The problem of reconstructing the initial condition in the Neumann problem for linear parabolic equations with space-and-time-dependent coefficients from noisy observation of the solution on a part of the boundary is studied. Three variational methods are suggested for solving the problem: 1) the least squares method which aims at the minimizing the misfit between the observation on the boundary by varying the initial condition, 2) J.-L. Lions’ method (proposed in 1968) which minimizes the gap between the solutions to the corresponding Neumann and Dirichlet problems when the observation is taken on whole boundary, and 3) energy space approach which minimizes an energy-like functional measuring the gap between the solutions to the corresponding Neumann and Dirichlet problems when the observation is taken on whole boundary. For these problems, we provide the gradient of the functionals to be minimized and derive the first optimality conditions. To solve the problems numerically, we discretize the problems either by the finite element method or by the boundary element method. The error estimates are proved and numerical examples are tested which show the efficiency of our approaches.

研究了从对一部分边界的解的嘈杂观测中，得出了具有时空相关系数的线性抛物方程的Neumann问题中的初始条件重构问题。建议使用三种变分方法来解决该问题：1）最小二乘法，旨在通过改变初始条件来最小化边界上的观测值之间的不匹配； 2）J.-L。Lions的方法（于1968年提出）在对整个边界进行观测时最小化了对应的Neumann和Dirichlet问题的解之间的差距，以及3）能量空间方法，这种方法最小化了类似能量的函数来测量解之间的差距当在整个边界上进行观察时，相应的Neumann和Dirichlet问题也会出现。对于这些问题，我们提供了要最小化的函数梯度，并得出了第一个最优性条件。为了用数值方法解决问题，我们通过有限元法或边界元法离散化问题。证明了误差估计并测试了数值示例，这些结果表明了我们方法的有效性。