{
  "id": "2012.11280",
  "version": "v1",
  "published": "2020-12-21T12:14:44.000Z",
  "updated": "2020-12-21T12:14:44.000Z",
  "title": "Sparsity regularization for inverse problems with nullspaces",
  "authors": [
    "Ole Løseth Elvetun",
    "Bjørn Fredrik Nielsen"
  ],
  "categories": [
    "math.NA",
    "cs.NA",
    "math.OC"
  ],
  "abstract": "We study a weighted $\\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter $\\alpha$ tends to zero. Furthermore, for positive values of $\\alpha$, we show that the regularized inverse solution equals the true source multiplied by a scalar $\\gamma$, where $\\gamma = 1 - c\\alpha$. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by a PDE-constrained optimization problem, but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-21T12:14:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparsity regularization",
      "regularized inverse solution equals",
      "regularization technique",
      "euclidean spaces",
      "solving inverse problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}