# An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs

## TL;DR

This paper presents a parallel-in-time method using PFASST to efficiently solve PDE-constrained optimization problems with parabolic equations, reducing computational costs and enabling faster solutions.

## Contribution

It introduces a fully time-parallel algorithm applying PFASST to both state and adjoint equations, improving efficiency in PDE-constrained optimization.

## Key findings

- Achieves significant parallel speedup in linear and nonlinear problems
- Demonstrates efficiency gains from reusing previous iteration information
- Validates approach with numerical experiments on reaction-diffusion problems

## Abstract

To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE constrained optimization. In order to develop an efficient fully time-parallel algorithm we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and adjoint equations simultaneously. We also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a non-linear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.