Quantitative estimates for a nonlinear inverse source problem in a coupled diffusion equations with uncertain measurements

TL;DR

This work analyzes a nonlinear inverse source problem in a coupled diffusion system, showing uniqueness and Lipschitz stability for recovering the source $q(x)$ from terminal data $g(x)=u_m(x,T)$. It introduces a monotone fixed-point operator $K$ linking fixed points to the true solution and establishes positivity, monotonicity, and energy-based stability results. For practical data with discrete noisy measurements, the problem is split into two subproblems $P1$ and $P2$, with stochastic error analysis in both expectation and exponential-tail forms, and a self-consistent, data-driven scheme to select the regularization parameter $\lambda$. The authors propose two numerical algorithms and validate them with 2D experiments, confirming the theoretical rates and illustrating robustness to noise, including different regularization penalties for smooth and non-smooth sources. Overall, the paper advances deterministic and probabilistic understanding of nonlinear inverse source problems in coupled diffusion models under realistic measurement constraints.

Abstract

This work considers a nonlinear inverse source problem in a coupled diffusion equation from the terminal observation. Theoretically, under some conditions on problem data, we build the uniqueness theorem for this inverse problem and show two Lipschitz-type stability results in $L^2$ and $(H^1(\cdot))^*$ norms, respectively. However, in practice, we could only observe the measurements at discrete sensors, which contain the noise. Hence, this work further investigates the recovery of the unknown source from the discrete noisy measurements. We propose a stable inversion scheme and provide probabilistic convergence estimates between the reconstructions and exact solution in two cases: convergence respect to expectation and convergence with an exponential tail. We provide several numerical experiments to illustrate and complement our theoretical analysis.

Figures (11)

• Figure 1: Numerical results for different regularization parameter $\lambda$ (Manual test, noise $10\%$).
• Figure 2: The optimal regularization parameter provided by self-consistent Algorithm \ref{['alg1']}. The empirical errors $\log Err1$, $\log Err2$ and $\log Err3$ by Algorithm \ref{['alg1']} with respect to iteration number ($s=0$ or $1$).
• Figure 3: The exact and reconstructed solutions by Algorithm \ref{['alg1']} for Example \ref{['ex1']}.
• Figure 4: (a), (c): The linear dependence of $\mathbb{E}(Err1)$ on $\lambda^{1/2}$ for $s=0$ and $s=1$; (b): The linear dependence of $\mathbb{E}(Err2)$ on $\lambda^{1/3}$ for $s=0$; (d): The linear dependence of $\mathbb{E}(Err3)$ on $\lambda^{1/6}$ for $s=1$.
• Figure 5: Profiles of exact solutions $q_1^*$ and $q_2^*$ in Example \ref{['ex2']}.

Theorems & Definitions (37)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.1
• Lemma 2.5
• proof
• Lemma 2.6