Multilevel Adaptive Sparse Grid Quadrature for Monte Carlo models

TL;DR

This paper introduces a multilevel adaptive sparse grid quadrature method tailored for Monte Carlo models, significantly reducing computational costs in high-dimensional stochastic integration tasks.

Contribution

The paper presents a novel multilevel adaptive sparse grid approach that leverages the multilevel structure of sparse grids, not based on telescoping sums, for efficient Monte Carlo model integration.

Findings

Significant computational savings over single-level methods

Effective in high-dimensional stochastic integration

Demonstrated on realistic kinetic Monte Carlo model

Abstract

Many problems require to approximate an expected value by some kind of Monte Carlo (MC) sampling, e.g. molecular dynamics (MD) or simulation of stochastic reaction models (also termed kinetic Monte Carlo (kMC)). Often, we are furthermore interested in some integral of the MC model's output over the input parameters. We present a Multilevel Adaptive Sparse Grid strategy for the numerical integration of such problems where the integrand is implicitly defined by a Monte Carlo model. In this approach, we exploit different levels of sampling accuracy in the Monte Carlo model to reduce the overall computational costs compared to a single level approach. Unlike existing approaches for Multilevel Numerical Quadrature, our approach is not based on a telescoping sum, but we rather utilize the intrinsic multilevel structure of the sparse grids and the employed locally supported, piecewise linear basis functions. Besides illustrative toy models, we demonstrate the methodology on a realistic kMC model for CO oxidation. We find significant savings compared to the single level approach - often orders of magnitude.