Solving the backward heat conduction problem by homotopy analysis method

The backward heat conduction problems aim to determine the temperature distribution in the past time from the present measurement data. For this linear ill-posed problem, we propose a homotopy-based iterative regularizing scheme for noisy input data. The advantages of the proposed scheme are, under general assumptions on the exact initial distribution, we can always ensure the convergence of the homotopy sequence with exact final data as initial guess. For noisy input data, we also establish the error analysis for the regularizing solution with noisy measurement data as our initial guess. Our algorithm is easily implementable with very low computational costs in the sense that we only need to do one iteration from initial guess using the final noisy data directly, while the error is still comparable to other regularizing methods. Numerical implementations are presented.