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Discretization and regularization for the reconstruction of inhomogeneities by scattering measurements

Daniela Di Donato, Luca Rondi

Abstract

We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem with regularization. Such a problem depends on a variety of parameters, that is, the number of measurements, the regularization parameter and the discretization parameter, namely the size of the mesh on which we discretize the unknown coefficients of the Helmholtz type equation modelling our physical system. We show, through a convergence analysis, that one can carefully choose these parameters in such a way that the solution to this discrete regularized minimum problem is a good approximation of the looked-for solution to the inverse problem.

Discretization and regularization for the reconstruction of inhomogeneities by scattering measurements

Abstract

We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem with regularization. Such a problem depends on a variety of parameters, that is, the number of measurements, the regularization parameter and the discretization parameter, namely the size of the mesh on which we discretize the unknown coefficients of the Helmholtz type equation modelling our physical system. We show, through a convergence analysis, that one can carefully choose these parameters in such a way that the solution to this discrete regularized minimum problem is a good approximation of the looked-for solution to the inverse problem.
Paper Structure (10 sections, 19 theorems, 121 equations)

This paper contains 10 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Lipschitz functions and Lipschitz open sets
  4. Discretizations of a domain and finite element spaces
  5. $\Gamma$-convergence
  6. The Helmholtz equation and radiating solutions
  7. Scattering problems
  8. Regularity results
  9. Continuity of the direct problem with respect to the inhomogeneity
  10. The approximation result

Key Result

Theorem 2.2

Let $E$ be any subset of $\mathbb{R}^M$ and let $f:E\to\mathbb{R}^{M_1}$ be a Lipschitz function on $E$. Then there exists a Lipschitz function $\tilde{f}:\mathbb{R}^M\to\mathbb{R}^{M_1}$ on $\mathbb{R}^M$ such that $\tilde{f}(x)=f(x)$ for any $x\in E$ and $\mathrm{Lip}(\tilde{f},\mathbb{R}^M)=\math

Theorems & Definitions (52)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 42 more