[Paper Review] Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map

Mourad Choulli, Yavar Kian+1 more|arXiv (Cornell University)|Jan 7, 2015

Numerical methods in inverse problems10 references3 citations

TL;DR

This paper establishes a double logarithmic stability estimate for the inverse problem of identifying a scalar potential in a stationary Schrödinger equation from a partial Dirichlet-to-Neumann map. Using complex geometric optics solutions and Alessandrini's identity, the authors derive stability bounds in $L^2$ and $H^{-1}$ norms, showing that the potential difference is controlled by a double-logarithmic term of the DN map norm, under Sobolev and $L^∞$ regularity constraints.

ABSTRACT

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.

Motivation & Objective

• To establish stability estimates for the inverse problem of recovering a scalar potential from partial boundary measurements in the Schrödinger equation.
• To quantify the dependence of the potential reconstruction error on the measurement error from a partial Dirichlet-to-Neumann map.
• To extend stability results to both $L^2$ and $H^{-1}$ norms of the potential difference under regularity constraints.
• To provide a theoretical foundation for the uniqueness and stability of inverse problems in partial data settings with applications to conductivity imaging.

Proposed method

• The authors use complex geometric optics (CGO) solutions that vanish on a portion of the boundary to probe the potential.
• They apply Alessandrini's identity to relate the difference of DN maps to the potential difference via integral identities.
• Fourier coefficient bounds are derived in a carefully chosen frequency set to control the solution behavior.
• Stability inequalities are established through energy estimates and interpolation techniques in Sobolev spaces.
• The method relies on transposition solutions in $H_{\Delta}(\Omega)$ and the boundedness of the DN map in $H^{-1/2}(\Gamma)$ and $H^{1/2}(G)$.
• The proof is extended to the conductivity problem via a change of variables and transformation of the DN map.

Experimental results

Research questions

• RQ1 Can a double logarithmic stability estimate be established for the inverse problem of identifying a scalar potential from partial Dirichlet-to-Neumann data?
• RQ2 How does the stability of the potential reconstruction depend on the regularity of the potential and the measurement error?
• RQ3 What is the optimal stability rate achievable when only partial boundary data is available?
• RQ4 Can the stability result be extended to the $H^{-1}$ norm and $L^\infty$ regularity of the potential?
• RQ5 How does the stability estimate relate to the conductivity inverse problem under partial measurements?

Key findings

• The paper establishes a double logarithmic stability estimate: $\|q_1 - q_2\|_{L^2(\Omega)} \leq c\left(\|\widetilde{\Lambda}_{q_1,q_2}\| + \left|\ln \tilde{c} \left|\ln \|\widetilde{\Lambda}_{q_1,q_2}\|\right|\right|^{-t}\right)$ for $t > 0$.
• For the $H^{-1}$ norm, the stability estimate is $\|q_1 - q_2\|_{H^{-1}(\Omega)} \leq c\left(\|\widetilde{\Lambda}_{q_1,q_2}\| + \left|\ln \tilde{c} \left|\ln \|\widetilde{\Lambda}_{q_1,q_2}\|\right|\right|^{-1}\right)$.
• The stability constants $c$ and $\tilde{c}$ depend only on the regularity bounds $\delta$ and the Sobolev index $t$.
• The result holds under the conditions that $q_1, q_2 \in \mathcal{Q} \cap \delta B_{L^\infty(\Omega)}$ and $(q_2 - q_1)\chi_\Omega \in \delta B_{H^t(\mathbb{R}^n)}$.
• An application to the conductivity problem yields a similar double logarithmic stability estimate for the difference of conductivities in $L^2$.
• The stability estimate is robust under the assumption that conductivities and their normal derivatives agree on the boundary and on the illuminated face, respectively.