A Galerkin Finite Element Method for the Fractional Calderón Problem

1. Problem Statement

The paper addresses the Fractional Calderón Problem, an inverse problem aimed at recovering an unknown potential $q$ supported in a bounded domain $\Omega \subset \mathbb{R}^d$ from exterior measurements.

• Forward Model: The fractional Schrödinger equation:
  $$\begin{cases} (-\Delta)^s u + qu = 0 & \text{in } \Omega, \\ u = f & \text{in } \Omega_e := \mathbb{R}^d \setminus \Omega, \end{cases}$$
  where $s \in (0,1)$ is the fractional order, $f$ is an exterior Dirichlet datum, and $(-\Delta)^s$ is the integral fractional Laplacian.
• Data: A single partial exterior measurement. Specifically, one prescribes a non-zero exterior datum $f$ supported in an open set $W_1 \subset \Omega_e$ and measures the resulting nonlocal flux (the Dirichlet-to-Neumann map) on a disjoint open set $W_2 \subset \Omega_e$.
• Challenges:
  1. Ill-posedness: The inverse problem is severely ill-posed, exhibiting logarithmic stability (exponential instability).
  2. Non-locality: The fractional Laplacian couples the interior and exterior, leading to dense system matrices in numerical discretizations, making assembly and linear algebra computationally expensive compared to local PDEs.
  3. Single Measurement: Unlike classical Calderón problems requiring full boundary data, this work focuses on recovering $q$ from a single exterior datum, relying on strong unique continuation properties.

2. Methodology

The authors propose a Galerkin Finite Element Method (FEM) combined with Tikhonov regularization. The strategy separates the reconstruction into two distinct steps: State Recovery and Coefficient Recovery.

A. Mathematical Framework

• Decomposition: The solution $u$ is decomposed as $u = \bar{u}_f + u_0$, where $\bar{u}_f$ is a smooth extension of the exterior datum $f$, and $u_0 \in \tilde{H}^s(\Omega)$ is the unknown interior correction (vanishing outside $\Omega$).
• Preprocessing: The measurement equation is transformed to isolate $u_0$. The observed flux $g$ is adjusted by the known contribution of the exterior datum to yield a preprocessed measurement $\mu = g - ((-\Delta)^s \bar{u}_f)|_W$.
• Unique Continuation: The core principle is that $u_0$ can be uniquely recovered from $\mu$ on $W$ due to the unique continuation property of the fractional Laplacian. This reduces the inverse problem to inverting the operator $L: u_0 \mapsto ((-\Delta)^s u_0)|_W$.

B. Discretization Strategy

1. Truncation: The unbounded domain $\mathbb{R}^d$ is truncated to a large bounded box $\Omega_R$.
2. Finite Element Spaces:
   • State Space: Continuous piecewise polynomials $V_h$ on a mesh of $\Omega_R$. The interior correction space is $V_{h,0} = \{v_h \in V_h : v_h = 0 \text{ on } \Omega_R \setminus \Omega\}$.
   • Observation Space: A separate mesh on the measurement set $W$ to sample the flux.
3. Handling Non-locality:
   • The operator $L$ is discretized via a dense matrix $B$ representing the integral fractional Laplacian.
   • Crucially, because the measurement set $W$ is separated from $\Omega$ by a distance $d_0 > 0$, the integral kernel is smooth on $W$. This allows for nonsingular quadrature and avoids the singularity issues typical of fractional Laplacian discretizations near the boundary.
4. Two-Step Reconstruction Algorithm:
   • Step 1 (State Recovery): Solve a discrete Tikhonov regularization problem to find $u_{0,h}^\alpha$:
     $$\min_{v_h \in V_{h,0}} \left( \| B v_h - \mu^\delta \|_Y^2 + \alpha \|v_h\|_{s,R,h}^2 \right)$$
     where $\mu^\delta$ is noisy data, $\alpha$ is the regularization parameter, and $\|\cdot\|_{s,R,h}$ is a discrete fractional energy norm.
   • Step 2 (Coefficient Recovery): Once $u_h^\alpha = \bar{u}_f + u_{0,h}^\alpha$ is recovered, the potential $q$ is reconstructed by solving a stabilized least-squares problem:
     $$\min_{q_h \in Q_h} \left( \| w_h^\alpha + q_h u_h^\alpha \|_{L^2(\Omega')}^2 + \alpha_q \|q_h\|_{L^2(\Omega')}^2 \right)$$
     where $w_h^\alpha$ approximates $(-\Delta)^s u_h^\alpha$. The stabilization term $\alpha_q$ prevents division by zero when $u_h^\alpha$ is small.

3. Key Contributions

• Theoretical Convergence: The authors prove the existence and uniqueness of the discrete reconstructor. They establish conditional convergence of the reconstructed state $u_h^\alpha$ to the true solution $u$ as the mesh size $h \to 0$, truncation $R \to \infty$, noise $\delta \to 0$, and regularization $\alpha \to 0$.
• Error Estimates:
  • Derivation of a priori error estimates for the state reconstruction.
  • Derivation of error bounds for the coefficient reconstruction, combining discretization errors with the logarithmic stability of the continuous inverse problem.
  • The total error bound scales as $O((\delta + \eta)^{\frac{2\nu}{2\nu+1}})$ for the state and involves a logarithmic term for the potential, reflecting the inherent ill-posedness.
• Computational Framework: The method cleanly separates the forward solver from the inverse step. The observation operator depends only on geometry and $s$, allowing it to be assembled once and reused.
• Handling Discontinuities: The paper introduces a Total Variation (TV) regularized variant for the coefficient step to better recover discontinuous potentials, demonstrating its effectiveness numerically.

4. Results and Numerical Experiments

The method was tested in 1D and 2D with various noise levels and potential types:

• Smooth Potentials: The method successfully reconstructed smooth, compactly supported potentials. The error decay followed the expected logarithmic stability trend ($\|q_h - q\| \sim |\log \delta|^{-\gamma}$).
• Discontinuous Potentials:
  • Standard $L^2$ regularization smoothed out jumps.
  • The TV-regularized approach (using an $\ell_1$ penalty on jumps) successfully recovered sharp interfaces and the correct support of the potential, even with significant noise.
• 2D Performance: In 2D, the method retained the central peak of a bump potential more clearly than previous finite difference methods (e.g., Li [31]), demonstrating the robustness of the Galerkin formulation.
• Stability: The reconstructions remained stable under noise levels ranging from $10^{-10}$ to $10^{-1}$, provided appropriate parameter choices ($\alpha \sim \delta^p$) were made.

5. Significance

• First FEM Realization: This work provides the first rigorous Galerkin finite element realization of the single-measurement reconstruction principle for the fractional Calderón problem.
• Bridging Theory and Practice: It translates abstract unique continuation theory into a practical, implementable numerical algorithm, addressing the specific challenges of non-local operators (dense matrices, truncation errors).
• Robustness: The separation of the forward and inverse steps, combined with nonsingular quadrature for the observation operator, offers a computationally efficient and robust framework for non-local inverse problems.
• Future Directions: The paper highlights the need for automated parameter selection rules (especially for TV regularization) and sharper total error estimates as key areas for future research.

In summary, the paper establishes a mathematically sound and numerically effective framework for solving the fractional Calderón problem with single measurements, proving convergence and demonstrating the ability to recover both smooth and discontinuous potentials in the presence of noise.