# On parareal algorithms for semilinear parabolic Stochastic PDEs

## TL;DR

This paper investigates the performance of parareal algorithms applied to semilinear parabolic stochastic PDEs, focusing on the choice of coarse integrator and noise regularity, supported by theoretical analysis and numerical experiments.

## Contribution

It introduces the use of exponential Euler schemes as coarse integrators in parareal algorithms for stochastic PDEs and analyzes their impact on convergence and efficiency.

## Key findings

- Exponential Euler as coarse integrator improves convergence.
- Noise regularity significantly affects algorithm performance.
- Number of iterations influences accuracy and computational cost.

## Abstract

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time' implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme. The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated, with theoretical analysis results and with extensive numerical experiments.