Multilevel Monte Carlo Approximation of Functions

Abstract

Many applications across sciences and technologies require a careful quantification of non-deterministic effects to a system output, for example when evaluating the system's reliability or when gearing it towards more robust operation conditions. At the heart of these considerations lies an accurate yet efficient characterization of uncertain system outputs. In this work we introduce and analyze novel multilevel Monte Carlo techniques for an efficient characterization of an uncertain system output's distribution. These techniques rely on accurately approximating general parametric expectations, i.e. expectations that depend on a parameter, uniformly on an interval. Applications of interest include, for example, the approximation of the characteristic function or of the cumulative distribution function of an uncertain system output. A further important consequence of the introduced approximation techniques for parametric expectations (i.e. for functions) is that they allow to construct multilevel Monte Carlo estimators for various robustness indicators, such as for quantiles (also known as value-at-risk) and for the conditional value-at-risk. These robustness indicators cannot be expressed as moments and are thus not easily accessible usually. In fact, here we provide a framework that allows to simultaneously estimate a cumulative distribution function, a quantile, and the associated conditional value-at-risk of an uncertain system output at the cost of a single multilevel Monte Carlo simulation, while each estimated quantity satisfies a prescribed tolerance goal.

Continuation Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design

“…where h p denotes an appropriate h-statistic of order p. One of the method's key ingredients is the use of such h-statistics [20] as unbiased central moment estimators with minimal variance for the level-wise contributions. In [21,22] we further extended the MLMC concept to accurately approximate the Cumulative distribution function (CDF) of a random system output and robustness measures such as quantiles (also known as Value at Risk, VaR) or coherent risk measures [23] such as the conditional value at risk (CVaR [24]). The α-quantile q α is given by:…”

“…The idea proposed in [21,22] to approximate simultaneously the CDF, quantiles q α and CV aR α of a certain output QoI Q is to first approximate the function H Q : Θ → R defined as:…”

“…Details on how to tune the MLMC estimator to obtain simultaneously accurate evaluations of F Q , q τ , CV aR τ are given in [21]. As underlined and extensively studied in [12], the Continuation-MLMC (C-MLMC) methodology is an extension of the classical MLMC approach that estimates, through a Bayesian update procedure, the optimal number of levels and simulations per level required to achieve a prescribed tolerance.…”

Accelerated Multilevel Monte Carlo With Kernel‐Based Smoothing and Latinized Stratification

“…Polynomial smoothing of the indicator function can improve computational efficiency for estimating CDFs (Giles et al., 2015; D. Lu et al., 2016), as can approximation of PDFs via a truncated moment sequence (Bierig & Chernov, 2016). Indirect estimation of a CDF via an appropriate primitive function (Krumscheid & Nobile, 2018) provides yet another tool to speed up the computation.…”

Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method

“…See also [16] for some related earlier works. Although the focus of this work is on scalar quantities of interest, it is noteworthy that the multilevel Monte Carlo method can also be applied to multidimensional quantities or reliability studies [19]. We will leave the application of these options in the context of contact problems for future works.…”