Analysis of the Monte-Carlo error in a hybrid semi-lagrangian scheme

TL;DR

This paper analyzes the Monte-Carlo error in a hybrid semi-lagrangian scheme for PDEs, showing it can be controlled under certain conditions and confirming results with numerical experiments.

Contribution

It provides a theoretical error estimate for Monte-Carlo discretizations combining semi-lagrangian schemes and probabilistic methods, with validation through numerical tests.

Findings

Monte-Carlo error is of order √(δt/N) under anti-CFL condition.

Error control is validated by numerical experiments.

The method applies to specific classes of PDEs.

Abstract

We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step $\delta t$ and on the mesh size $\delta x$ and for $N$ - the number of realizations - reasonably large, we control this error by a term of order $\mathcal{O}(\sqrt{\delta t /N})$. We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.