Two-dimensional nonlinear inverse heat conduction problem

Y and T: Temperatures. Y: Solution of the direct problem, in K. T: Solution of the inverse problem, in K. k: Iteration of the conjugate gradient method. The inverse problem is solved using this method. x: Space. t: Time. Inverse problem in Case 1.1: Temperatures on the inner boundary of an annulus are estimated from data (namely, temperatures and heat fluxes) on the outer boundary of the annulus. Inverse problem in Case 2.1: Temperatures on the outer boundary of an annulus are estimated from data (namely, temperatures and heat fluxes) on the inner boundary of the annulus. In both cases, the initial condition is known. In both cases, the redundant boundary data and initial conditions for the inverse problem are perturbed with Gaussian noise. Case_1.1/DHCP: Y(x, t) for Case 1.1. Case_1.1/11: T(x, t) for Case 1.1 at k = 11. Case_1.1/Case_1.1_vid.mp4: Y(x, t) for Case 1.1 is compared with T(x, t) for Case 1.1 at k = 11. Upper-left corner: t, in s. Case_2.1/DHCP: Y(x, t) for Case 2.1. Case_2.1/3: T(x, t) for Case 2.1 at k = 3. Case_2.1/Case_2.1_vid.mp4: Y(x, t) for Case 2.1 is compared with T(x, t) for Case 2.1 at k = 3. Upper-left corner: t, in s. Y(x, t) is saved to file T.*. T(x, t) is saved to file dir_IHCP.*. HDF5 files (*.h5) can be read using, for example, the h5py package in Python. XDMF files (*.xdmf) can be read using, for example, ParaView. MP4 files (*.mp4) can be played using, for example, VLC.

变量Y与T均代表温度，单位为开尔文（K）。其中Y为正问题（direct problem）的解，T为反问题（inverse problem）的解。k为共轭梯度法（conjugate gradient method）的迭代次数，本次反问题即通过该方法求解。x代表空间坐标，t代表时间坐标。

案例1.1的反问题：通过环形空腔外边界的观测数据（即温度与热通量（heat fluxes）），估算该环形空腔内边界的温度分布。
案例2.1的反问题：通过环形空腔内边界的观测数据（即温度与热通量），估算该环形空腔外边界的温度分布。

两个案例均已知初始条件，且反问题所需的冗余边界数据与初始条件均被施加了高斯噪声（Gaussian noise）扰动。

数据集文件结构与说明如下：
- Case_1.1/DHCP：存储案例1.1的Y(x, t)数据
- Case_1.1/11：存储案例1.1在迭代次数k=11时的T(x, t)数据
- Case_1.1/Case_1.1_vid.mp4：用于对比展示案例1.1的Y(x, t)与k=11时的T(x, t)，视频左上角标注时间t，单位为秒（s）
- Case_2.1/DHCP：存储案例2.1的Y(x, t)数据
- Case_2.1/3：存储案例2.1在迭代次数k=3时的T(x, t)数据
- Case_2.1/Case_2.1_vid.mp4：用于对比展示案例2.1的Y(x, t)与k=3时的T(x, t)，视频左上角标注时间t，单位为秒（s）

数据存储规则：Y(x, t)保存于以T.*为命名前缀的文件中，T(x, t)保存于以dir_IHCP.*为命名前缀的文件中。
各类文件读取方式：
- HDF5格式文件（*.h5）可通过Python的h5py库读取；
- XDMF格式文件（*.xdmf）可通过ParaView等可视化工具读取；
- MP4格式视频文件（*.mp4）可通过VLC等播放器播放。