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On Exponential Instability of an Inverse Problem for the Wave Equation

Leonard Busch, Matti Lassas, Lauri Oksanen, Mikko Salo

TL;DR

The paper studies the inverse problem of recovering a time-independent potential $q$ in a wave equation with an obstacle from the source-to-solution map. It proves that, under partial data where measurements are restricted to a region $\Omega$ and $q$ is supported in the obstacle's shadow, the recovery is exponentially unstable; this is traced to Gevrey-3 smoothing of $S_q-S_0$ caused by the obstacle geometry and propagation of singularities. The authors formulate a Banach-space framework on a closed manifold and construct an operator family $F_q$ with uniform Gevrey-boundedness, then apply a general instability theorem to obtain a lower bound on the continuity modulus $\omega(s)$: $\omega(s) \gtrsim |\log s|^{-\delta\frac{6n+7}{n}}$. The work connects to Dirichlet-to-Neumann data stability results and boundary-control methods, clarifying that partial-data configurations can yield exponential, rather than logarithmic or Hölder, instability in inverse wave problems with obstacles.

Abstract

For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data at $t=0$ and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential $q$ from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of $q$ lies in the `shadow region' of the obstacle, we show that recovery of $q$ is exponentially unstable.

On Exponential Instability of an Inverse Problem for the Wave Equation

TL;DR

The paper studies the inverse problem of recovering a time-independent potential in a wave equation with an obstacle from the source-to-solution map. It proves that, under partial data where measurements are restricted to a region and is supported in the obstacle's shadow, the recovery is exponentially unstable; this is traced to Gevrey-3 smoothing of caused by the obstacle geometry and propagation of singularities. The authors formulate a Banach-space framework on a closed manifold and construct an operator family with uniform Gevrey-boundedness, then apply a general instability theorem to obtain a lower bound on the continuity modulus : . The work connects to Dirichlet-to-Neumann data stability results and boundary-control methods, clarifying that partial-data configurations can yield exponential, rather than logarithmic or Hölder, instability in inverse wave problems with obstacles.

Abstract

For a time-independent potential , consider the source-to-solution operator that maps a source to the solution of in Euclidean space with an obstacle, where we impose on vanishing Cauchy data at and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of lies in the `shadow region' of the obstacle, we show that recovery of is exponentially unstable.

Paper Structure

This paper contains 3 sections, 9 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Proof
  3. Existence of Solutions

Key Result

Theorem 1.1

Let $r\in (0,1)$ be fixed, $\Sigma \Subset B_{r}(2e_1), \Xi \Subset B_{r}(-2e_1)$ be open, $T' \in (0,T)$, and define $\Omega \coloneqq (0,T')\times \Xi$. Let $\mu\in\mathbb{R}, \delta>0$ be fixed so that $\mu+\delta > n/2$ and define $K \coloneqq \left\{q \in H^{\mu+\delta}(U)\colon \mathrm{supp} \ then $\omega(s) \gtrsim \left\lvert\log s\right\rvert^{-\delta\frac{6n+7}{n}}$ for $s$ small.

Figures (1)

  • Figure 1: A pictorial representation of $B_{r}(\pm 2e_1), \mathcal{O}$ in $U$ considered the ambient space with $\partial B_{T+4}\subset \partial U$ outside of frame. The following discussion is in the context of the proof of \ref{['lem:bich']}. The dotted line represents case 1: a line segment with endpoints in $B_{r}(\pm2e_1)$ must pass through the obstacle $\mathcal{O}$ (in particular intersecting $\partial\mathcal{O}$); a ray cannot follow this path. The continuous lines are the two possible cases 2a, 2b: a ray starting in $B_{r}(-2e_1)$ intersects $\partial\mathcal{O}$ either glancingly and continues as the same line segment, staying away from $B_{r}(2e_1)$. Or, the ray intersects $\partial\mathcal{O}$ transversally, being reflected according to Snell's law off the tangential plane $H$ at the intersection point, represented here by the dashed line. In the latter case one sees that the ray always stays on one side of $H$, whereas $B_{r}(2e_1)$ lies on the other side of $H$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6: zbMATH07465842
  • ...and 10 more