{
  "id": "2007.11377",
  "version": "v1",
  "published": "2020-07-22T12:40:09.000Z",
  "updated": "2020-07-22T12:40:09.000Z",
  "title": "$α\\ell_{1}-β\\ell_{2}$ sparsity regularization for nonlinear ill-posed problems",
  "authors": [
    "Liang Ding",
    "Weimin Han"
  ],
  "comment": "33 pages, 4 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper, we consider the $\\alpha\\| \\cdot\\|_{\\ell_1}-\\beta\\| \\cdot\\|_{\\ell_2}$ sparsity regularization with parameter $\\alpha\\geq\\beta\\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $\\alpha>\\beta\\geq0$, the results for the case $\\alpha=\\beta\\geq0$ are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain condition on the nonlinearity of $F$, we prove that every minimizer of $ \\alpha\\| \\cdot\\|_{\\ell_1}-\\beta\\| \\cdot\\|_{\\ell_2}$ regularization is sparse. For the case $\\alpha>\\beta\\geq0$, if the exact solution is sparse, we derive convergence rate $O(\\delta^{\\frac{1}{2}})$ and $O(\\delta)$ of the regularized solution under two commonly adopted conditions on the nonlinearity of $F$, respectively. In particular, it is shown that the iterative soft thresholding algorithm can be utilized to solve the $ \\alpha\\| \\cdot\\|_{\\ell_1}-\\beta\\| \\cdot\\|_{\\ell_2}$ regularization problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T12:40:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K10",
      "G.1.6"
    ],
    "keywords": [
      "nonlinear ill-posed problems",
      "sparsity regularization",
      "nonlinear ill-posed inverse problems",
      "nonlinear ill-posed equations",
      "regularization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}