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Serge Gratton(serge.grattonenseeiht.fr) Abstract: When solving nonlinear leastsquares 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 trustregion GaussNewton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the leastsquares 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 conjugategradientlike method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugategradients 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 (Physicalspace Statistical Analysis System), another wellknown 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 trustregion solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a SteihaugToint truncated conjugategradient method in the primal space. Keywords: Data assimilation, dualspace minimization, preconditioning, conjugategradient methods, globalization, trustregion methods Category 1: Applications  Science and Engineering (Optimization of Systems modeled by PDEs ) Category 2: Nonlinear Optimization (Nonlinear Systems and LeastSquares ) Category 3: Other Topics (Optimization of Simulated Systems ) Citation: S. Gratton, S. Gurol and Ph. L. Toint, "Preconditioning and Globalizing Conjugate Gradients in Dual Space for Quadratically Penalized NonlinearLeast Squares Problems", Report NAXYS102010, University of Namur, Namur (Belgium) 2010. Download: [PDF] Entry Submitted: 12/16/2010 Modify/Update this entry  
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