Stable reconstruction of the initial condition in parabolic equations from boundary observations

Abstract 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.