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Aylwin, R., JerezHanckes, C., Schwab, C., & Zech, J. (2020). Domain Uncertainty Quantification in Computational Electromagnetics. SIAMASA J. Uncertain. Quantif., 8(1), 301–341.
Abstract: We study the numerical approximation of timeharmonic, electromagnetic fields inside a lossy cavity of uncertain geometry. Key assumptions are a possibly highdimensional parametrization of the uncertain geometry along with a suitable transformation to a fixed, nominal domain. This uncertainty parametrization results in families of countably parametric, Maxwelllike cavity problems that are posed in a single domain, with inhomogeneous coefficients that possess finite, possibly low spatial regularity, but exhibit holomorphic parametric dependence in the differential operator. Our computational scheme is composed of a sparse grid interpolation in the highdimensional parameter domain and an Hcurl conforming edge element discretization of the parametric problem in the nominal domain. As a steppingstone in the analysis, we derive a novel Strangtype lemma for Maxwelllike problems in the nominal domain, which is of independent interest. Moreover, we accommodate arbitrary small Sobolev regularity of the electric field and also cover uncertain isotropic constitutive or material laws. The shape holomorphy and edgeelement consistency error analysis for the nominal problem are shown to imply convergence rates for multilevel Monte Carlo and for quasiMonte Carlo integration, as well as sparse grid approximations, in uncertainty quantification for computational electromagnetics. They also imply expression rate estimates for deep ReLU networks of shapetosolution maps in this setting. Finally, our computational experiments confirm the presented theoretical results.

Bergen, M., & Munoz, F. D. (2018). Quantifying the effects of uncertain climate and environmental policies on investments and carbon emissions: A case study of Chile. Energy Econ., 75, 261–273.
Abstract: In this article we quantify the effect of uncertainty of climate and environmental policies on transmission and generation investments, as well as on CO2 emissions in Chile. We use a twostage stochastic planning model with recourse or corrective investment options to find optimal portfolios of infrastructure both under perfect information and uncertainty. Under a series of assumptions, this model is equivalent to the equilibrium of a much more complicated bilevel market model, where a transmission planner chooses investments first and generation firms invest afterwards. We find that optimal investment strategies present important differences depending on the policy scenario. By changing our assumption of how agents will react to this uncertainty we compute bounds on the cost that this uncertainty imposes on the system, which we estimate ranges between 3.2% and 5.7% of the minimum expected system cost of $57.6B depending on whether agents will consider or not uncertainty when choosing investments. We also find that, if agents choose investments using a stochastic planning model, uncertain climate policies can result in nearly 18% more CO2 emissions than the equilibrium levels observed under perfect information. Our results highlight the importance of credible and stable longterm regulations for investors in the electric power industry if the goal is to achieve climate and environmental targets in the most costeffective manner and to minimize the risk of asset stranding. (C) 2018 Elsevier B.V. All rights reserved.

Fuenzalida, C., JerezHanckes, C., & McClarren, R. G. (2019). Uncertainty Quantification For Multigroup Diffusion Equations Using Sparse Tensor Approximations. SIAM J. Sci. Comput., 41(3), B545–B575.
Abstract: We develop a novel method to compute first and second order statistical moments of the neutron kinetic density inside a nuclear system by solving the energydependent neutron diffusion equation. Randomness comes from the lack of precise knowledge of external sources as well as of the interaction parameters, known as cross sections. Thus, the density is itself a random variable. As Monte Carlo simulations entail intense computational work, we are interested in deterministic approaches to quantify uncertainties. By assuming as given the first and second statistical moments of the excitation terms, a sparse tensor finite element approximation of the first two statistical moments of the dependent variables for each energy group can be efficiently computed in one run. Numerical experiments provided validate our derived convergence rates and point to further research avenues.

