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Identifying the source term in a viscoelastic membrane with a Riemann-Liouville time derivative by the partial interior observation

Zhiwei Yang, Yikan Liu

TL;DR

This work addresses the inverse source problem for a viscoelastic membrane governed by a fractional Riemann-Liouville time derivative with order $\alpha\in(1,2)$, aiming to recover the spatial source $q(\bm x)$ from partial interior data. The authors cast the problem as a PDE-constrained, Tikhonov-regularized optimization and derive an adjoint-based gradient, enabling an efficient finite-element conjugate gradient algorithm that solves both the forward and adjoint problems. Numerical experiments demonstrate robust reconstructions under noisy, limited data and across varying fractional orders and source forms, highlighting the method's stability and versatility. The results suggest the RL derivative offers a physically meaningful memory model for viscoelastic membranes and pave the way for extensions to variable-order, multiscale, and data-driven fractional modeling with potential applications in nondestructive testing and structural health monitoring.

Abstract

This paper studies an inverse source problem for a viscoelastic membrane, where the material's memory effect is characterized by the Riemann-Liouville fractional derivative. The problem is to recover the unknown source term from the limited interior observation data. We propose an optimal control framework to address this ill-posed inverse problem. The first-order optimality condition leads to a coupled system of forward and backward fractional partial differential equations. A numerical algorithm combining the finite element method and a conjugate gradient iterative scheme is then developed for the reconstruction of the source term. Several numerical examples are provided to demonstrate the effectiveness and robustness of the proposed method.

Identifying the source term in a viscoelastic membrane with a Riemann-Liouville time derivative by the partial interior observation

TL;DR

This work addresses the inverse source problem for a viscoelastic membrane governed by a fractional Riemann-Liouville time derivative with order , aiming to recover the spatial source from partial interior data. The authors cast the problem as a PDE-constrained, Tikhonov-regularized optimization and derive an adjoint-based gradient, enabling an efficient finite-element conjugate gradient algorithm that solves both the forward and adjoint problems. Numerical experiments demonstrate robust reconstructions under noisy, limited data and across varying fractional orders and source forms, highlighting the method's stability and versatility. The results suggest the RL derivative offers a physically meaningful memory model for viscoelastic membranes and pave the way for extensions to variable-order, multiscale, and data-driven fractional modeling with potential applications in nondestructive testing and structural health monitoring.

Abstract

This paper studies an inverse source problem for a viscoelastic membrane, where the material's memory effect is characterized by the Riemann-Liouville fractional derivative. The problem is to recover the unknown source term from the limited interior observation data. We propose an optimal control framework to address this ill-posed inverse problem. The first-order optimality condition leads to a coupled system of forward and backward fractional partial differential equations. A numerical algorithm combining the finite element method and a conjugate gradient iterative scheme is then developed for the reconstruction of the source term. Several numerical examples are provided to demonstrate the effectiveness and robustness of the proposed method.
Paper Structure (10 sections, 2 theorems, 50 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 10 sections, 2 theorems, 50 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model formulation
  4. Numerical discretization of the forward problem
  5. Optimization formulation of the inverse source problem
  6. Finite element conjugate gradient method
  7. Numerical validation and performance analysis
  8. Robustness analysis under noisy measurements
  9. Discussion of numerical findings
  10. Concluding remarks

Key Result

Theorem 3.1

The Fréchet derivative of the Tikhonov-regularized cost functional $\mathcal{J}(q)$ defined in ocp:e0 with respect to the source parameter $q \in L^2(\Omega)$ is given by the following explicit formula: where the adjoint state variable $v(q)$ satisfies the backward problem where $\bm1_{\Omega_0}$ denotes the characteristic function of $\Omega_0$.

Figures (2)

  • Figure 1: Reconstruction results for $\alpha=1.5$ with $\Omega_0 = \Omega\setminus[0.05,0.95]^2$, $T=1.5$, and $\delta=1\%$ noise. Top: reconstructed (left) and true (right) solutions. Bottom: absolute error distribution (left) and the corresponding loss with respect to iterations (right).
  • Figure 2: Reconstruction results for three different source term functional forms ($\alpha=1.5$, $\Omega_0=\Omega\setminus[0.05,0.95]^2$, $\delta=1\%$). Each row shows reconstructed (left) and true (right) solutions for cases (1)--(3) respectively.

Theorems & Definitions (4)

  • Theorem 3.1: Gradient of the regularized cost functional
  • Lemma 3.1: Fractional integration by parts
  • proof
  • proof : Proof of Theorem $\ref{['thm:gradient']}$