{
  "id": "2303.14473",
  "version": "v1",
  "published": "2023-03-25T14:02:01.000Z",
  "updated": "2023-03-25T14:02:01.000Z",
  "title": "A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube",
  "authors": [
    "Bernd Hofmann",
    "Hans-Jürgen Fischer",
    "Robert Plato"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Numerical differentiation of a function, contaminated with noise, over the unit interval $[0,1] \\subset \\mathbb{R}$ by inverting the simple integration operator $J:L^2([0,1]) \\to L^2([0,1])$ defined as $[Jx](s):=\\int_0^s x(t) dt$ is discussed extensively in the literature. The complete singular system of the compact operator $J$ is explicitly given with singular values $\\sigma_n(J)$ asymptotically proportional to $1/n$, which indicates a degree {\\sl one} of ill-posedness for this inverse problem. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case with operator $J$, there is little material available about the analysis of the d-dimensional case, where the compact integral operator $J_d:L^2([0,1]^d) \\to L^2([0,1]^d)$ defined as $[J_d\\,x](s_1,\\ldots,s_d):=\\int_0^{s_1}\\ldots\\int_0^{s_d} x(t_1,\\ldots,t_d)\\, dt_d\\ldots dt_1$ over unit $d$-cube is to be inverted. This inverse problem of mixed differentiation $x(s_1,\\ldots,s_d)=\\frac{\\partial^d}{\\partial s_1 \\ldots \\partial s_d} y(s_1,\\ldots ,s_d)$ is of practical interest, for example when in statistics copula densities have to be verified from empirical copulas over $[0,1]^d \\subset \\mathbb{R}^d$. In this note, we prove that the non-increasingly ordered singular values $\\sigma_n(J_d)$ of the operator $J_d$ have an asymptotics of the form $\\frac{(\\log n)^{d-1}}{n}$, which shows that the degree of ill-posedness stays at one, even though an additional logarithmic factor occurs. Some more discussion refers to the special case $d=2$ for characterizing the range $\\mathcal{R}(J_2)$ of the operator $J_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-25T14:02:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A52",
      "47B06",
      "65R30",
      "45C05",
      "45P05"
    ],
    "keywords": [
      "d-dimensional unit cube",
      "mixed differentiation",
      "ill-posedness",
      "additional logarithmic factor occurs",
      "inverse problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}