Climate change impact and risk assessment is per definition a highly interdisciplinary task. Collaboration across disciplines is, unfortunately, often complicated by different perspectives, approaches, and terminology. To help building bridges, we propose a generalized mathematical framework for impact and risk assessment. In an unprecedented community effort, we have derived a generally applicable risk equation for spatially-distributed and dynamic systems. We start off with a general framing and then refine individual parts of the equation as much as needed. We will show how the individual terms of our unified risk equation explicitly relate to concepts of frequency, intensity, duration, exposure, vulnerability and asset worth. The rigorous mathematical treatment allows investigating the importance of risk factors and serves as a basis for risk management and reduction. Yet, the actual quantification of risk is not our primary goal – rather, the proposed framework forces us to be very precise in definitions and terminology. Thereby, it effectively improves communication and collaboration across disciplines. Indeed, we even learn greatly in cases where we identify limitations that seem to spoil such a mathematically rigorous treatment. We have successfully applied the framework to various disciplines of civil and environmental engineering, such as flood risk assessment, seismic risk assessment and reliability analysis of critical infrastructure. Users of the equation praise the structured common ground for discussion and highly recommend at least hypothetically applying this framework to gain a more unified understanding of the problem at hand. In this presentation, we discuss the potential of our proposed framework for risk assessment under climate change. Our transparent and rigorous approach is ideally suited to inform stakeholders and policymakers. Further, we are confident that our approach will serve as a catalyst for interdisciplinary advances toward effective adaptation and mitigation strategies.

Given a set of competing models of some phenomenon together with measurement data, Bayesian model selection (BMS) is a process of finding the model that is the best candidate for being the true data-generating process. BMS relies on the computation of Bayesian model evidence, which is defined as the marginal likelihood of the measurement data (i.e., the average likelihood over a model's parameter space). In this article, we introduce a new method for computing Bayesian model evidence. Our method consists of three key elements. First, all competing model functions are emulated by Gaussian processes. Model evaluations for training the Gaussian processes are chosen one by one in a sequential manner. Second, a model-time allocation strategy decides how many model evaluations are spent on each of the competing models. Third, a sequential sampling strategy selects design points in each model's parameter space. In numerical experiments, the method shows a speed-up of more than 1,000 compared to Monte Carlo estimation. While, in lower-dimensional cases, the use of Gaussian processes alone is very effective, in higher-dimensional cases, the modeltime allocation strategy and the sampling strategy become more important as they focus the effort on the right model and in the right areas of the parameter domains.

Currently, the export regime of a catchment is often characterized by the relationship between compound concentration and discharge in the catchment outlet or, more specifically, by the regression slope in log-concentrations versus log-discharge plots. However, the scattered points in these plots usually do not follow a plain linear regression representation because of different processes (e.g., hysteresis effects). This work proposes a simple stochastic time-series model for simulating compound concentrations in a river based on river discharge. Our model has an explicit transition parameter that can morph the model between chemostatic behavior and chemodynamic behavior. As opposed to the typically used linear regression approach, our model has an additional parameter to account for hysteresis by including correlation over time. We demonstrate the advantages of our model using a high-frequency data series of nitrate concentrations collected with in situ analyzers in a catchment in Germany. Furthermore, we identify event-based optimal scheduling rules for sampling strategies. Overall, our results show that (i) our model is much more robust for estimating the export regime than the usually used regression approach, and (ii) sampling strategies based on extreme events (including both high and low discharge rates) are key to reducing the prediction uncertainty of the catchment behavior. Thus, the results of this study can help characterize the export regime of a catchment and manage water pollution in rivers at lower monitoring costs. We propose a simple stochastic time-series model to represent the export regime of a catchment beyond simple regression. We propose how to get the required data with the least effort when the use of high-frequency in situ analyzers is not feasible or restricted. Sampling strategies based on extreme events are essential for reducing the prediction uncertainty of the catchment behavior.

Methods for sequential design of computer experiments typically consist of two phases. In the first phase, the exploratory phase, a space-filling initial design is used to estimate hyperparameters of a Gaussian process emulator (GPE) and to provide some initial global exploration of the model function. In the second phase, more design points are added one by one to improve the GPE and to solve the actual problem at hand (e.g., Bayesian optimization, estimation of failure probabilities, solving Bayesian inverse problems). In this article, we investigate whether hyperparameters can be estimated without a separate exploratory phase. Such an approach will leave hyperparameters uncertain in the first iterations, so the acquisition function (which tells where to evaluate the model function next) and the GPE-based estimator need to be adapted to non-Gaussian random fields. Numerical experiments are performed exemplarily on a sequential method for solving Bayesian inverse problems. These experiments show that hyperparameters can indeed be estimated without an exploratory phase and the resulting method works almost as efficient as if the hyperparameters had been known beforehand. This means that the estimation of hyperparameters should not be the reason for including an exploratory phase. Furthermore, we show numerical examples, where these results allow us to eliminate the exploratory phase to make the sequential design method both faster (requiring fewer model evaluations) and easier to use (requiring fewer choices by the user).

