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Reconstruction of a potential parameter in subdiffusion via a Kohn--Vogelius type functional: Theory and computation

Hamza Kahlaoui, Mourad Hrizi, Abdessamad Oulmelk, Xiangcheng Zheng, Mahmoud A. Zaky, Ahmed Hendy

TL;DR

The paper tackles the ill-posed inverse problem of reconstructing a space-dependent potential in subdiffusion from boundary measurements by casting it as a regularized optimization of a Kohn–Vogelius-type functional, $\mathcal{K}_\rho$, which balances Dirichlet/Neumann discrepancy with a $L^2$-norm penalty on $q$.A rigorous analytical framework establishes the forward problem's well-posedness, continuity and Fréchet differentiability of the forward maps, and the Fréchet differentiability and Lipschitz continuity of the Kohn–Vogelius functional's gradient, enabling a convergent conjugate gradient method with an explicit gradient via adjoint equations.Existence, uniqueness (for large enough $\rho$), stability with respect to boundary data, and convergence as $\rho\to0$ are proved, and the method is validated numerically in 1D and 2D, including noisy data and both smooth and non-smooth potentials.Compared to least-squares, the regularized Kohn–Vogelius approach provides improved robustness to noise and discontinuities, with accurate reconstructions demonstrated in diverse scenarios and fractional orders $0<\alpha<1$.

Abstract

This work considers the reconstruction of a space-dependent potential from boundary observations in subdiffusion by a stable and robust recovery method. Specifically, we develop an algorithm to minimize the Kohn-Vogelius cost function, which measures the difference between the solutions of two excitations. The inverse potential problem is recast into an optimization problem, where the objective is to minimize a Kohn-Vogelius-type functional within a set of admissible potentials. We establish the well-posedness of this optimization problem by proving the existence and uniqueness of a minimizer and demonstrating its stability with respect to perturbations in the boundary data. Furthermore, we analyze the Fréchet differentiability of the Kohn-Vogelius functional and prove the Lipschitz continuity of its gradient. These theoretical results enable the development of a convergent conjugate gradient algorithm for numerical reconstruction. The effectiveness and robustness of the proposed method are confirmed through several numerical examples in both one and two dimensions, including cases with noisy data.

Reconstruction of a potential parameter in subdiffusion via a Kohn--Vogelius type functional: Theory and computation

TL;DR

The paper tackles the ill-posed inverse problem of reconstructing a space-dependent potential in subdiffusion from boundary measurements by casting it as a regularized optimization of a Kohn–Vogelius-type functional, $\mathcal{K}_\rho$, which balances Dirichlet/Neumann discrepancy with a $L^2$-norm penalty on $q$.A rigorous analytical framework establishes the forward problem's well-posedness, continuity and Fréchet differentiability of the forward maps, and the Fréchet differentiability and Lipschitz continuity of the Kohn–Vogelius functional's gradient, enabling a convergent conjugate gradient method with an explicit gradient via adjoint equations.Existence, uniqueness (for large enough $\rho$), stability with respect to boundary data, and convergence as $\rho\to0$ are proved, and the method is validated numerically in 1D and 2D, including noisy data and both smooth and non-smooth potentials.Compared to least-squares, the regularized Kohn–Vogelius approach provides improved robustness to noise and discontinuities, with accurate reconstructions demonstrated in diverse scenarios and fractional orders $0<\alpha<1$.

Abstract

This work considers the reconstruction of a space-dependent potential from boundary observations in subdiffusion by a stable and robust recovery method. Specifically, we develop an algorithm to minimize the Kohn-Vogelius cost function, which measures the difference between the solutions of two excitations. The inverse potential problem is recast into an optimization problem, where the objective is to minimize a Kohn-Vogelius-type functional within a set of admissible potentials. We establish the well-posedness of this optimization problem by proving the existence and uniqueness of a minimizer and demonstrating its stability with respect to perturbations in the boundary data. Furthermore, we analyze the Fréchet differentiability of the Kohn-Vogelius functional and prove the Lipschitz continuity of its gradient. These theoretical results enable the development of a convergent conjugate gradient algorithm for numerical reconstruction. The effectiveness and robustness of the proposed method are confirmed through several numerical examples in both one and two dimensions, including cases with noisy data.

Paper Structure

This paper contains 33 sections, 26 theorems, 294 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Well-posedness of the forward problem
  4. Notations and auxiliary results
  5. Well-posedness of the forward problem
  6. Problem reformulation -A Kohn-Vogelius formulation
  7. Existence and uniqueness of optimal solutions
  8. Continuity of the Dirichlet and Neumann solution operators
  9. Existence of optimal solutions
  10. Uniqueness of optimal solutions
  11. Lipschitz continuity analysis of the forward operators
  12. Fréchet differentiability analysis of the forward operators
  13. The Fréchet derivative of $\mathcal{K}_\rho$
  14. Uniqueness
  15. Stability and convergence of the Kohn and Vogelius method
  16. ...and 18 more sections

Key Result

Lemma 2

(see li2009space). Let $\alpha\in(0,1)$. Then for any $w\in {}_0H^{\alpha}(0,T)$ and $v\in {}_0H^{\frac{\alpha}{2}}(0,T)$, we have

Figures (10)

  • Figure 1: The results obtained with different fractional orders for Example \ref{['example0']}.
  • Figure 2: Numerical results of the conjugate gradient algorithm for regular Examples.
  • Figure 3: Numerical results of the conjugate gradient algorithm for complex Examples.
  • Figure 4: Numerical solutions for Example \ref{['example1']} with varying values of the regularization parameter $\rho$.
  • Figure 5: Numerical results of the conjugate gradient algorithm for regular Examples with noise.
  • ...and 5 more figures

Theorems & Definitions (53)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 5
  • Theorem 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 43 more