{
  "id": "1912.06225",
  "version": "v1",
  "published": "2019-12-12T21:29:02.000Z",
  "updated": "2019-12-12T21:29:02.000Z",
  "title": "Forward-backward approximation of evolution equations in finite and infinite horizon",
  "authors": [
    "Andres Contreras",
    "Juan Peypouquet"
  ],
  "categories": [
    "math.OC",
    "math.FA"
  ],
  "abstract": "This research is concerned with evolution equations and their forward-backward discretizations. Our first contribution is an estimation for the distance between iterates of sequences generated by forward-backward schemes, useful in the convergence and robustness analysis of iterative algorithms of widespread use in variational analysis and optimization. Our second contribution is the approximation, on a bounded time frame, of the solutions of evolution equations governed by accretive (monotone) operators with an additive structure, by trajectories defined using forward-backward sequences. This provides a short, simple and self-contained proof of existence and regularity for such solutions; unifies and extends a number of classical results; and offers a guide for the development of numerical methods. Finally, our third contribution is a mathematical methodology that allows us to deduce the behavior, as the number of iterations tends to $+\\infty$, of sequences generated by forward-backward algorithms, based solely on the knowledge of the behavior, as time goes to $+\\infty$, of the solutions of differential inclusions, and viceversa.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-12T21:29:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A60",
      "37L05",
      "49M25"
    ],
    "keywords": [
      "evolution equations",
      "infinite horizon",
      "forward-backward approximation",
      "differential inclusions",
      "algorithms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}