# A multigrid solver for PDE-constrained optimization with uncertain inputs

## TL;DR

This paper introduces a multigrid solver tailored for large saddle-point systems in PDE-constrained optimization under uncertainty, demonstrating optimal complexity and robustness across various problem types.

## Contribution

The paper develops a novel collective multigrid algorithm with convergence analysis, enabling efficient solutions of large saddle-point systems in uncertain PDE-constrained optimization.

## Key findings

- Optimal $O(N)$ complexity for reduced saddle-point systems
- Effective as a solver and preconditioner in diverse PDE optimization problems
- Robust performance across different problem settings

## Abstract

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.