Unique continuation for the wave equation: the stability landscape

TL;DR

This work studies the ill-posed problem of unique continuation for the wave equation with data supported in a volumetric subset, proving conditional Hölder stability in a subdomain and Lipschitz stability on the full domain when the lateral boundary trace lies in a finite-dimensional subspace. It then designs a stabilized space-time dG finite element method that inherits these stability properties, providing error bounds that reflect the continuous stability regime. Numerical experiments validate conditional Hölder stability, quantify the effect of boundary-trace information, and illustrate the critical role of the trace space dimension in achieving Lipschitz-type convergence. The results lay a rigorous foundation for stable numerical reconstruction under partial data and offer guidance on selecting trace spaces to ensure accurate, reliable simulations of wave propagation problems in imaging and control contexts.

Abstract

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be continued with Hölder stability into a certain proper subset of the space-time domain. Additionally, we show that unique continuation of the solution to the entire space-time cylinder with Lipschitz stability is possible given the knowledge of a suitable finite dimensional space in which the trace of the solution on the lateral boundary is contained. These results allow us to design a finite element method that provably converges to the exact solution at a rate that mirrors the stability properties of the continuous problem.

Figures (8)

• Figure 1: The spacetime geometry. Time is the vertical axis. The curved parts of the boundaries of $(-T,T) \times \Omega$ and $(-T,T) \times \omega$ in green and blue, respectively. The surface $\psi = \rho$ in orange. Set $B \setminus \omega$ is the region between the blue and orange surfaces.
• Figure 2: The plots show the error $\lVert u - \underline{u}_1 \rVert_{L^2(D)}$ for $D \in \{ B, \omega_T \}$ under $h$-refinement.
• Figure 3: Plots of the difference $u - \underline{u}_1$ in $B$ and its complement $Q\setminus B$. The central plot studies the convergence in $Q\setminus B$. Here $k=q$.
• Figure 4: Convergence of the error $\lVert u - \underline{u}_1 \rVert_{L^2(B_{\kappa})}$ in the sets $B_{\kappa} = \mho(\kappa \rho)$ for $\kappa \in \{1,3/4,1/2,1/4 \}$ using $q=k=q_{\ast}=k_{\ast} = 1$. The figures on the left and right show the space-time domains $\mho(\kappa \rho)$ for $\kappa = 1/2$, respectively $\kappa = 1/4$.
• Figure 5: Upper panel: Distribution of the mass of the mode $\underline{u_1}^{\lambda}$ over time. Lower panel: Convergence of the error $\lVert u - \underline{u}_1 \rVert_{L^2(B)}$ for different types of noise $\delta u \in \{ \delta u^s, \delta u^{\lambda} \}$ defined in \ref{['eq:smooth-noise']}, respectively \ref{['eq:bad-mode-noise']}. The noise is scaled such that $\lVert \delta u \rVert_{L^2(\omega_T) } \sim h^{\theta}$ for $\theta \in [1,2]$.

Theorems & Definitions (21)

• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 3
• Lemma 4: Norm
• proof
• Lemma 5: Continuity
• Lemma 6: Interpolation
• Lemma 7