Numerical Recovery of the Diffusion Coefficient in Diffusion Equations from Terminal Measurement
Bangti Jin, Xiliang Lu, Qimeng Quan, Zhi Zhou
TL;DR
The paper tackles the inverse problem of identifying a spatially varying diffusion coefficient in a time-fractional diffusion equation from terminal data. It establishes a Hölder-type conditional stability result applicable to both normal and subdiffusive regimes, based on weighted energy estimates and decay properties of the fractional derivative. A fully discrete regularized output least-squares scheme is analyzed, employing Galerkin FEM in space and backward Euler convolution quadrature in time, with an a priori error bound in $L^2(\Omega)$ that relates the reconstruction error to the noise level, discretization, and regularization, and provides practical parameter guidelines (e.g., $\gamma \sim \delta^2$, $h \sim \delta^{1/2}$). The framework extends prior terminal-data results to the fractional setting and offers a unified analysis for $\alpha=1$ and $0<\alpha\le1$, supported by numerical experiments in 1D and 2D that corroborate the theoretical findings and illustrate the method’s effectiveness for terminal-data-based diffusion coefficient recovery. This contributes a rigorous foundation for reliable diffusion-coefficient identification from late-time measurements in anomalous diffusion models and informs parameter choices in practical implementations.
Abstract
In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{ö}lder type stability estimate for a large terminal time $T$. This is achieved by novel decay estimates of the (fractional) time derivative of the solution. To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an $H^1(Ω)$-seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived $L^2(Ω)$ error bound depends explicitly on the noise level, regularization parameter and discretization parameter(s), which gives a useful guideline of the \textsl{a priori} choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.