Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution 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 parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ 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 areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).