Stable reconstruction of a spatial source term in a time-space fractional diffusion equation using Tikhonov regularization and final-time measurements

Technical Summary: Stable Reconstruction of a Spatial Source Term in a Time-Space Fractional Diffusion Equation

Problem Statement
This research addresses an inverse source problem associated with a time-space fractional diffusion equation. The objective is to uniquely and stably reconstruct a space-dependent source term, $f(x)$, given noisy final-time measurements, $y(x, T)$, at a terminal time $T$. The governing equation models anomalous diffusion processes characterized by memory effects (via the Caputo time-fractional derivative $\partial_t^\alpha$) and long-range interactions (via the fractional Laplacian $(-\Delta)^s$). The source term is assumed to be separable, taking the form $F(x, t) = g(t)f(x)$, where $g(t)$ is a known function and $f(x)$ is the unknown spatial component to be identified. The problem is inherently ill-posed, meaning that small perturbations in the measurement data can lead to large deviations in the reconstructed source, necessitating regularization.

Methodology
The authors employ a combined theoretical and numerical approach:

1. Theoretical Analysis of the Direct Problem:

   • The well-posedness of the forward problem is established using functional analysis in fractional Sobolev spaces ($K^s(\Omega)$).
   • The existence and uniqueness of the weak solution are proven, relying on spectral representations involving Mittag-Leffler functions.
   • Uniqueness of the Inverse Problem: The paper proves that the space-dependent source $f(x)$ is uniquely determined by the final-time observation $y(\cdot, T)$, provided the time-weighting function $g(t)$ is non-negative and non-zero.

2. Optimization and Regularization:

   • The inverse problem is reformulated as a minimization problem using a Tikhonov regularization functional:
     $$L_\eta(f) = \|y(\cdot, T) - Z^\delta\|_{L^2(\Omega)}^2 + \eta \|f\|_{L^2(\Omega)}^2$$
     where $Z^\delta$ represents noisy data and $\eta > 0$ is the regularization parameter.
   • The authors prove the existence of a minimizer for this functional and establish the stability of the solution with respect to perturbations in the data.

3. Numerical Algorithm:

   • A gradient descent algorithm (specifically a conjugate gradient method) is developed to solve the minimization problem.
   • To compute the gradient of the cost functional, an adjoint problem is introduced. The adjoint equation involves a backward Riemann-Liouville fractional derivative ($D_{T-}^\alpha$) and a terminal condition derived from the residual between the computed and observed data.
   • The gradient is explicitly derived as $\nabla L_\eta(f) = 2(\int_0^T g(t)p(\cdot, t)dt + \eta f)$, where $p$ is the solution to the adjoint problem.
   • The numerical implementation utilizes finite difference methods for discretizing both the forward and adjoint equations.

Key Results
The paper presents four numerical experiments in a one-dimensional domain ($\Omega = (0, 1)$) to validate the proposed method:

• Smooth Sources: The algorithm successfully reconstructs smooth source terms (e.g., polynomial and sinusoidal functions) with high accuracy, both in noise-free scenarios and with added noise levels of 0.1%, 5%, and 10%.
• Complex/Discontinuous Sources: The method is tested on piecewise constant and piecewise linear (sawtooth) sources. While the algorithm demonstrates stability in the presence of noise, the results indicate limitations in capturing sharp discontinuities and singularities. The reconstructed solutions exhibit smoothing artifacts near jump discontinuities.
• Convergence: The relative error between the reconstructed and exact source terms decreases exponentially with the number of iterations, confirming the convergence of the iterative scheme.

Significance and Claims
The authors claim that the primary contributions of this work are:

1. Theoretical Rigor: Establishing the well-posedness of the direct problem and the uniqueness of the inverse source problem for the specific time-space fractional model.
2. Stable Reconstruction Framework: Formulating the ill-posed inverse problem as a stable optimization problem via Tikhonov regularization and proving the existence and stability of the minimizer.
3. Algorithmic Implementation: Developing a gradient-based reconstruction algorithm that explicitly handles the nonlocal nature of the fractional operators through an adjoint state approach.

The paper concludes that the proposed approach is effective and stable for reconstructing regular source terms from noisy final-time data. However, the authors modestly acknowledge that the gradient descent method struggles to accurately capture the singularities inherent in complex, discontinuous source terms, suggesting that further improvements or alternative approaches are needed to address these specific limitations. The study does not propose new applications beyond the scope of the mathematical modeling but validates the method's utility for the specific class of inverse problems defined.