Multilevel Monte Carlo Covariance Estimation for the Computation of Sobol' Indices

Abstract

Crude and quasi Monte Carlo (MC) sampling techniques are common tools dedicated to estimating statistics (expectation, variance, covariance) of a random quantity of interest. We focus here on the uncertainty quantification framework where the quantity of interest is the output of a numerical simulator fed with uncertain input parameters. Then, sampling the output involves running the simulator for different samples of the inputs, which may be computationally time-consuming. To reduce the cost of sampling, a first approach consists in replacing the numerical simulator by a surrogate model that is cheaper to evaluate, thus making it possible to generate more samples of the output and therefore leading to a lower sampling error. However, this approach adds to the sampling error an unavoidable model error. Another approach, which does not introduce any model error, is the so-called multilevel MC (MLMC) method. Given a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost, MLMC combines samples obtained at different levels to construct an estimator at a reduced cost compared to standard MC sampling. In this paper, we derive and analyze multilevel covariance estimators and adapt the MLMC convergence theorem in terms of the corresponding covariances and fourth order moments. We propose a multilevel algorithm driven by a target cost as an alternative to typical algorithms driven by a target accuracy. These results are used in a sensitivity analysis context in order to derive a multilevel estimation of Sobol' indices, whose building blocks can be written as covariance terms in a pick-and-freeze formulation. These contributions are successfully tested on an initial value problem with random parameters.

Global and local sensitivity analysis of the Emission Dispersion Model input parameters

“…Because SA is associated with uncertainty analysis, Monte Carlo method (Derwent et al, 2018;Lemieux, 2009;Mycek and De Lozzo, 2019;Nezaratian et al, 2018) is usually used to conduct both of them. It comprises the following steps:…”

Multi-fidelity covariance estimation in the log-Euclidean geometry

“…Loss of definiteness in Euclidean multi-fidelity covariance estimation Direct application of multi-level and multi-fidelity Monte Carlo estimation [15,11,43,32] based on control variates to (co)variance estimation has been proposed in, e.g., [7,40,31]. These multi-fidelity estimators, however, rely on differences of singlefidelity Monte Carlo estimators, which can lead to a loss of definiteness of the estimated covariance matrix, as we will detail in Section 2.…”

“…. , L. The classical approach to multi-fidelity estimation, i.e., in the Euclidean geometry, is to use the surrogate samples to define control variates that reduce the variance of the standard Monte Carlo estimator (1) [7,40,31]. This approach yields the Euclidean multi-fidelity (EMF) covariance estimator…”

Multilevel Monte Carlo estimation of background error covariances in ensemble variational data assimilation

“…MOSAP The semidefinite programming approach to the MOSAP does not extends naturally here, as the matrix φ(m) no longer depends linearly on the sample sizes m. Note this will be the case for all multilevel estimators of the covariance introduced in this note, including multilevel estimators of covariance matrices. This could be circumvented by minimizing an upper bound of the variance that linearly depends on the inverses of the samples sizes, as done ny Mycek and De Lozzo (2019).…”