Recovering surface fluxes on the boundary of the domain from pointwise measurements

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. The prolbem is not well-posed in the Hadamard sense. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed and the numerical method relying on the finite element method in the space variables and a finite difference method in time is constructed. The results of numerical experiments are quite satisfactory and the procedure is stable under small perturbations.