[Paper Review] Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

Xinlin Cao, Yi‐Hsuan Lin+1 more|arXiv (Cornell University)|Dec 4, 2017

Numerical methods in inverse problems20 references7 citations

TL;DR

This paper establishes the first simultaneous uniqueness results for recovering both an embedded obstacle and a surrounding potential in anisotropic fractional Schrödinger operators using a single exterior Dirichlet-to-Neumann (DtN) measurement. The key contribution is proving that the obstacle is uniquely recoverable from one measurement, independent of the potential, and that multiple measurements uniquely determine the potential, resolving long-standing open problems in the local case (s=1) via nonlocal analysis tools.

ABSTRACT

Let $A\in\mathrm{Sym}(n imes n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(- abla\cdot(A(x) abla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $Ω$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s=1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

Motivation & Objective

• To resolve the longstanding open problem of uniquely recovering both an embedded obstacle and surrounding potential in nonlocal Schrödinger operators.
• To establish that a single exterior measurement can uniquely determine the embedded obstacle, regardless of the surrounding potential.
• To prove that multiple exterior measurements uniquely recover the surrounding potential in $ L^\infty $, extending results beyond the local case.
• To develop a framework based on strong uniqueness and Runge approximation for nonlocal inverse problems with embedded obstacles.
• To address the simultaneous recovery problem in the anisotropic fractional setting, where the local case (s=1) remains unresolved.

Proposed method

• Utilizes the strong uniqueness property of anisotropic fractional Schrödinger operators to propagate vanishing solutions through the domain.
• Employs the Runge approximation property to construct solutions with dense range in $ L^2 $, enabling density arguments in the inverse problem.
• Applies integral identities derived from the weak formulation of the nonlocal equation to relate DtN maps to the potential and obstacle.
• Uses the Dirichlet-to-Neumann map $ \Lambda_{D,q} $ as the sole exterior measurement, defined via the nonlocal operator $ \mathscr{L}_A^s $.
• Implements a limiting argument with sequences of solutions converging in $ L^2 $ to test arbitrary $ L^2 $ functions, enabling density-based uniqueness proofs.
• Relies on the well-posedness of the nonlocal problem under the eigenvalue condition (1.3), ensuring the existence and uniqueness of weak solutions.

Experimental results

Research questions

• RQ1 Can a single exterior measurement uniquely determine an embedded obstacle in anisotropic fractional Schrödinger operators, independent of the surrounding potential?
• RQ2 Can multiple exterior measurements uniquely recover the surrounding potential $ q \in L^\infty $ in the presence of an embedded obstacle?
• RQ3 Does the nonlocal nature of the fractional operator enable stronger uniqueness results than in the local case (s=1), where these problems remain open?
• RQ4 How do the strong uniqueness and Runge approximation properties facilitate the solution of simultaneous inverse problems in nonlocal PDEs?
• RQ5 What role does the anisotropy of the diffusion tensor $ A(x) $ play in the uniqueness of obstacle and potential recovery?

Key findings

• The embedded obstacle $ D $ is uniquely recoverable from a single exterior Dirichlet-to-Neumann measurement, regardless of the surrounding potential $ q $, as shown in Theorem 1.1.
• Multiple exterior measurements uniquely determine the potential $ q \in L^\infty(\Omega \setminus \overline{D}) $, as proven in Theorem 1.2 and Theorem 5.1.
• The uniqueness of $ q $ is established via a limiting argument using sequences of solutions that converge in $ L^2 $, enabling the use of dense test functions.
• The strong uniqueness property ensures that if a solution vanishes on an open set in the exterior, it vanishes identically, which is critical for proving uniqueness.
• The Runge approximation property allows constructing solutions with prescribed behavior in the interior, which is essential for proving the uniqueness of $ q $.
• The results are robust under soft or hard obstacle conditions, as the boundary condition $ \mathcal{B}u=0 $ does not affect the core uniqueness arguments.