A weighted reduced basis method for parabolic PDEs with random data
For researchers solving parametric PDEs with random inputs, this work offers an incremental improvement to reduced basis methods by incorporating a probability measure into the basis selection process.
This paper proposes a weighted POD-greedy method for constructing reduced bases for parabolic PDEs with random data, aiming to improve the accuracy of statistical outputs. Numerical results for a thermal conduction problem show that the weighted method achieves errors closer to the optimal POD projection compared to the non-weighted version, with improvements in mean absolute error for an adjoint-corrected functional.
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.