Accuracy controlled data assimilation for parabolic problems

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that strict subset computational domain under consideration. Unlike previous approaches this related our starting point regularized least squares formulation in continuous infinite-dimensional setting based stable variational formulations PDE. allows us derive priori as well posteriori error bounds for recovered states respect certain reference solution. In these regularization parameter disentangled underlying discretization. An important ingredient derivation construction suitable Fortin operators which allow control oscillation errors stemming discretization dual norms. Moreover, framework contrive preconditioners discrete whose application can be performed linear time, condition numbers preconditioned systems are uniformly proportional problem. particular, we provide stopping criteria iterative solvers bounds. The presented numerical experiments quantify theoretical findings demonstrate performance scheme relation regularization.