Guevara, E., Babonneau, F., HomemdeMello, T., & Moret, S. (2020). A machine learning and distributionally robust optimization framework for strategic energy planning under uncertainty. Appl. Energy, 271, 18 pp.
Abstract: This paper investigates how the choice of stochastic approaches and distribution assumptions impacts strategic investment decisions in energy planning problems. We formulate a twostage stochastic programming model assuming different distributions for the input parameters and show that there is significant discrepancy among the associated stochastic solutions and other robust solutions published in the literature. To remedy this sensitivity issue, we propose a combined machine learning and distributionally robust optimization (DRO) approach which produces more robust and stable strategic investment decisions with respect to uncertainty assumptions. DRO is applied to deal with ambiguous probability distributions and Machine Learning is used to restrict the DRO model to a subset of important uncertain parameters ensuring computational tractability. Finally, we perform an outofsample simulation process to evaluate solutions performances. The Swiss energy system is used as a case study all along the paper to validate the approach.

Lagos, G., Espinoza, D., Moreno, E., & Vielma, J. P. (2015). Restricted risk measures and robust optimization. Eur. J. Oper. Res., 241(3), 771–782.
Abstract: In this paper we consider characterizations of the robust uncertainty sets associated with coherent and distortion risk measures. In this context we show that if we are willing to enforce the coherent or distortion axioms only on random variables that are affine or linear functions of the vector of random parameters, we may consider some new variants of the uncertainty sets determined by the classical characterizations. We also show that in the finite probability case these variants are simple transformations of the classical sets. Finally we present results of computational experiments that suggest that the risk measures associated with these new uncertainty sets can help mitigate estimation errors of the Conditional ValueatRisk. (C) 2014 Elsevier B.V. All rights reserved.

O' Ryan, R., Benavides, C., Diaz, M., San Martin, J. P., & Mallea, J. (2019). Using probabilistic analysis to improve greenhouse gas baseline forecasts in developing country contexts: the case of Chile. Clim. Policy, 19(3), 299–314.
Abstract: In this paper, initial steps are presented toward characterizing, quantifying, incorporating and communicating uncertainty applying a probabilistic analysis to countrywide emission baseline forecasts, using Chile as a case study. Most GHG emission forecasts used by regulators are based on bottomup deterministic approaches. Uncertainty is usually incorporated through sensitivity analysis and/or use of different scenarios. However, much of the available information on uncertainty is not systematically included. The deterministic approach also gives a wide range of variation in values without a clear sense of probability of the expected emissions, making it difficult to establish both the mitigation contributions and the subsequent policy prescriptions for the future. To improve on this practice, we have systematically included uncertainty into a bottomup approach, incorporating it in key variables that affect expected GHG emissions, using readily available information, and establishing expected baseline emissions trajectories rather than scenarios. The resulting emission trajectories make explicit the probability percentiles, reflecting uncertainties as well as possible using readily available information in a manner that is relevant to the decision making process. Additionally, for the case of Chile, contradictory deterministic results are eliminated, and it is shown that, whereas under a deterministic approach Chile's mitigation ambition does not seem high, the probabilistic approach suggests this is not necessarily the case. It is concluded that using a probabilistic approach allows a better characterization of uncertainty using existing data and modelling capacities that are usually weak in developing country contexts. Key policy insights Probabilistic analysis allows incorporating uncertainty systematically into key variables for baseline greenhouse gas emission scenario projections. By using probabilistic analysis, the policymaker can be better informed as to future emission trajectories. Probabilistic analysis can be done with readily available data and expertise, using the usual models preferred by policymakers, even in developing country contexts.

Pereira, J. (2016). The robust (minmax regret) single machine scheduling with interval processing times and total weighted completion time objective. Comput. Oper. Res., 66, 141–152.
Abstract: Single machine scheduling is a classical optimization problem that depicts multiple real life systems in which a single resource (the machine) represents the whole system or the bottleneck operation of the system. In this paper we consider the problem under a weighted completion time performance metric in which the processing time of the tasks to perform (the jobs) are uncertain, but can only take values from closed intervals. The objective is then to find a solution that minimizes the maximum absolute regret for any possible realization of the processing times. We present an exact branchandbound method to solve the problem, and conduct a computational experiment to ascertain the possibilities and limitations of the proposed method. The results show that the algorithm is able to optimally solve instances of moderate size (2540 jobs depending on the characteristics of the instance). (c) 2015 Elsevier Ltd. All rights reserved.
Keywords: Scheduling; Single machine; Uncertainty; Robust optimization; Branchandbound

