# A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty

## TL;DR

This paper introduces a mesh refinement stochastic gradient method for PDE-constrained optimization under uncertainty, effectively balancing discretization and stochastic errors to improve numerical efficiency.

## Contribution

It develops a novel mesh refinement strategy integrated with stochastic gradient methods for PDE-constrained optimization with uncertain parameters.

## Key findings

- Mesh refinement improves convergence in stochastic PDE optimization.
- The method effectively balances discretization and stochastic errors.
- Numerical experiments validate the approach across different random fields.

## Abstract

Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth tracking-type functional subject to a linear partial differential equation with random coefficients and box constraints. The approach we take is based on stochastic approximation where, in place of a true gradient, a stochastic gradient is chosen using one sample from a known probability distribution. Feasibility is maintained by performing a projection at each iteration. In the application of this method to PDE-constrained optimization under uncertainty, new challenges arise. We observe the discretization error made by approximating the stochastic gradient using finite elements. Analyzing the interplay between PDE discretization and stochastic error, we develop a mesh refinement strategy coupled with decreasing step sizes. Additionally, we develop a mesh refinement strategy for the modified algorithm using iterate averaging and larger step sizes. The effectiveness of the approach is demonstrated numerically for different random field choices.