A Weighted POD Method for Elliptic PDEs with Random Inputs
For researchers solving parametric PDEs with random inputs, this method offers a computationally cheaper alternative to weighted greedy methods when error bounds are unavailable.
The authors propose a weighted proper orthogonal decomposition (POD) method for elliptic PDEs with random inputs, which minimizes error in L2 norm without requiring an error bound, and uses sparse sampling to reduce computational costs. Numerical tests show the method achieves good accuracy compared to unweighted POD and weighted greedy approaches.
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a $L^2$ norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.