Commonly, chemical catchment regimes are described by a simple regression slope of logconcentrations versus log-discharges measured in the catchment outlet river. The slope value of these plots defines the chemical regime of a catchment. A slope=-1 corresponds to a constant contaminant release subject to dilution by rainfall (an unrealistic extreme but needed as a base flow); whereas a slope=0 means that there are chemostatic effects in the catchment or a washout of contaminants at a constant concentration. However, reality shows that actual time-series measurements of discharge and concentrations conflict with this naive representation since the measurements show temporal hysteresis that defies regression assumptions (i.e. that regression residuals must be uncorrelated). To represent this time interaction beyond regression, we design a simple stochastic time-series model that accounts for fluctuating concentration release and transport with memory. In this work, we also establish how to get the observation data required for a robust estimation of the slope with the least effort. To show the capability of our proposed model and method, we apply a retrospective optimal design of experiments to a high-frequency data series of nitrate concentration (collected by online probes) and discharge of a real catchment in Germany. We thin out the data by applying frequency and event-based monitoring strategies to find out the key components of the strategies that best predict the catchment behavior. Results indicate that our catchment under study (the Ammer catchment in southwestern Germany) is relatively close to a chemostatic type catchment and that our stochastic model, in fact, provides more accurate results for small data sets. Also, optimal data collection schemes for this purpose should be event-based, considering both high and low extremes of discharge that are spread out over time.

We present a sequential design strategy for efficient sampling of model functions during the solution of Bayesian inverse problems. The model function is assumed to be computationally expensive and therefore is described by a random field (such as a Gaussian process emulator). The sequential design strategy is a greedy one-step look ahead method, minimizing the Bayes risk with respect to a loss function measuring the quadratic L-2-error in the likelihood estimate. Four numerical examples demonstrate that the proposed sampling method is more efficient than space-filling, prior-based designs.

In this work we present the optimized stochastic collocation method (OSC). OSC is a new sampling rule that can be applied to polynomial chaos expansions (PCE) for uncertainty quantification. Given a model function, the goal of PCE is to find the polynomial from a given polynomial space that is closest to the model function with respect to the L2-norm induced by a given probability measure. Many PCE methods approximate the involved projection integral by discretization with a finite set of integration points. Our key idea is to choose these integration points through numerical optimization based on an operator norm derived from the discretized projection operator. OSC is a generalization of Gaussian quadrature: both methods coincide for one-dimensional integration and under appropriate problem settings in multidimensional problems. As opposed to many established integration rules, OSC does not generally lead to tensor grids in multidimensional problems. With OSC, the user can specify the number of integration points independently of the problem dimension and PCE expansion order. This allows one to reduce the number of model evaluations and still achieve a high accuracy. The input parameters can follow any kind of probability distribution, as long as the statistical moments up to a certain order are available. Even statistically dependent parameters can be handled in a straightforward and natural fashion. Moreover, OSC allows reusing integration points, if results from earlier model evaluations are available. Gauss-Kronrod and Stroud integration rules can be reproduced with OSC for the respective special cases.

Observable phenomena can often be described by alternative models with different degrees of fidelity. Such models typically contain uncertain parameters and forcings, rendering predictions of the state variables uncertain as well. Within the probabilistic framework, solutions of these models are given in terms of their probability density functions (PDFs). In the presence of data, the latter can be treated as prior distributions. Uncertainty and assimilation of measurements into model predictions, e.g., via Bayesian updating of solution PDFs, pose a question of model selection: Given a significant difference in computational cost, is a lower-fidelity model preferable to its higher-fidelity counterpart? We investigate this question in the context of multiphase flow in heterogeneous porous media whose hydraulic properties are uncertain. While low-fidelity (reduced-complexity) models introduce a model error, their moderate computational cost makes it possible to generate more realizations, which reduces the (e.g., Monte Carlo) sampling error. These two errors determine the model with the smallest total error. Our analysis suggests that assimilation of measurements of a quantity of interest (a medium's saturation, in our example) influences both types of errors, increasing the probability that the predictive accuracy of a reduced-complexity model exceeds that of its higher-fidelity counterpart.