{
  "id": "1204.0771",
  "version": "v1",
  "published": "2012-04-03T19:19:29.000Z",
  "updated": "2012-04-03T19:19:29.000Z",
  "title": "Regularization of Linear Ill-posed Problems by the Augmented Lagrangian Method and Variational Inequalities",
  "authors": [
    "Klaus Frick",
    "Markus Grasmair"
  ],
  "categories": [
    "math.NA",
    "math.OC"
  ],
  "abstract": "We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a standard source condition. Using the method of variational inequalities, we extend these results in this paper to convergence rates of lower order, both for the case of an a priori parameter choice and an a posteriori choice based on Morozov's discrepancy principle. In addition, our approach allows the derivation of convergence rates with respect to distance measures different from the Bregman distance. As a particular application, we consider sparsity promoting regularization, where we derive a range of convergence rates with respect to the norm under the assumption of restricted injectivity in conjunction with generalized source conditions of H\\\"older type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-03T19:19:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65J20",
      "47A52"
    ],
    "keywords": [
      "augmented lagrangian method",
      "linear ill-posed problems",
      "variational inequalities",
      "regularization",
      "bregman distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012InvPr..28j4005F"
    }
  }
}