[Paper Review] On Single Measurement Stability for the Fractional Calder\'on Problem.

Angkana Rüland|arXiv (Cornell University)|Jul 27, 2020

Numerical methods in inverse problems30 references4 citations

TL;DR

This paper establishes a single measurement logarithmic stability estimate for the fractional Calderón problem by combining quantitative propagation of smallness for the Caffarelli-Silvestre extension with boundary doubling estimates to control the order of vanishing of solutions. The key result is a stability bound of the form $\|q_1 - q_2\|_{L^\infty(\Omega)} \leq \omega(\|\Lambda_{q_1}f - \Lambda_{q_2}f\|_{H^{-s}(W)})$, where the modulus of continuity $\omega(t)$ decays like $|\log(Ct)|^{-\gamma}$, confirming the optimality of logarithmic stability for this inverse problem.

ABSTRACT

In this short note we prove the logarithmic stability of the single measurement uniqueness result for the fractional Calder\\'on problem which had been derived in \\cite{GRSU18}. To this end, we use the quantitative uniqueness results established in \\cite{RS20a} and complement these bounds with a boundary doubling estimate. The latter yields control of the order of vanishing of solutions to the fractional Schr\\"odinger equation. Then, following a scheme introduced in \\cite{S10,ASV13} in the context of the determination of a surface impedance from far field measurements, this allows us to deduce logarithmic stability of the potential $q$.

Motivation & Objective

• To establish a single measurement stability estimate for the fractional Calderón problem, complementing the existing uniqueness result from [GRSU20].
• To quantify the stability of the inverse problem by deriving a logarithmic decay rate for the potential reconstruction error in terms of measurement error.
• To control the zero set of solutions to the fractional Schrödinger equation using boundary doubling estimates, enabling quantitative unique continuation from a single measurement.
• To extend the propagation of smallness principle from the Caffarelli-Silvestre extension to the nonlocal inverse problem setting under minimal regularity assumptions.

Proposed method

• Utilizes the Caffarelli-Silvestre extension to convert the nonlocal fractional Schrödinger equation into a degenerate elliptic problem in one higher dimension.
• Applies quantitative propagation of smallness estimates to transfer information from the measurement data on $W \subset \mathbb{R}^n \setminus \overline{\Omega}$ to the domain $\Omega$.
• Employs boundary doubling estimates to control the order of vanishing of solutions near the boundary, which is essential for reconstructing the potential from a single measurement.
• Combines interpolation inequalities and a priori bounds on the potentials $q_j$ in $C^{0,s}(\Omega)$ to estimate the $L^2$-norm of $u_1(q_1 - q_2)$.
• Uses the weak formulation of the Dirichlet-to-Neumann map $\Lambda_q f = (-\Delta)^s u|_{W}$ to relate measurement error to solution error.
• Optimizes the resulting stability estimate by balancing the decay from logarithmic terms and powers of the radius in the boundary doubling argument.

Experimental results

Research questions

• RQ1 Can the single measurement uniqueness result for the fractional Calderón problem be strengthened to a quantitative stability estimate?
• RQ2 What is the optimal rate of stability for the inverse problem when only one measurement is available?
• RQ3 How can the order of vanishing of solutions to the fractional Schrödinger equation be quantitatively controlled to enable reconstruction of the potential?
• RQ4 To what extent can the propagation of smallness principle be extended to degenerate elliptic equations arising from the Caffarelli-Silvestre extension?
• RQ5 Is logarithmic stability the optimal possible rate for single-measurement recovery of the potential in the fractional Calderón problem?

Key findings

• The paper establishes a single measurement logarithmic stability estimate for the fractional Calderón problem, with the error in the potential bounded by a modulus of continuity $\omega(t) \leq C|\log(Ct)|^{-\gamma}$.
• The stability rate is optimal up to the exponent $\gamma$, consistent with known results for the classical Calderón problem and the unique continuation property.
• Boundary doubling estimates are used to control the order of vanishing of solutions, which is the central step in quantifying the unique continuation from a single measurement.
• The stability estimate depends on a priori bounds on the potentials in $C^{0,s}(\Omega)$, the data $f$ in $\widetilde{H}^{s+\epsilon}(W)$, and geometric parameters such as $\operatorname{dist}(\Omega', \partial\Omega)$.
• The propagation of smallness from the Caffarelli-Silvestre extension allows transfer of measurement error to solution error in $H^s(\Omega)$, which is then used to estimate the potential difference.
• The final stability bound is derived via interpolation and energy estimates, with the optimal radius in the boundary doubling argument chosen to balance logarithmic and power-law terms.