# Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

## TL;DR

This paper presents a novel recursive multilevel Picard method that effectively addresses the curse of dimensionality in high-dimensional Allen-Cahn PDEs, enabling more efficient numerical solutions for complex reaction-diffusion equations.

## Contribution

The authors develop and analyze truncated full-history recursive multilevel Picard schemes to overcome computational challenges in high-dimensional nonlinear PDEs like Allen-Cahn equations.

## Key findings

- Successfully reduces computational complexity in high dimensions
- Demonstrates convergence of the proposed schemes
- Applicable to reaction-diffusion PDEs with Lipschitz nonlinearities

## Abstract

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen-Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.