Preconditioning and globalizing conjugate gradients in dual space for quadratically penalized nonlinear‐least squares problems

When solving nonlinear least‐squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust‐region Gauss‐Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least‐squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent conjugate‐gradient‐like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugate‐gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical‐space Statistical Analysis System), another well‐known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust‐region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug‐Toint truncated conjugate‐gradient method in the primal space.