  We consider a one-dimensional one-phase inverse Stefan problem for the heat
equation. It consists in recovering a boundary influx condition from the
knowledge of the position of the moving front, and the initial state. We
derived a logarithmic stability estimate that shows that the inversion is
ill-posed. The proof is based on integral equations and unique continuation for
holomorphic functions. We also proposed a direct algorithm with a
regularization term to solve the nonlinear inverse problem. Several numerical
tests using noisy data are provided with relative errors analysis.
