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A numerical method for reconstructing the potential in fractional Calderón problem with a single measurement

Xinyan Li

Abstract

In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calderón problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.

A numerical method for reconstructing the potential in fractional Calderón problem with a single measurement

Abstract

In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calderón problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.
Paper Structure (9 sections, 4 theorems, 74 equations, 4 figures)

This paper contains 9 sections, 4 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview and algorithms of the forward problem
  3. Overview of the well-posedness
  4. The finite difference scheme for forward problem
  5. The inversion algorithm
  6. One dimension inversion
  7. Two dimension inversion
  8. Numerical inversions
  9. Conclusions

Key Result

Lemma 2.1

GhoSal Let $n\geq1,s\in(0,1)$, $\Omega$ be a bounded open set and $q\in L^{\infty}(\Omega)$. Suppose that Assumption ass1 holds. Let Then for any $f\in H^s(\mathbb{R}^n)$ and $g\in (\tilde{H}^s(\Omega))^*$, there is a unique solution $u\in H^s(\mathbb{R}^n)$ in fseghosh satisfying Moreover, the norm estimate holds with $C$ independent of $g$ and $f$.

Figures (4)

  • Figure 1: The numerical results for Example 4.1 with different noise level $\delta$, (a) the reconstruction results $q_{\alpha}^{\delta}(x)$ and real solution $q(x)$; (b) the relationship between $|log(\delta)|^{-1}$ and $\|q_{\alpha}^{\delta}-q\|_{L^{\infty}(\Omega)}$.
  • Figure 2: The numerical results for Example 4.2 with different noise level $\delta$, (a) the reconstruction results $q_{\alpha}^{\delta}(x)$ and real solution $q(x)$; (b) the relationship between $|log(\delta)|^{-1}$ and $\|q_{\alpha}^{\delta}-q\|_{L^{\infty}(\Omega)}$.
  • Figure 3: The numerical results of estimated potential for Example 4.3 with noise level $\delta={\rm 1E-7}$ and regularization parameter $\alpha={\rm 1E-14}$ along with the real potential $q(x)$.
  • Figure 4: The numerical results of two-dimensional fraction Calderón problem for Example 4.4, (a) the real potential $q(x,y)$; (b) the reconstruction potential $q_{\alpha}^{\delta}(x,y)$ with $\delta$=1E-6,$\alpha$=1E-13; (c) the reconstruction error $|q_{\alpha}^{\delta}(x,y)-q(x,y)|$; (d) the relationship between $|log(\delta)|^{-1}$ and $\|q_{\alpha}^{\delta}-q\|_{L^{\infty}(\Omega)}$ with $\alpha=0.1\delta^2$.

Theorems & Definitions (11)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.2
  • ...and 1 more