{
  "id": "1810.08775",
  "version": "v1",
  "published": "2018-10-20T09:29:47.000Z",
  "updated": "2018-10-20T09:29:47.000Z",
  "title": "Tikhonov regularization with l^0-term complementing a convex penalty: l^1 convergence under sparsity constraints",
  "authors": [
    "Wei Wang",
    "Shuai Lu",
    "Bernd Hofmann",
    "Jin Cheng"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations Ax = y with an injective and bounded linear operator A mapping between l^2 and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the l^0-term and the complementing convex penalty, the important special case of the l^2-norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-20T09:29:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65J20",
      "47A52"
    ],
    "keywords": [
      "convex penalty",
      "tikhonov regularization",
      "sparsity constraints",
      "general stabilizing convex functional",
      "important special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}