{
  "id": "2001.04369",
  "version": "v1",
  "published": "2020-01-13T16:03:40.000Z",
  "updated": "2020-01-13T16:03:40.000Z",
  "title": "Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification: Extensions to $L^p$",
  "authors": [
    "Troy Butler",
    "Tim Wildey",
    "Wenjuan Zhang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "A previous study analyzed the convergence of probability densities for forward and inverse problems when a sequence of approximate maps between model inputs and outputs converges in $L^\\infty$. This work generalizes the analysis to cases where the approximate maps converge in $L^p$ for any $1\\leq p < \\infty$. Specifically, under the assumption that the approximate maps converge in $L^p$, the convergence of probability density functions solving either forward or inverse problems is proven in $L^q$ where the value of $1\\leq q<\\infty$ may even be greater than $p$ in certain cases. This greatly expands the applicability of the previous results to commonly used methods for approximating models (such as polynomial chaos expansions) that only guarantee $L^p$ convergence for some $1\\leq p<\\infty$. Several numerical examples are also included along with numerical diagnostics of solutions and verification of assumptions made in the analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-13T16:03:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inverse problems",
      "approximate models",
      "uncertainty quantification",
      "convergence",
      "approximate maps converge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}