The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner

We consider a backward heat conduction problem (BHCP) in a slab, subject to noisy data at final time. The BHCP is known to be highly ill-posed. In order to stably solve the BHCP by a numerical method, we employ a new post-conditioner in the linear system obtained by the method of fundamental solutions (MFS), and then we use the conjugate gradient method (CGM) to solve the post-conditioned linear system to determine the unknown coefficients used in the expansion by the MFS. The method can retrieve the initial data rather well, with a certain degree of accuracy. Several numerical examples of the BHCP demonstrate that the present method is applicable, even for those of strongly ill-posed problems with a large value of final time and with large noise. We also demonstrate that the CGM alone is not enough to accurately recover the initial temperature.