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Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

Kuang Huang, Bangti Jin, Yavar Kian, Faouzi Triki

Abstract

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.

Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

Abstract

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.
Paper Structure (12 sections, 9 theorems, 113 equations, 4 figures, 1 table)

This paper contains 12 sections, 9 theorems, 113 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Source recovery in the space
  4. Source recovery in the plane
  5. Source recovery in an interval
  6. Preliminary properties
  7. Proofs of the main results
  8. Proof of Theorem \ref{['t2']}
  9. Proof of Theorems \ref{['t3']} and \ref{['t100']}
  10. Proof of Theorem \ref{['t5']}
  11. Numerical results and discussions
  12. Conclusion

Key Result

Theorem 2.1

Let $d=3$ and, for $j=1,2$, let $\lambda^j\in L^2(0,T)$, satisfy condition lambda with $N=1$, $\lambda^j\not\equiv0$, $x^j\in\Omega$ and $u_0\in H^1(\Omega)$. Assume also that there exists $s\in(0,\frac{1}{2})$ such that $\lambda^1\in H^s(0,T)$ and let the following conditions be fulfilled. Let $S$ be an open nonempty subset of the boundary $\partial\Omega$ and let $u^j$, $j=1,2$, be the solution

Figures (4)

  • Figure 1: The numerical results for Example \ref{['ex:1']} (i) (top) and (ii) (bottom): the recovered source amplitude $\lambda$ (for $\delta=0.5\%$) (left), the error $e$ versus iteration $k$ (for $\delta=0.5\%$), and the error $e$ versus the noise level $\delta$ on a log-log scale (right).
  • Figure 2: The numerical results for Example \ref{['ex:2']} (i) (top) and (ii) (bottom): the recovered source amplitude (for $\delta=0.5\%$) (left); the error $e$ versus iteration $k$ (for $\delta=0.5\%$) (middle), and the error $e$ versus the noise level $\delta$ on a log-log scale (right).
  • Figure 3: The numerical results for Example \ref{['ex:3']}: the recovered source amplitudes (for $\delta=0.5\%$) (top), the error $e$ versus iteration $k$ (for $\delta=0.5\%$) (bottom left), and the error $e$ versus the noise level $\delta$ on log-log scale (bottom right).
  • Figure 4: Numerical results for Example \ref{['ex:4']}: the recovered source amplitudes (for $\delta=0.5\%$) (top), the error $e$ versus iteration $k$ (for $\delta=0.5\%$) (bottom left), and the error $e$ versus the noise level $\delta$ on a log-log scale (bottom right).

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Theorem 3.1
  • Proposition 3.1
  • proof
  • proof
  • Proposition 3.2
  • ...and 9 more