# A Weighted POD Method for Elliptic PDEs with Random Inputs

## TL;DR

This paper introduces a weighted proper orthogonal decomposition method for efficiently solving elliptic PDEs with random inputs, reducing computational costs while maintaining high accuracy in stochastic problems.

## Contribution

The paper presents a novel weighted POD approach that does not require error bounds and effectively handles high-dimensional stochastic PDEs with sparse input discretization.

## Key findings

- The weighted POD method achieves comparable accuracy to high-fidelity solutions.
- It outperforms unweighted reduced models in high-dimensional settings.
- Numerical tests confirm its efficiency and accuracy across various problem dimensions.

## Abstract

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.