Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using concepts from random set theory, we propose to adapt the Vorob’ev expectation and deviation to capture the variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob’ev deviation can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front.

Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.

A Nonstationary Space-Time Gaussian Process Model for Partially Converged Simulations

In the context of expensive numerical experiments, a promising solution for alleviating the computational costs consists of using partially converged simulations instead of exact solutions. The gain in computational time is at the price of precision in the response. This work addresses the issue of fitting a Gaussian process model to partially converged simulation data for further use in prediction. The main challenge consists of the adequate approximation of the error due to partial convergence, which is correlated in both design variables and time directions. Here, we propose fitting a Gaussian process in the joint space of design parameters and computational time. The model is constructed by building a nonstationary covariance kernel that reflects accurately the actual structure of the error. Practical solutions are proposed for solving parameter estimation issues associated with the proposed model. The method is applied to a computational fluid dynamics test case and shows significant improvement in prediction compared to a classical kriging model.

Methane emissions from floodplains in the Amazon Basin: challenges in developing a process-based model for global applications

Tropical wetlands are estimated to represent about 50% of the natural wetland methane (CH4) emissions and explain a large fraction of the observed CH4 variability on timescales ranging from glacial–interglacial cycles to the currently observed year-to-year variability. Despite their importance, however, tropical wetlands are poorly represented in global models aiming to predict global CH4 emissions. This publication documents a first step in the development of a process-based model of CH4 emissions from tropical floodplains for global applications. For this purpose, the LPX-Bern Dynamic Global Vegetation Model (LPX hereafter) was slightly modified to represent floodplain hydrology, vegetation and associated CH4 emissions. The extent of tropical floodplains was prescribed using output from the spatially explicit hydrology model PCR-GLOBWB. We introduced new plant functional types (PFTs) that explicitly represent floodplain vegetation. The PFT parameterizations were evaluated against available remote-sensing data sets (GLC2000 land cover and MODIS Net Primary Productivity). Simulated CH4 flux densities were evaluated against field observations and regional flux inventories. Simulated CH4 emissions at Amazon Basin scale were compared to model simulations performed in the WETCHIMP intercomparison project. We found that LPX reproduces the average magnitude of observed net CH4 flux densities for the Amazon Basin. However, the model does not reproduce the variability between sites or between years within a site. Unfortunately, site information is too limited to attest or disprove some model features. At the Amazon Basin scale, our results underline the large uncertainty in the magnitude of wetland CH4 emissions. Sensitivity analyses gave insights into the main drivers of floodplain CH4 emission and their associated uncertainties. In particular, uncertainties in floodplain extent (i.e., difference between GLC2000 and PCR-GLOBWB output) modulate the simulated emissions by a factor of about 2. Our best estimates, using PCR-GLOBWB in combination with GLC2000, lead to simulated Amazon-integrated emissions of 44.4 ± 4.8 Tg yr−1. Additionally, the LPX emissions are highly sensitive to vegetation distribution. Two simulations with the same mean PFT cover, but different spatial distributions of grasslands within the basin, modulated emissions by about 20%. Correcting the LPX-simulated NPP using MODIS reduces the Amazon emissions by 11.3%. Finally, due to an intrinsic limitation of LPX to account for seasonality in floodplain extent, the model failed to reproduce the full dynamics in CH4 emissions but we proposed solutions to this issue. The interannual variability (IAV) of the emissions increases by 90% if the IAV in floodplain extent is accounted for, but still remains lower than in most of the WETCHIMP models. While our model includes more mechanisms specific to tropical floodplains, we were unable to reduce the uncertainty in the magnitude of wetland CH4 emissions of the Amazon Basin. Our results helped identify and prioritize directions towards more accurate estimates of tropical CH4 emissions, and they stress the need for more research to constrain floodplain CH4 emissions and their temporal variability, even before including other fundamental mechanisms such as floating macrophytes or lateral water fluxes.

Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set

Stepwise uncertainty reduction (SUR) strategies aim at constructing a sequence of points for evaluating a function  f in such a way that the residual uncertainty about a quantity of interest progressively decreases to zero. Using such strategies in the framework of Gaussian process modeling has been shown to be efficient for estimating the volume of excursion of f above a fixed threshold. However, SUR strategies remain cumbersome to use in practice because of their high computational complexity, and the fact that they deliver a single point at each iteration. In this article we introduce several multipoint sampling criteria, allowing the selection of batches of points at which f can be evaluated in parallel. Such criteria are of particular interest when f is costly to evaluate and several CPUs are simultaneously available. We also manage to drastically reduce the computational cost of these strategies through the use of closed form formulas. We illustrate their performances in various numerical experiments, including a nuclear safety test case. Basic notions about kriging, auxiliary problems, complexity calculations, R code, and data are available online as supplementary materials.

Policy Instrument Selection under Uncertainty: The Case of Micropollution Regulation

The paper addresses the question of which factors drive the formation of policy preferences when there are remaining uncertainties about the causes and effects of the problem at stake. To answer this question we examine policy preferences reducing aquatic micropollutants, a specific case of water protection policy and different actor groups (e.g. state, science, target groups). Here, we contrast two types of policy preferences: a) preventive or source-directed policies, which mitigate pollution in order to avoid contact with water; and b) reactive or end-of-pipe policies, which filter water already contaminated by pollutants. In a second step, we analyze the drivers for actors’ policy preferences by focusing on three sets of explanations, i.e. participation, affectedness and international collaborations. The analysis of our survey data, qualitative interviews and regression analysis of the Swiss political elite show that participation in the policy-making process leads to knowledge exchange and reduces uncertainties about the policy problem, which promotes preferences for preventive policies. Likewise, actors who are affected by the consequences of micropollutants, such as consumer or environmental associations, opt for anticipatory policies. Interestingly, we find that uncertainties about the effectiveness of preventive policies can promote preferences for end-of-pipe policies. While preventive measures often rely on (uncertain) behavioral changes of target groups, reactive policies are more reliable when it comes to fulfilling defined policy goals. Finally, we find that in a transboundary water management context, actors with international collaborations prefer policies that produce immediate and reliable outcomes.