标题： 非线性反源问题在耦合扩散方程中的定量估计及不确定测量值 显示英文标题

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

摘要： 本文研究了一类耦合扩散方程终端观测下的非线性逆源问题。 从理论上讲，在问题数据满足某些条件下，我们建立了该反问题的唯一性定理，并分别给出了在 $L^2$ 和 $(H^1(\cdot))^*$ 范数下的两个Lipschitz型稳定性结果。 然而，在实际应用中，我们只能在离散传感器处观测到含有噪声的数据。 因此，本文进一步研究了从离散含噪测量值恢复未知源的问题。 我们提出了一种稳定的反演方案，并在两种情况下提供了重构与精确解之间的概率收敛估计：期望意义上的收敛和具有指数尾部的收敛。 我们还提供了一些数值实验来说明并补充我们的理论分析。 显示英文摘要

摘要： 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.