# Stochastic methods for solving high-dimensional partial differential equations

## TL;DR

This paper introduces novel stochastic algorithms that leverage probabilistic representations and sparse interpolation techniques to efficiently solve high-dimensional PDEs, addressing computational challenges in complex systems.

## Contribution

It develops new algorithms combining control variates and adaptive sparse interpolation for high-dimensional PDEs, enhancing computational efficiency and accuracy.

## Key findings

- Algorithms effectively solve high-dimensional PDEs
- Numerical examples demonstrate improved accuracy
- Methods outperform traditional approaches in complexity

## Abstract

We propose algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and time-integration schemes are used to estimate pointwise evaluations of the solution of a PDE. We use a sequential control variates algorithm, where control variates are constructed based on successive approximations of the solution of the PDE. Two different algorithms are proposed, combining in different ways the sequential control variates algorithm and adaptive sparse interpolation. Numerical examples will illustrate the behavior of these algorithms.