# Stochastic Galerkin reduced basis methods for parametrized linear elliptic partial differential equations

## TL;DR

This paper introduces a stochastic Galerkin reduced basis (SGRB) method to efficiently estimate parameter-dependent statistics of elliptic PDE solutions, significantly reducing computational costs for multiple parameter queries.

## Contribution

The paper develops a novel SGRB approach that combines stochastic Galerkin finite element methods with reduced basis techniques, enabling fast statistical output estimation for parametrized elliptic PDEs.

## Key findings

- SGRB achieves similar accuracy with fewer basis functions compared to standard methods.
- A single SGRB run estimates statistical outputs for new parameters, unlike Monte Carlo approaches.
- Numerical tests demonstrate the efficiency of SGRB in a convection-diffusion-reaction problem.

## Abstract

We consider the estimation of parameter-dependent statistics of functional outputs of elliptic boundary value problems (BVPs) with parametrized random and deterministic inputs. For a given value of the deterministic paremeter, a stochastic Galerkin finite element (SGFE) method can estimate the corresponding expectation and variance of a linear output at the cost of a single solution of a large block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced order model for a spatial-stochastic weak formulation of the parameter-dependent BVP. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a POD reduced basis generated from snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds.~We test the SGRB method numerically for a convection-diffusion-reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.