Adaptive stratified sampling for non-smooth problems

TL;DR

This paper introduces an adaptive stratification sampling method tailored for nonsmooth problems, significantly reducing variance and computational cost compared to traditional Monte Carlo methods, especially in uncertainty quantification tasks.

Contribution

The paper presents a novel adaptive stratification approach that effectively handles nonsmooth problems, with theoretical analysis and practical algorithms for variance reduction.

Findings

Achieves up to 1000x speedup over Monte Carlo methods.

Provides theoretical estimates for performance and failure probability.

Numerical experiments validate the method's efficiency and effectiveness.

Abstract

Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature nonsmooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty quantification methods relying on strict smoothness assumptions. To remedy these challenges, we propose an adaptive stratification method suitable for nonsmooth problems and with significantly reduced variance compared to Monte Carlo sampling. The stratification is iteratively refined and samples are added sequentially to satisfy an allocation criterion combining the benefits of proportional and optimal sampling. Theoretical estimates are provided for the expected performance and probability of failure to correctly estimate essential statistics. We devise a practical adaptive stratification method with strata of the same kind of geometrical shapes, cost-effective refinement satisfying a greedy variance reduction criterion. Numerical experiments corroborate the theoretical findings and exhibit speedups of up to three orders of magnitude compared to standard Monte Carlo sampling.