RamirezSagner, G., & Munoz, F. D. (2019). The effect of headsensitive hydropower approximations on investments and operations in planning models for policy analysis. Renew. Sust. Energ. Rev., 105, 38–47.
Abstract: Planning for new generation infrastructure in hydrothermal power systems requires consideration of a series of nonlinearities that are often ignored in planning models for policy analysis. In this article, three different capacity planning models are used, one nonlinear and two linear ones, with different degrees of complexity, to quantify the impact of simplifying the head dependency of hydropower generation on investments in both conventional and renewable generators and system operations. It was found that simplified investment models can bias the optimal generation portfolios by, for example, understating the need for coal and combinedcycle gas units and overstating investments in wind capacity with respect to a more accurate nonlinear formulation, which could affect policy recommendations. It was also found that the economic cost of employing a simplified model can be below 10% of total system cost for most of the scenarios and system configurations analyzed, but as high as nearly 70% of total system cost for specific applications. Although these results are not general, they suggest that for certain system configurations both linear models can provide reasonable approximations to more complex nonlinear formulations. Uncertain water inflows were also considered using stochastic variants of all three planning models. Interestingly, if due to time or computational limitations only one of these two features could be accounted for, these results indicate that explicit modeling of the nonlinearhead effect in a deterministic model could yield better results (up to 0.6% of economic regret) than a stochastic linear model (up to 9.6% of economic regret) that considers the uncertainty of water inflows.
Keywords: Generation planning; Hydropower; Policy analysis; Simplifications; Uncertainty

Reus, L., Belbeze, M., Feddersen, H., & Rubio, E. (2018). Extraction Planning Under Capacity Uncertainty at the Chuquicamata Underground Mine. Interfaces, 48(6), 543–555.
Abstract: We propose an extraction schedule for the Chuquicamata underground copper mine in Chile. The schedule maximizes profits while adhering to all operational and geomechanical requirements involved in proper removal of the material. We include extraction capacity uncertainties due to failure in equipment, specifically to the overland conveyor, which we find to be the most critical component in the extraction process. First we present the extraction plan based on a deterministic model, which does not assume uncertainty in the extraction capacity and represents the solution that the mine can implement without using the results of this study. Then we extend this model to a stochastic setting by generating different scenarios for capacity values in subsequent periods. We construct a multistage model that handles economic downside risk arising from this uncertainty by penalizing plans that deviate from an ex ante profit target in one or more scenarios. Simulation results show that a stochasticbased solution can achieve the same expected profits as the deterministicbased solution. However, the earnings of the stochasticbased solution average 5% more for scenarios in which earnings are below the 10th percentile. If we choose a target 2% below the expected profit obtained by the deterministicbased solution, this average increases from 5% to 9%.

VogtGeisse, K., Lorenzo, C., & Feng, Z. L. (2013). Impact Of AgeDependent Relapse And Immunity On Malaria Dynamics. J. Biol. Syst., 21(4), 49 pp.
Abstract: An agestructured mathematical model for malaria is presented. The model explicitly includes the human and mosquito populations, structured by chronological age of humans. The infected human population is divided into symptomatic infectious, asymptomatic infectious and asymptomatic chronic infected individuals. The original partial differential equation (PDE) model is reduced to an ordinary differential equation (ODE) model with multiple age groups coupled by aging. The basic reproduction number R0 is derived for the PDE model and the age group model in the case of general n age groups. We assume that infectiousness of chronic infected individuals gets triggered by bites of even susceptible mosquitoes. Our analysis points out that this assumption contributes greatly to the R0 expression and therefore needs to be further studied and understood. Numerical simulations for n = 2 age groups and a sensitivity/uncertainty analysis are presented. Results suggest that it is important not only to consider asymptomatic infectious individuals as a hidden cause for malaria transmission, but also asymptomatic chronic infections (>60%), which often get neglected due to undetectable parasite loads. These individuals represent an important reservoir for future human infectiousness. By considering agedependent immunity types, the model helps generate insight into effective control measures, by targeting age groups in an optimal way.
