Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion

Xu Wu, Jiang Yang, Zhi Zhou

TL;DR

This work tackles the inverse problem of simultaneously identifying a spatially varying potential $q(x)$ and the initial condition $v(x)$ in a time-fractional subdiffusion model from two terminal observations. It combines a constructive fixed-point reconstruction framework with a regularized backward problem (quasi-boundary value) and a fully discrete numerical scheme that uses finite elements in space and convolution quadrature in time, plus a two-grid approach for handling ill-posed initial data recovery. The authors prove existence, uniqueness, and conditional stability of the inverse problem, establish linear convergence of the iterative method, and derive an a priori error estimate that prescribes how to balance regularization and discretization parameters to achieve optimal rates under noise. Numerical experiments in 2D validate the theory, showing that the proposed method attains rates consistent with the theory and remains robust across smooth, piecewise-smooth, and discontinuous coefficients. The results offer a practical and theoretically sound framework for solving coupled inverse problems in subdiffusion, with potential applications to heterogeneous media and biological transport where fractional dynamics are relevant.

Abstract

This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings.

Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion

TL;DR

This work tackles the inverse problem of simultaneously identifying a spatially varying potential and the initial condition in a time-fractional subdiffusion model from two terminal observations. It combines a constructive fixed-point reconstruction framework with a regularized backward problem (quasi-boundary value) and a fully discrete numerical scheme that uses finite elements in space and convolution quadrature in time, plus a two-grid approach for handling ill-posed initial data recovery. The authors prove existence, uniqueness, and conditional stability of the inverse problem, establish linear convergence of the iterative method, and derive an a priori error estimate that prescribes how to balance regularization and discretization parameters to achieve optimal rates under noise. Numerical experiments in 2D validate the theory, showing that the proposed method attains rates consistent with the theory and remains robust across smooth, piecewise-smooth, and discontinuous coefficients. The results offer a practical and theoretically sound framework for solving coupled inverse problems in subdiffusion, with potential applications to heterogeneous media and biological transport where fractional dynamics are relevant.

Abstract

This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings.

Paper Structure

This paper contains 11 sections, 26 theorems, 194 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Uniqueness and conditional stability of the inverse problem
  3. Preliminary
  4. Conditional stability of the inverse problem.
  5. Regularization and convergence analysis for backward problem
  6. Fully discretization and its error analysis
  7. Semidiscrete scheme to the problem
  8. Error analysis of fully discrete reconstruction
  9. Numerical reconstruction and error estimates
  10. Numerical Experiments
  11. Conclusion

Key Result

Lemma 2.1

Let $u$ be the solution of eqn:pde with potential $q\in \mathcal{A}$ and $\bar{v},\ f\in L^2({\Omega})$ with $\bar{v}\ge 0,\ f-qD\ge 0,\ a.e.$ in $\Omega$. Then there holds $u(t)\ge 0$, for $t> 0$.

Figures (4)

  • Figure 1: Convergence rates of $e_v$ and $e_q$ for recovering $(v_i^\dag, q_j^\dag)$ against $\delta$.
  • Figure 2: Convergence rates for recovering $(v_2^\dag, q_2^\dag)$ using the scheme zhang2022identification.
  • Figure 3: Convergence histories of Algorithm \ref{['alg']} to recover $( v^\dag_2,q^\dag_2)$ for with different $\alpha$ or different $T_1$, $T_2$, where $\delta=10^{-3}$, $\tau=0.005$.
  • Figure 4: Profiles of numerical reconstruction with $\alpha=0.5$, $\tau =0.0025$, $\delta=10^{-3}$. (a) exact initial condition and potential $(v_2^\dag,q_2^\dag)$; (b) 353 iterations, $\|q^{353}-q^{352}\|_{L^2(\Omega)}\le 10^{-8}$; (c) 21 iterations, $\|q^{21}-q^{20}\|_{L^2(\Omega)}\le 10^{-8}$.

Theorems & Definitions (51)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 41 more