Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Abstract

Summary In this paper, we present an adaptive algorithm to construct response surface approximations of high‐fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi‐index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high‐fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multiphysics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input‐output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

Ensembles Are Required to Handle Aleatoric and Parametric Uncertainty in Molecular Dynamics Simulation

“…The general idea is to transform the adaptive SC expansion into a polynomial chaos expansion (PCE), which facilitates an easy computation of the Sobol indices. As this is already well documented, and not critical for our discussion, we refer to refs and for more details.…”

“…To compute the Sobol sensitivity indices, we employ the method described in ref ( 39 ). The general idea is to transform the adaptive SC expansion into a polynomial chaos expansion (PCE), which facilitates an easy computation of the Sobol indices.…”

Global Sensitivity Analysis Using the Ultra‐Low Resolution Energy Exascale Earth System Model

“…The Sobol indices (Equation ) can be computed using a number of different methods, for example, via (Quasi) Monte Carlo sampling (Saltelli et al., 2010), using surrogates (such as polynomial chaos expansions (Sudret, 2008)), or with sparse grids (J. Jakeman et al., 2019). Herein, we employ the software library PyApprox (J. D. Jakeman, 2022), a flexible and efficient open‐source tool for high‐dimensional approximation and UQ, which utilizes Gaussian processes (Harbrecht et al., 2020; Rasmussen & Williams, 2006).…”

Modeling Water Quality in Watersheds: From Here to the Next Generation

“…Unfortunately, due to the computational expense of many water quality models (e.g., Buahin & Horsburgh, 2018 ), building an accurate surrogate can still be intractable. Recently, in other fields such as aerospace engineering, multifidelity methods ( Gorodetsky et al, 2020 ; Jakeman et al, 2020 ; Peherstorfer et al, 2018 ) have been used to reduce the computational burden of uncertainty analysis. Multifidelity methods use an ensemble of models of varying complexity, speed, and accuracy (fidelity).…”