{
  "id": "1505.02566",
  "version": "v1",
  "published": "2015-05-11T11:07:40.000Z",
  "updated": "2015-05-11T11:07:40.000Z",
  "title": "Reconstruction of the solution and the source of hyperbolic equations from boundary measurements: mixed formulations",
  "authors": [
    "Nicolae Cindea",
    "Arnaud Munch"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1502.00114",
  "categories": [
    "math.OC"
  ],
  "abstract": "We introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\\Omega\\times (0,T)$ - $\\Omega$ a bounded subset of $\\mathbb{R}^N$ - from a partial boundary observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discuss several examples for $N=1$ and $N=2$. The problem of the reconstruction of both the state and the source term is also addressed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-11T11:07:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mixed formulation",
      "boundary measurements",
      "reconstruction",
      "space-time finite elements discretization",
      "usual geometric optic conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150502566C"
    }
  }
}