Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D

1. Problem Statement

The paper addresses the linearised Calderón problem in three dimensions. The goal is to reconstruct an unknown conductivity perturbation $\eta$ inside a unit ball $B \subset \mathbb{R}^3$, given linearised boundary measurements.

• Context: The standard Calderón problem seeks to recover a conductivity coefficient $\gamma$ from the Neumann-to-Dirichlet (ND) map $\Lambda(\gamma)$. The linearised version considers a perturbation $\eta$ around a known background conductivity (assumed to be $\gamma \equiv 1$).
• The Operator: The linearised forward map $F = D\Lambda(1; \eta)$ maps the perturbation $\eta$ to the change in boundary potential. It is defined by the relation:
  $$\langle (F\eta)f, g \rangle_{L^2(\partial B)} = -\int_B \eta \nabla u_f \cdot \nabla u_g \, dx$$
  where $u_f$ and $u_g$ are harmonic functions with Neumann traces $f$ and $g$.
• Key Challenge: Previous work (specifically [Garde & Hyvönen, 2024]) solved this for $L^2$ perturbations in 2D. This paper extends the problem to 3D and, crucially, allows for unbounded perturbations in the space $L^3(B)$.
  • Significance of $L^3$: The operator $F$ is shown to be continuous from $L^3(B)$ to the boundary data space. This is a significant generalization because standard $L^\infty$ assumptions are too restrictive for many physical applications involving singularities or sharp transitions.

2. Methodology

The authors develop an exact direct reconstruction algorithm based on a 3D Zernike basis. The method relies on a "forward substitution" strategy, allowing coefficients to be computed sequentially without solving a large linear system.

A. Basis Functions

The reconstruction expands the perturbation $\eta$ in an orthonormal basis for $L^2(B)$ (which contains $L^3(B)$):
$$\eta = \sum_{k, \ell, m} c_{\ell}^{k,m} \psi_{\ell}^{k,m}$$
where the basis functions are products of 3D radial Zernike polynomials $R_\ell^k(r)$ and spherical harmonics $Y_\ell^m(\theta, \phi)$:
$$\psi_{\ell}^{k,m}(r, \theta, \phi) = R_\ell^k(r) Y_\ell^m(\theta, \phi)$$

• $k, \ell \in \mathbb{N}_0$ (radial and angular degrees).
• $m \in \{-\ell, \dots, \ell\}$ (azimuthal order).

B. The Reconstruction Algorithm (Theorem 1.1)

The core contribution is a formula that allows the inductive reconstruction of the coefficients $c_{\ell}^{k,m}$.

1. Triangular Structure: The method exploits the fact that the interaction between the perturbation and specific boundary measurements creates a triangular system.
2. Specific Measurements: To reconstruct the coefficients for a fixed radial index $k$, the method uses boundary currents corresponding to the spherical harmonic $Y_0^{k+1}$ (specifically, the $m=0$ mode of degree $k+1$).
3. Forward Substitution:
   • The coefficients for $k=0$ (harmonic functions) are reconstructed directly from data $(F\eta)Y_1^0$.
   • The coefficients for $k=1$ are reconstructed using data $(F\eta)Y_2^0$ and the previously computed $k=0$ coefficients.
   • In general, $\eta_k$ (the projection onto the subspace of index $k$) is reconstructed from $(F\eta)Y_{k+1}^0$ and all previous $\eta_0, \dots, \eta_{k-1}$.

The explicit formula for the coefficient $c_{\ell}^{k,m}$ is:
$$c_{\ell}^{k,m} = (Q_{\ell,0}^{k,m,k})^{-1} \left( \langle (F\eta)Y_{k+1}^0, Y_{\ell+k+1}^m \rangle - \sum_{q=0}^{k-1} \sum_{s=0}^{k-q} Q_{\ell,s}^{k,m,q} c_{\ell+2s}^{q,m} \right)$$
where $Q$ terms are scalars involving Gaunt coefficients (integrals of three spherical harmonics) and Pochhammer symbols, which are independent of $\eta$.

3. Key Contributions

1. Extension to $L^3(B)$: The paper proves that the linearised Calderón problem is well-posed for unbounded perturbations in $L^3(B)$ in 3D. This is achieved by showing the Fréchet derivative extends continuously to this space, leveraging Sobolev embeddings and the specific regularity of harmonic functions with $L^2$ Neumann data.
2. 3D Zernike Basis Reconstruction: It generalizes the 2D Zernike-based reconstruction method to 3D, providing an exact, non-iterative formula for recovering perturbations.
3. Data Efficiency: A major finding is that the 3D method requires less data than a full basis expansion would suggest.
   • In 2D, all Fourier modes are typically needed.
   • In 3D, the reconstruction of the $k$-th radial layer only requires the $m=0$ mode ($Y_0^{k+1}$) of the boundary current. This avoids the redundancy of measuring all $2\ell+1$ modes for every $\ell$, significantly reducing the measurement burden.
4. Handling Unbounded Perturbations: The method is theoretically capable of reconstructing perturbations that are not in $L^\infty$, provided they are in $L^3$.

4. Results and Numerical Validation

• Numerical Example: The authors tested the method on a localized Gaussian-type perturbation.
  • Exact Data: Using simulated data with 16-digit precision, the method achieved "nearly perfect" reconstructions even with truncated indices.
  • Noisy/Rough Data: Using a rough Finite Element (FE) model (P1 elements on a tetrahedral mesh) to simulate measurements, the method still produced stable and accurate reconstructions for the first few radial layers ($k=0$ to $4$).
• Stability: The authors note that while the 3D method is likely less numerically stable than the 2D counterpart due to the extra summation term (arising from the product of spherical harmonics not being a single harmonic), it remains robust enough for practical applications with noisy data.
• Regularization: The method naturally accommodates regularization via $k$-dependent truncation of the $\ell$-indices (limiting the angular resolution as the radial depth increases), which is a unique feature compared to other reconstruction methods.

5. Significance and Implications

• Theoretical Advancement: This work bridges a gap between 2D and 3D linearised inverse conductivity problems, specifically addressing the functional analytic requirements for unbounded perturbations ($L^3$ vs $L^2$).
• Computational Efficiency: The "forward substitution" nature of the algorithm means it does not require solving large, ill-conditioned linear systems (common in iterative methods like Landweber iteration). It is computationally cheap and easy to implement.
• Practical Applicability: The reduction in required boundary measurements (using only $m=0$ modes) makes the method more feasible for experimental setups where acquiring full spherical harmonic data is difficult or expensive.
• Foundation for Future Work: By establishing the extension to $L^3$, the paper opens the door for reconstructing conductivities with singularities, which are common in real-world materials but often excluded by standard $L^\infty$ assumptions.

In summary, Garde and Hirvensalo provide a rigorous, efficient, and exact algorithm for 3D conductivity reconstruction that handles unbounded perturbations and minimizes the required boundary data, representing a significant step forward in the numerical solution of the linearised Calderón problem.