Iterative solution methods
Recipes for adjoint code construction
ACM Transactions on Mathematical Software (TOMS)
ICS '89 Proceedings of the 3rd international conference on Supercomputing
Trust-region methods
Automatic Preconditioning by Limited Memory Quasi-Newton Updating
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Approximate Gauss-Newton Methods for Nonlinear Least Squares Problems
SIAM Journal on Optimization
Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization
ACM Transactions on Mathematical Software (TOMS)
Combination Preconditioning and the Bramble-Pasciak$^{+}$ Preconditioner
SIAM Journal on Matrix Analysis and Applications
Mathematical Programming: Series A and B
Accelerating the LSTRS Algorithm
SIAM Journal on Scientific Computing
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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.