# An adaptive multilevel Monte Carlo algorithm for the stochastic drift-diffusion-Poisson system

## TL;DR

This paper introduces an adaptive multilevel Monte Carlo method for efficiently solving the stochastic drift-diffusion-Poisson system, combining goal-oriented mesh refinement and stochastic sampling to improve accuracy and computational efficiency.

## Contribution

The paper develops a novel adaptive multilevel Monte Carlo algorithm with error estimation techniques for the stochastic drift-diffusion-Poisson system, enhancing convergence and efficiency over uniform refinement.

## Key findings

- Achieves linear convergence of the H^1 error.
- Improves error control through efficient error indicator estimation.
- Demonstrates advantages over uniform mesh refinement in numerical examples.

## Abstract

We present an adaptive multilevel Monte Carlo algorithm for solving the stochastic drift-diffusion-Poisson system with non-zero recombination rate. The a-posteriori error is estimated to enable goal-oriented adaptive mesh refinement for the spatial dimensions, while the a-priori error is estimated to guarantee \red{linear} convergence of the $H^1$ error. In the adaptive mesh refinement, efficient estimation of the error indicator gives rise to better error control. For the stochastic dimensions, we use the multilevel Monte Carlo method to solve this system of stochastic partial differential equations. Finally, the advantage of the technique developed here compared to uniform mesh refinement is discussed using a realistic numerical example.