[Paper Review] Hyperbolic inverse problem with data on disjoint sets

Yavar Kian, Yaroslav Kurylev+2 more|arXiv (Cornell University)|Feb 11, 2016

Numerical methods in inverse problems26 references3 citations

TL;DR

This paper establishes the unique recovery of lower-order coefficients $A$ and $q$ in a hyperbolic wave equation from partial Dirichlet-to-Neumann data on disjoint sets $\mathcal{S}$ and $\mathcal{R}$, using strict convexity of $\mathcal{R}$ and exact controllability from $\mathcal{S}$ in time $T/2$. The key result shows that $A$ and $q$ are uniquely determined up to gauge equivalence in a neighborhood of $\mathcal{R}$, even when $\mathcal{S}$ and $\mathcal{R}$ are disjoint.

ABSTRACT

We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

Motivation & Objective

• To address the inverse problem of recovering lower-order coefficients $A$ and $q$ in a wave equation from partial boundary measurements on disjoint sets $\mathcal{S}$ and $\mathcal{R}$.
• To overcome the challenge of non-overlapping data domains where standard methods fail due to lack of overlap in observation and excitation regions.
• To establish uniqueness of $A$ and $q$ up to gauge invariance under geometric conditions: strict convexity of $\mathcal{R}$ and exact controllability from $\mathcal{S}$ in time $T/2$.
• To develop a novel technique based on localized wave solutions near $\mathcal{R}$ using its strict convexity, enabling recovery despite disjoint data sets.

Proposed method

• Construct boundary sources $f$ supported on $(0,T)\times\mathcal{S}$ such that the solution $u_f(T,\cdot)$ concentrates near $\mathcal{R}$, exploiting the strict convexity of $\mathcal{R}$ to localize energy at time $T$.
• Use the response operator $\Lambda_{\mathcal{S},\mathcal{R}}^T$ defined as $f \mapsto (\partial_\nu u_f - \frac{1}{2}(A,\nu)_g u_f)|_{(0,T)\times\mathcal{R}}$ to relate boundary sources to Neumann traces.
• Apply a gauge transformation framework: $A_\kappa = A + 2\kappa^{-1}\operatorname{grad}_g \kappa$, $q_\kappa = q + \kappa(A - \Delta_g)\kappa^{-1}$, under which $\Lambda_{\mathcal{S},\mathcal{R}}^T$ remains invariant.
• Use exact controllability from $\mathcal{S}$ in time $T/2$ to ensure that solutions can reach any point in $M$ by time $T$, enabling the construction of approximating sequences.
• Leverage weak convergence and density arguments in $L^2(M)$ to analyze limits of solutions at time $T$, showing convergence to Dirac-like measures near $\mathcal{R}$.
• Use the fact that $v_{1,\phi}(T,\cdot)$ and $v_{2,\phi}(T,\cdot)$ solve adjoint problems to relate solutions from two different coefficient sets and derive uniqueness via inner product identities.

Experimental results

Research questions

• RQ1 Can the lower-order coefficients $A$ and $q$ in a hyperbolic wave equation be uniquely recovered from partial boundary data when the source and observation sets $\mathcal{S}$ and $\mathcal{R}$ are disjoint?
• RQ2 What geometric conditions on $\mathcal{R}$ and control-theoretic conditions on $\mathcal{S}$ are sufficient to ensure uniqueness of $A$ and $q$ up to gauge invariance?
• RQ3 How can one construct solutions that concentrate near $\mathcal{R}$ at time $T$ when $\mathcal{S}$ and $\mathcal{R}$ are disjoint, using only boundary sources on $\mathcal{S}$?
• RQ4 To what extent does the presence of non-self-adjoint terms ($A$, $q$) affect the inverse problem, and how is gauge invariance handled in this setting?

Key findings

• The response operator $\Lambda_{\mathcal{S},\mathcal{R}}^T$ uniquely determines the coefficients $A$ and $q$ up to gauge transformations in a neighborhood of $\mathcal{R}$, provided $\mathcal{R}$ is strictly convex and the wave equation is exactly controllable from $\mathcal{S}$ in time $T/2$.
• The solution $u_f(T,\cdot)$ can be made to concentrate near $\mathcal{R}$ at time $T$ by constructing appropriate boundary sources $f$ supported on $\mathcal{S}$, leveraging the strict convexity of $\mathcal{R}$.
• The inverse problem remains solvable even when $\mathcal{S}$ and $\mathcal{R}$ are disjoint, which contrasts with the impossibility of recovering geometry from fixed-frequency data in such cases.
• The method relies on weak limits of solutions at time $T$ converging to Dirac-type measures, enabling the recovery of pointwise values of $A$ and $q$ via inner product identities with solutions of adjoint problems.
• The result holds for complex-valued $A$ and $q$, allowing for non-self-adjoint systems with energy dissipation.
• A global uniqueness result is established under a convex foliation condition, extending the local result to the full manifold under stronger geometric assumptions.