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Quantifying uncertain system outputs via the multi-level Monte Carlo method --- distribution and robustness measures

Sundar Subramaniam Ganesh, Sebastian Krumscheid, Fabio Nobile

In this work, we consider the problem of estimating the probability distribution, the quantile or the conditional expectation above the quantile, the so called conditional-value-at-risk, of output quantities of complex random differential models by the MLMC method. We follow the approach of (reference), which recasts the estimation of the above quantities to the computation of suitable parametric expectations. In this work, we present novel computable error estimators for the estimation of such quantities, which are then used to optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We demonstrate the efficiency and robustness of our adaptive continuation-MLMC in an array of numerical test cases.

2022

Complexity analysis of stochastic gradient methods for PDE-constrained optimal control problems with uncertain parameters

Sebastian Krumscheid, Matthieu Claude Martin, Fabio Nobile

We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squared L2 misfit between the state (i.e. solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a Stochastic Gradient method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.

2021

Quantifying uncertain system outputs via the multilevel Monte Carlo method — Part I: Central moment estimation

Sebastian Krumscheid, Fabio Nobile, Michele Pisaroni

In this work we introduce and analyze a novel multilevel Monte Carlo (MLMC) estimator for the accurate approximation of central moments of system outputs affected by uncertainties. Central moments play a central role in many disciplines to characterize a random system output's distribution and are of primary importance in many prediction, optimization, and decision making processes under uncertainties. We detail how to effectively tune the MLMC algorithm for central moments of any order and present a complete practical algorithm that is implemented as part of a Python library [1]. In fact, we validate the methodology on selected reference problems and apply it to an aerodynamic relevant test case, namely the transonic RAE 2822 airfoil affected by operating and geometric uncertainties.

2020