Cite this article as: Joseph B. Nagel and Bruno Sudret, A unified framework for multilevel uncertainty quantification in bayesian inverse problems, Probabilistic Engineering Mechanics, http://dx.

A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain Monte Carlo simulation and allows one to make use of linear least squares instead. The pros and cons of spectral Bayesian inference are discussed and demonstrated on the basis of simple applications from classical statistics and inverse modeling

This paper presents an efficient surrogate modeling strategy for the uncertainty quantification and Bayesian calibration of a hydrological model. In particular, a process-based dynamical urban drainage simulator that predicts the discharge from a catchment area during a precipitation event is considered. The goal is to perform a global sensitivity analysis and to identify the unknown model parameters as well as the measurement and prediction errors. These objectives can only be achieved by cheapening the incurred computational costs, that is, lowering the number of necessary model runs. With this in mind, a regularity-exploiting metamodeling technique is proposed that enables fast uncertainty quantification. Principal component analysis is used for output dimensionality reduction and sparse polynomial chaos expansions are used for the emulation of the reduced outputs. Sensitivity measures such as the Sobol indices are obtained directly from the expansion coefficients. Bayesian inference via Markov chain Monte Carlo posterior sampling is drastically accelerated.The quantification of uncertainty has become an integral aspect of computational science and engineering in the last two decades [1,2]. Algorithmic advances on the one hand and hardware improvements on the other hand allow for an increasingly detailed simulation of complex systems.That the underlying models are hardly perfect and the model parameters are barely known with certainty prompts scientist and engineers to conduct an end-to-end analysis of the encountered errors. This process, for which one commonly relies on probability theory, is known as uncertainty quantification (UQ).

Hierarchical or multilevel modeling establishes a convenient framework for solving complex inverse problems [1,2] in the presence of uncertainty. In the last two decades it has been studied from a frequentist [3] and a Bayesian perspective [4]. We will adopt a Bayesian point of view to statistical inversion and uncertainty quantification and present a Bayesian multilevel framework that allows for inversion and optimal analysis of "perfect" or noise-free data in the presence of aleatory and epistemic types of uncertainty and in experimental situations when data is scarce or expensive to acquire. In this contribution to the annual MascotNum workshop we will discuss the abovementioned framework on the basis of an application example within the domain of aerospace engineering [5]. We will not only illustrate the very potential of Bayesian multilevel modeling as well as ways to overcome its immanent major challenges, but more importantly we will discuss the main observations, considerations and key questions that the practical problem solution [6] has given rise to.

Dam breach models are commonly used to predict outflow hydrographs of potentially failing dams and are key ingredients for evaluating flood risk. In this paper a new dam breach modeling framework is introduced that shall improve the reliability of hydrograph predictions of homogeneous earthen embankment dams. Striving for a small number of parameters, the simplified physics‐based model describes the processes of failing embankment dams by breach enlargement, driven by progressive surface erosion. Therein the erosion rate of dam material is modeled by empirical sediment transport formulations. Embedding the model into a Bayesian multilevel framework allows for quantitative analysis of different categories of uncertainties. To this end, data available in literature of observed peak discharge and final breach width of historical dam failures were used to perform model inversion by applying Markov chain Monte Carlo simulation. Prior knowledge is mainly based on noninformative distribution functions. The resulting posterior distribution shows that the main source of uncertainty is a correlated subset of parameters, consisting of the residual error term and the epistemic term quantifying the breach erosion rate. The prediction intervals of peak discharge and final breach width are congruent with values known from literature. To finally predict the outflow hydrograph for real case applications, an alternative residual model was formulated that assumes perfect data and a perfect model. The fully probabilistic fashion of hydrograph prediction has the potential to improve the adequate risk management of downstream flooding.