论文标题
与时间相关系数的稳定性和数值分析
Stability and numerical analysis of backward problem for subdiffusion with time-dependent coefficients
论文作者
论文摘要
我们的目的是研究向后问题，即从终端观察中恢复初始数据，并通过时间依赖系数进行延伸。首先，通过使用解决方案操作员的平滑属性和冻结扩散系数的扰动参数，我们在终端时间的某些较小/宽敞的条件下，在Sobolev空间中显示了稳定性估计值。此外，在观察嘈杂的情况下，我们采用了一种准边界价值方法来正规化问题，然后显示正规化方案的收敛性。最后，为了通过数值重建初始数据，我们通过在空间和向后的Euler卷积正交正交中应用有限元方法来提出一个完全离散的方案。然后建立一个\ textsl {先验}错误估计。该证明是基于涉及时间依赖系数的扰动参数和直接问题的一些非标准误差估计。误差估计为平衡离散参数，正则化参数和噪声水平提供了有用的指南。提出了一些数值实验，以说明我们的理论结果。
Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a perturbation argument of freezing the diffusion coefficients, we show a stability estimate in Sobolev spaces, under some smallness/largeness condition on the terminal time. Moreover, in case of noisy observation, we apply a quasi-boundary value method to regularize the problem and then show the convergence of the regularization scheme. Finally, to numerically reconstruct the initial data, we propose a completely discrete scheme by applying the finite element method in space and backward Euler convolution quadrature in time. An \textsl{a priori} error estimate is then established. The proof is heavily built on a perturbation argument dealing with time dependent coefficients and some nonstandard error estimates for the direct problem. The error estimate gives a useful guide for balancing discretization parameters, regularization parameter and noise level. Some numerical experiments are presented to illustrate our theoretical results.
