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A Galerkin Finite Element Method for the Fractional Calderón Problem

Mukul Dwivedi, Jesse Railo, Andreas Rupp

Abstract

We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.

A Galerkin Finite Element Method for the Fractional Calderón Problem

Abstract

We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.

Paper Structure

This paper contains 17 sections, 12 theorems, 194 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and single-measurement reconstruction
  3. Discrete unique continuation and reconstruction of the potential
  4. Discrete setting: truncation domain, finite element spaces, and discrete norms
  5. Truncated fractional energy and truncation error
  6. Deterministic consistency of the preprocessed data and total consistency error
  7. Discrete Tikhonov functional and the normal equations
  8. Convergence of the reconstructed state
  9. Reconstruction of q
  10. Computing $(-\Delta)^s u_h^\alpha$ in $\Omega$
  11. Convergence and error analysis for the reconstructed potential
  12. Logarithmic stability for the reconstructed potential (single measurement)
  13. Numerical experiments
  14. Example 4.1 (1D, compactly supported smooth potential).
  15. Example 4.2 (1D, discontinuous potential).
  16. ...and 2 more sections

Key Result

Lemma 2.3

Assume eq:W-sep and $W$ is bounded. If $v\in L^2(\Omega)$ is extended by $0$ outside $\Omega$, then for every $x\in W$ the quantity is well-defined and $\mathcal{L}v$ is smooth on $W$. Moreover, one has the bound and for every integer $m\ge 0$ there exists $C_m>0$ such that In particular, $\mu$ in eq:mu-def is a pointwise-defined function on $W$ whenever $u_0\in L^2(\Omega)$.

Figures (4)

  • Figure 1: Example \ref{['subsec:ex41']}. One-dimensional reconstruction: panel (A) shows the recovered interior component $u_{0,h}^{\alpha,\delta}$ for the representative noise levels, panel (B) shows the decomposition of the state-step error into total, bias, and propagated-noise parts, panel (C) shows the recovered coefficient $q_h^{\alpha,\delta}$, and panel (D) shows the potential error $\|q_h^{\alpha,\delta}-q\|_{L^\infty(\Omega')}$ as a function of the relative noise level $\delta$.
  • Figure 2: Example \ref{['subsec:ex42']}: discontinuous potential reconstructions on $\Omega'$.
  • Figure 3: Example \ref{['sec:ex43-2d']} (2D): state recovery for the representative noise level $\delta=10^{-8}$ with the same parameters as Fig. \ref{['fig:ex43-2d-q']}(B).
  • Figure 4: Example \ref{['sec:ex43-2d']} (2D bump potential). Setup: $\Omega = (-1,1)^2$, $\Omega_R = (-3,3)^2$, $\Omega' = (-0.75,0.75)^2$, $W=\Omega_R\setminus(-1-\varepsilon,1+\varepsilon)^2$ with $\varepsilon=0.05$, fractional order $s=0.5$, and grid size $h=0.05$. Panel (A) shows true potential $q$. Panels (B)--(C) use $\delta=10^{-8}$, $\alpha=0.1\,\delta^{3/2}$, and $\alpha_q=0.01\,\delta$. Panel (D) shows stability trend on the tested noise range $\delta\in[10^{-10},10^{-1}]$.

Theorems & Definitions (28)

  • Remark 2.2
  • Lemma 2.3: Pointwise meaning of the preprocessed measurement
  • proof
  • Lemma 3.3: Discrete data norm converges to $L^2(W)$
  • proof
  • Lemma 3.4: Tail integral bound
  • proof
  • Theorem 3.5: Existence, uniqueness, and normal equations
  • proof
  • Lemma 3.6
  • ...and 18 more