Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

TL;DR

This paper extends the multilevel Picard approximation method to high-dimensional semilinear PDEs with locally monotone coefficients, overcoming the curse of dimensionality under broader conditions than before.

Contribution

It generalizes the previous method to include locally monotone coefficient functions, broadening applicability to polynomial coefficient semilinear PDEs.

Findings

Method overcomes curse of dimensionality for a wider class of PDEs.

Applicable to polynomial coefficient functions with locally monotone properties.

Maintains provable convergence guarantees.

Abstract

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.