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A mixed finite elements approximation of inverse source problems for the wave equation with variable coefficients using observability

Carlos Castro, Sorin Micu

TL;DR

We address the inverse source problem for a 1D wave equation with variable coefficients observed at a boundary. A mixed finite element semi-discretization is developed to preserve a uniform observability property with respect to the mesh parameter $h$, enabling a stable least-squares reconstruction of the unknown source $f$. The authors prove a uniform discrete observability inequality via spectral analysis and Ingham-type arguments, establish stability and convergence of the discrete inverse problem, and validate the theory through numerical experiments, including noisy data and discontinuous potentials. The results show that the mixed FE approach avoids the deterioration of standard discretizations and yields convergent, robust reconstructions in practical settings.

Abstract

We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem using a space discretization based on a mixed finite element method is proposed and analyzed. Its stability and convergence relay on a new uniform boundary observability property with respect to the discretization parameter.

A mixed finite elements approximation of inverse source problems for the wave equation with variable coefficients using observability

TL;DR

We address the inverse source problem for a 1D wave equation with variable coefficients observed at a boundary. A mixed finite element semi-discretization is developed to preserve a uniform observability property with respect to the mesh parameter , enabling a stable least-squares reconstruction of the unknown source . The authors prove a uniform discrete observability inequality via spectral analysis and Ingham-type arguments, establish stability and convergence of the discrete inverse problem, and validate the theory through numerical experiments, including noisy data and discontinuous potentials. The results show that the mixed FE approach avoids the deterioration of standard discretizations and yields convergent, robust reconstructions in practical settings.

Abstract

We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem using a space discretization based on a mixed finite element method is proposed and analyzed. Its stability and convergence relay on a new uniform boundary observability property with respect to the discretization parameter.
Paper Structure (11 sections, 19 theorems, 203 equations, 2 figures, 3 tables)

This paper contains 11 sections, 19 theorems, 203 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A mixed finite element method and main results
  3. Spectral analysis
  4. Proof of the uniform observability result
  5. Inverse source problem. Proof of Theorems \ref{['te:estab0']} and \ref{['te:coninv']}
  6. Numerical experiments
  7. Smooth potential
  8. Noisy observation
  9. Discontinuous potential
  10. Smaller time observation
  11. Appendix

Key Result

Theorem 1

Assume that eq:pot holds. There exist two constants $T_0,\, \kappa_0>0$, independent of $h$, such that, for any $T> T_0$, the following inequality is verified: where $U_h=\left[u_{j}\right]_{1\leq j\leq N}$ is the solution of the discrete homogeneous system

Figures (2)

  • Figure 1: The two different source terms considered: a discontinuous one $f(x)$ (left) and a smooth one $g(x)$ (right).
  • Figure 2: Reconstruction of the source $g(x)$ for different time observations $T$.

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 34 more