This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate according to a probability measure in the set-up of the reduced space. The error of stochastic finite element solutions is usually measured in a root mean square sense regarding their dependence on the stochastic input parameters. An orthogonal projection of a snapshot set onto a corresponding POD basis defines an optimum reduced approximation in terms of a Monte Carlo discretization of the root mean square error. The errors of a weighted POD-greedy Galerkin solution are compared against an orthogonal projection of the underlying snapshots onto a POD basis for a numerical example involving thermal conduction. In particular, it is assessed whether a weighted POD-greedy solutions is able to come significantly closer to the optimum than a non-weighted equivalent. Additionally, the performance of a weighted POD-greedy Galerkin solution is considered with respect to the mean absolute error of an adjoint-corrected functional of the reduced solution.

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Abstract.A controllable repairman model consists of L machines subject to failures and two repair servers working at different speeds. The problem of optimal allocation of failed machines between the servers is examined. The optimal control policy is calculated versus cost structures. As a result the optimal policy can be of threshold type, hysteretic type or have more complicated form. It is shown that the corresponding Markov process for hysteretic control policy belongs to the class of the Quasi-Birth-and-Death processes (QBD) with three diagonal block infinitesimal matrix. The stationary characteristics in this case are derived in matrix analytic form. Some numerical results are used to illustrate a number of features of the controlled model under study.