We aim at the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $\alpha\in(1,2)$. From terminal observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the "mildly" ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying finite element method in space and convolution quadrature in time. We establish error bounds of the discrete solution in both cases of smooth and nonsmooth data. The error estimate is very useful in practice since it indicates the way to choose discretization parameters and regularization parameter, according to the noise level. The theoretical results are supported by numerical experiments.

中文翻译：

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后向扩散波问题：稳定性、正则化和逼近

我们的目标是开发和分析近似求解后向扩散波问题的数值方案，该问题涉及时间阶为 $\alpha\in(1,2)$ 的分数阶导数。从两个时间级别的终端观察，即 $u(T_1)$ 和 $u(T_2)$，我们同时恢复了两个初始数据 $u(0)$ 和 $u_t(0)$，因此解决方案 $u( t)$ 表示所有 $t > 0$。首先，在$T_1$和$T_2$左右的一些条件下，建立了后向扩散波问题的存在性、唯一性和Lipschitz稳定性。此外，对于噪声数据，我们提出了一种准边界值方案来正则化“轻度”不适定问题，并显示正则化解的收敛性。接下来，为了数值求解正则化问题，通过在空间上应用有限元方法和在时间上应用卷积求积，提出了一种完全离散的方案。我们在平滑和非平滑数据的两种情况下建立离散解的误差界限。误差估计在实践中非常有用，因为它指示了根据噪声水平选择离散化参数和正则化参数的方式。理论结果得到数值实验的支持。