A Fully Parallelized and Budgeted Multi-Level Monte Carlo Method and the Application to Acoustic Waves

TL;DR

This paper introduces a new parallelized multi-level Monte Carlo method that optimally allocates computational resources to reduce error efficiently, demonstrated through acoustic wave simulations in complex media.

Contribution

It combines continuation MLMC, dynamic programming, and a novel parallelization strategy, providing theoretical error bounds and empirical validation for high-dimensional acoustic problems.

Findings

Achieves significant error reduction with optimized resource allocation

Demonstrates scalability and efficiency on high-performance computing systems

Validates theoretical bounds through empirical experiments on acoustic wave problems

Abstract

We present a novel variant of the multi-level Monte Carlo method that effectively utilizes a reserved computational budget on a high-performance computing system to minimize the mean squared error. Our approach combines concepts of the continuation multi-level Monte Carlo method with dynamic programming techniques following Bellman's optimality principle, and a new parallelization strategy based on a single distributed data structure. Additionally, we establish a theoretical bound on the error reduction on a parallel computing cluster and provide empirical evidence that the proposed method adheres to this bound. We implement, test, and benchmark the approach on computationally demanding problems, focusing on its application to acoustic wave propagation in high-dimensional random media.