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Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D

Henrik Garde, Markus Hirvensalo

TL;DR

This work extends the linearised Calderón problem to three dimensions in the unit ball, showing that any $L^3$ perturbation $\eta$ can be exactly reconstructed from the Fréchet derivative data $F\eta$ using a 3D Zernike basis. The authors develop a forward-substitution reconstruction with a triangular coefficient system by expanding perturbations in $\psi_\ell^{k,m}(r,\theta,\varphi) = R_\ell^k(r) Y_\ell^m(\theta,\varphi)$ and exploiting Gaunt coefficients and 3D harmonic analysis. The method achieves exact reconstruction with fewer boundary measurements than a full $L^2$-basis approach and remains numerically efficient, even under truncation and noise; the paper also discusses extension to $L^d$ perturbations and provides a numerical demonstration. Overall, the approach offers a robust, analytic reconstruction framework for 3D linearised inverse problems with unbounded perturbations and practical computational advantages.

Abstract

Recently an algorithm was given in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L^2$ perturbation from linearised data in the two-dimensional linearised Calderón problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calderón problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any $L^3$ perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full $L^2$ basis of current densities.

Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D

TL;DR

This work extends the linearised Calderón problem to three dimensions in the unit ball, showing that any perturbation can be exactly reconstructed from the Fréchet derivative data using a 3D Zernike basis. The authors develop a forward-substitution reconstruction with a triangular coefficient system by expanding perturbations in and exploiting Gaunt coefficients and 3D harmonic analysis. The method achieves exact reconstruction with fewer boundary measurements than a full -basis approach and remains numerically efficient, even under truncation and noise; the paper also discusses extension to perturbations and provides a numerical demonstration. Overall, the approach offers a robust, analytic reconstruction framework for 3D linearised inverse problems with unbounded perturbations and practical computational advantages.

Abstract

Recently an algorithm was given in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any perturbation from linearised data in the two-dimensional linearised Calderón problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calderón problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full basis of current densities.
Paper Structure (8 sections, 4 theorems, 64 equations, 3 figures)

This paper contains 8 sections, 4 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Article structure
  3. A simple numerical example
  4. Constructing the 3D Zernike orthonormal basis
  5. On spherical harmonics
  6. On 3D radial Zernike polynomials
  7. Proof of Theorem
  8. Extension of the linearised problem

Key Result

Theorem 1.1

For any $\eta \in L^3(B)$, expanded as for an $\ell^2$-sequence of coefficients $c_{\ell}^{k,m}$, then The scalars $Q_{\ell,s}^{k,m,q}$ are independent of $\eta$, and defined as using Pochhammer symbols (rising factorials) and the Gaunt coefficient

Figures (3)

  • Figure 2.1: The perturbation $\eta$ from \ref{['eq:etaexample']} and $\omega_K$ from \ref{['eq:omegaK']} for $K = 0,\dots,7$. The plots are in the $xy$-plane.
  • Figure 2.2: The perturbation $\eta$ from \ref{['eq:etaexample']} and approximations $\widetilde{\omega}_K$ from \ref{['eq:omegatildeK']} for $K = 0,\dots,7$ based on accurate measurements from Mathematica. The plots are in the $xy$-plane.
  • Figure 2.3: The perturbation $\eta$ from \ref{['eq:etaexample']} and approximations $\widetilde{\omega}_K$ from \ref{['eq:omegatildeK']} for $K = 0,\dots,4$ based measurements from a rough FE model. The plots are in the $xy$-plane.

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Proposition A.1
  • proof