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Reconstruction algorithms for the fractional Laplacian and applications to inverse problems

Ethan Rinaldo, Mahamadi Warma

Abstract

We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space $\mathbb{R}^n$ from local data $(u|_W, [(-Δ)^s u]|_W)$, where $W$ is an arbitrarily small nonempty open subset of $\mathbb R^n$ and $(-Δ)^s$ denotes the fractional Laplace operator of order $s\in (0,1)$. These procedures rely crucially on the weak Unique Continuation Property (UCP) for the fractional Laplacian. We apply these schemes to two distinct inverse problems. Following the seminal work from Ghosh et al., the first one concerns the recovery of a potential (Calderón-type problem) from the fractional Schrödinger equation under nonlocal Robin-type exterior conditions. The second one involves recovering the solution of the space-fractional heat equation in $\mathbb{R}^n$ from localized time-dependent measurements within a ball. To tackle these problems, we introduce new analytical tools such as a generalized weak Kelvin transform and a fractional Robin-to-Robin map. Finally, we provide numerical simulations for one of the reconstruction methods, illustrating the stability issues and the severe ill-posedness inherent to such inverse problems.

Reconstruction algorithms for the fractional Laplacian and applications to inverse problems

Abstract

We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space from local data , where is an arbitrarily small nonempty open subset of and denotes the fractional Laplace operator of order . These procedures rely crucially on the weak Unique Continuation Property (UCP) for the fractional Laplacian. We apply these schemes to two distinct inverse problems. Following the seminal work from Ghosh et al., the first one concerns the recovery of a potential (Calderón-type problem) from the fractional Schrödinger equation under nonlocal Robin-type exterior conditions. The second one involves recovering the solution of the space-fractional heat equation in from localized time-dependent measurements within a ball. To tackle these problems, we introduce new analytical tools such as a generalized weak Kelvin transform and a fractional Robin-to-Robin map. Finally, we provide numerical simulations for one of the reconstruction methods, illustrating the stability issues and the severe ill-posedness inherent to such inverse problems.
Paper Structure (19 sections, 30 theorems, 150 equations, 6 figures, 1 table)

This paper contains 19 sections, 30 theorems, 150 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Functional settings
  3. Fractional order function spaces
  4. The Fractional Laplace operator and its properties
  5. Direct and inverse fractional Robin problems
  6. $L^{\infty}$ and $L^{1}$-weak solutions
  7. Convergence to the Dirichlet problem
  8. The inverse problem using the Robin-to-Robin map
  9. The inverse problem for $L^{\infty}$-weak solutions
  10. The inverse problem for $L^1$-weak solutions
  11. A new notion of Kelvin transform
  12. Reconstruction algorithms and applications
  13. Reconstruction algorithms
  14. Reconstruction with Kelvin transform
  15. Reconstruction with Green functions
  16. ...and 4 more sections

Key Result

Lemma 2.2

The space $V$ endowed with the inner product with associated norm given in norm-V is a Hilbert space.

Figures (6)

  • Figure 1: Fractional Heat Equation
  • Figure 2: Smooth extension outside the ball
  • Figure 3: $(-\Delta)^s K_\alpha[h]$ vs. $\frac{1}{|x|^{2\alpha}} K_\alpha[(-\Delta)^s h]$ for $s=0.3$ (on the left) and $s=0.75$ (on the right).
  • Figure 4: Relative $L^2$ errors in reconstruction for $h$ with $u(x) = \frac{\sin(x)}{x}$ and truncation interval is $(-5, 5)$. Each curve represents a different value of $\alpha := 2s$.
  • Figure 5: Relative $L^2$ errors in reconstruction for $h$ with $u(x) = e^{-x^2}$ where truncation interval is $(-5, 5)$ and $\alpha=0.5$. Each curve represents a different regularization coefficient $\lambda$.
  • ...and 1 more figures

Theorems & Definitions (67)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Proposition 2.8
  • ...and 57 more