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A preconditioning technique for a class of PDE-constrained optimization problems

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Abstract

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss–Newton scheme to PDE-constrained optimization problems with a hyperbolic constraint. The preconditioner is of block triangular form and involves diagonal perturbations of the (approximate) Hessian to insure nonsingularity and an approximate Schur complement. We establish some properties of the preconditioned saddle point systems and we present the results of numerical experiments illustrating the performance of the preconditioner on a model problem motivated by image registration.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Michele Benzi & Eldad Haber

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Eldad Haber

  3. Quantitative Analytics Research Group, Standard & Poor’s, New York, NY, 10041, USA

    Lauren Taralli

Authors
  1. Michele Benzi
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  2. Eldad Haber
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  3. Lauren Taralli
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Corresponding author

Correspondence to Michele Benzi.

Additional information

Communicated by Rafael Bru.

The work of M. Benzi was supported in part by NSF grant DMS-0511336.

The work of E. Haber was supported in part by DOE under grant DEFG02-05ER25696, and by NSF under grants CCF-0427094, CCF-0728877, DMS-0724759 and DMS-0915121.

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Benzi, M., Haber, E. & Taralli, L. A preconditioning technique for a class of PDE-constrained optimization problems. Adv Comput Math 35, 149–173 (2011). https://doi.org/10.1007/s10444-011-9173-8

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