Stochastic spectral embedding in forward and inverse uncertainty quantification

Abstract

Uncertainties are an important ingredient in the analysis of real-world systems by means of computational models. The scientific discipline that develops methods for modelling uncertainties in this context is called uncertainty quantification (UQ). Methods of this field are frequently grouped into forward and inverse UQ methods, where the former use information about the model input to quantify uncertainties in the model output, and the latter use information about the model output to indirectly quantify uncertainties in the model input.

In this thesis we first give an overview of the current state-of-the-art in UQ by providing an in-depth literature review and introduction to the most widely used methods. For forward UQ we introduce techniques for uncertainty propagation, surrogate modelling, sensitivity analysis, and reliability analysis. For inverse UQ we present the powerful Bayesian inference framework, specifically in the context of model calibration.

As a main methodological contribution we propose a new surrogate modelling technique called stochastic spectral embedding (SSE). Surrogate models enable many forward and inverse UQ analyses on high-fidelity computational models, by approximating the original model with a cheap-to-evaluate replacement model. This allows for repeated model evaluations that are necessary for many classes of UQ analyses. SSE is particularly powerful for approximating computational models with non-homogeneous complexity. This refers to models with outputs whose complexity strongly depends on the region of the input space that they are evaluated at. The technique consists of a series of local residual spectral expansions and, because it preserves the local spectral properties, enables the exact computation of the first two model output moments and variance-based Sobol' sensitivity indices. Applying SSE to likelihood functions typical of Bayesian inference, we show that it is possible to analytically compute many quantities of interest and effectively solve this type of inverse problem in a ``sampling-free'' manner. We call this approach stochastic spectral likelihood embedding (SSLE) and present it as a generalisation of the previously presented spectral likelihood expansion (SLE) technique. To improve the efficiency of this approach, we further introduce an adaptive experimental design enrichment scheme that predominantly evaluates the likelihood function in its informative regions. We then apply SSE to the limit-state function in reliability problems. By proposing modifications to its construction, this yields a powerful active-learning reliability method that we call stochastic spectral embedding-based reliability (SSER). We test this technique on a series of benchmark functions revealing its competitiveness with existing reliability methods.

To disseminate advanced UQ techniques, we finally show two applications to complex engineering problems. The first problem studies a model for transient heat transfer in gypsum fire insulation boards that relies on temperature-dependent effective material properties (TEMPs). We propose a low-dimensional parametrisation and analyse the model with polynomial chaos expansions (PCE) and sensitivity analysis techniques. Using available measurements of the heat evolution, we also calibrate this model and validate our calibration against another set of measurements. In the second application we analyse a sediment transport model of a section of the Rhône river. We derive a probabilistic model for the uncertainties in the transport model input and construct a surrogate model. Using the latter we conduct a sensitivity analysis and show that only a small fraction of input parameters significantly influence the model output. Neglecting the unimportant parameter uncertainties, we calibrate the model using a measurement series of the riverbed height.

Keywords

Stochastic spectral embedding, uncertainty quantification, surrogate modelling, reliability analysis, inverse problems, Bayesian model calibration.

BibTeX cite

@PHDTHESIS{WagnerThesis,
author = {Wagner, P.-R.},
title = {Stochastic spectral embedding in forward and inverse uncertainty quantification},
school = {ETH Z\"urich, Z\"urich, Switzerland},
year = {2021},
doi = {10.3929/ethz-b-000513631}
}