Reduced-basis output bound methods for parabolic problems

In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by parametrized parabolic partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to efficiently compute accurate approximations to the solution of the parabolic problem and relevant outputs of interest. In addition, we develop an error estimation procedure to a posteriori validate the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analyzed for the semidiscrete case. In both cases the reduced-basis is constructed by taking snapshots of the solution in both time and parameters. In that sense the method is close to POD.