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Covariance estimation using h-statistics in Monte Carlo and multilevel Monte Carlo methods

Sharana Kumar Shivanand

TL;DR

This paper addresses unbiased covariance estimation for random fields arising in forward uncertainty quantification, introducing h-statistics to obtain closed-form mean-square-error expressions for both Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) estimators. By deriving unbiased estimators of central moments and their multivariate extensions, the authors formulate MC and MLMC covariance estimators with provably unbiased sampling-error terms and explicit power-sum representations. They develop an h-statistics-based MLMC framework that preserves positive definiteness and provides optimal sample allocations across mesh levels, including a detailed complexity analysis. The methods are demonstrated on a 1D Poisson/steady-state heat equation with log-normal conductivity, showing substantial gains in accuracy and especially in computational efficiency: h-statistics MLMC can be significantly faster (e.g., ~16×) and cost-cutting (e.g., ~22% vs conventional MLMC) while maintaining high accuracy (relative errors within 1%). Overall, the work offers unbiased, closed-form error control for covariance estimation in UQ applications and demonstrates practical benefits for PDE-based uncertainty propagation.

Abstract

We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. The numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.

Covariance estimation using h-statistics in Monte Carlo and multilevel Monte Carlo methods

TL;DR

This paper addresses unbiased covariance estimation for random fields arising in forward uncertainty quantification, introducing h-statistics to obtain closed-form mean-square-error expressions for both Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) estimators. By deriving unbiased estimators of central moments and their multivariate extensions, the authors formulate MC and MLMC covariance estimators with provably unbiased sampling-error terms and explicit power-sum representations. They develop an h-statistics-based MLMC framework that preserves positive definiteness and provides optimal sample allocations across mesh levels, including a detailed complexity analysis. The methods are demonstrated on a 1D Poisson/steady-state heat equation with log-normal conductivity, showing substantial gains in accuracy and especially in computational efficiency: h-statistics MLMC can be significantly faster (e.g., ~16×) and cost-cutting (e.g., ~22% vs conventional MLMC) while maintaining high accuracy (relative errors within 1%). Overall, the work offers unbiased, closed-form error control for covariance estimation in UQ applications and demonstrates practical benefits for PDE-based uncertainty propagation.

Abstract

We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. The numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.
Paper Structure (20 sections, 2 theorems, 70 equations, 7 figures, 6 tables)

This paper contains 20 sections, 2 theorems, 70 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem description
  3. Monte Carlo method
  4. Classical Monte Carlo estimator
  5. Monte Carlo estimation by h-statistics
  6. h-statistics: unbiased estimation of central moments
  7. Monte Carlo estimation of covariance by h-statistics
  8. Multilevel Monte Carlo method
  9. Classical multilevel Monte Carlo estimator
  10. Multilevel Monte Carlo estimator using h-statistics
  11. MSE of h-statistics based MLMC covariance estimator
  12. Computational Complexity
  13. Application to Poisson's equation: 1D steady state heat equation
  14. Numerical results
  15. Specifications
  16. ...and 5 more sections

Key Result

Proposition 3.1

Let us consider $c_\alpha$, $c_\gamma$, $\alpha$, and $\gamma$ to be positive constants. The error bounds are defined as follows: Then, for any $0 < \epsilon < e^{-1}$, the Monte Carlo (MC) covariance estimator $\widehat{\mathrm{h}}_{1,1}^{\mathrm{MC}}$ with $N = \mathcal{O}(\epsilon^{-2})$ and $h = \mathcal{O}(\epsilon^{1/\alpha})$ satisfies $\mathrm{MSE}(\widehat{\mathrm{h}}_{1,1}^{\mathrm{MC}}

Figures (7)

  • Figure 1: Geometry and boundary conditions
  • Figure 2: Four levels of nested mesh grids
  • Figure 3: Screening results of MLMC covariance estimators $\widehat{\mathrm{h}}_{1,1}^{\mathrm{ML}}$and $\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}^{\mathrm{ML}}$
  • Figure 4: Comparison of h-statistics-based MLMC (top left), classical MLMC (top right) and h-statistics-based MC (bottom) covariance estimation of the stochastic temperature field $\boldsymbol U^{h_L}(\omega)$ on finest level $L=3$
  • Figure 5: Comparison of absolute errors (in $\mathrm{K}^4$) between h-statistics-based MLMC and classical MLMC covariance estimators, with respect to h-statistics-based MC estimation
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 3.1
  • Proposition 4.1