizing some measure of the size of the misfit or residual. The most commonly used measure is the
pan class="inline-formula" id="inline-formula-2">![Formula](A13/embed/mml-math-2.gif)
pan> norm (squared), leading to least squares
problems. The advantage of least squares
problems for linear systems is that they can be solved by methods (for exam
ple,
pan class="inline-formula" id="inline-formula-3">![Formula](A13/embed/mml-math-3.gif)
pan> factorization) that retain the linear behavior of the
problem. The disadvantage of least squares solutions is that the solution is sensitive to outliers. More robust norms, a
pproximating the
pan class="inline-formula" id="inline-formula-4">![Formula](A13/embed/mml-math-4.gif)
pan> norm, can be used to reduce the sensitivity to outliers. Unfortunately, these more robustnorms lead to nonlinear minimization
problems, even for linear systems, and many efficient algorithms for nonlinear minimization
problems require line searches. One iterative method for solving linear
problems in these more robust norms is iteratively reweighted least squares (IRLS). Recently, the limited-memory Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm (LBFGS) has been a
pplied efficiently to these
problems. A variety of nonlinear conjugate gradient algorithms (NLCG) can also be a
pplied. LBFGS and NLCG methods require a line search in each iteration. We show that exact line searches for these methods can be
performed very efficiently for com
puting solutions to linear systems in these robust norms, thereby
promoting fast convergence of these methods. We also com
pare LBFGS and NLCG (with and without exact line searches) to IRLS for a small number of iterations.
p>