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Armstrong, M., Valencia, J., Lagos, G., & Emery, X. (2022). Constructing Branching Trees of Geostatistical Simulations. Math. Geosci., Early Access.
Abstract: This paper proposes the use of multistage stochastic programming with recourse for optimised strategic openpit mine planning. The key innovations are, firstly, that a branching tree of geostatistical simulations is developed to take account of uncertainty in ore grades, and secondly, scenario reduction techniques are applied to keep the trees to a manageable size. Our example shows that different mine plans would be optimal for the downside case when the deposit turns out to be of lower grade than expected compared to when it is of higher grade than expected. Our approach further provides th

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.

Azar, M., Carrasco, R. A., & Mondschein, S. (2022). Dealing with Uncertain Surgery Times in Operating Room Scheduling. Eur. J. Oper. Res., 299(1), 377–394.
Abstract: The operating theater is one of the most expensive units in the hospital, representing up to 40% of the total expenses. Because of its importance, the operating room scheduling problem has been addressed from many different perspectives since the early 1960s. One of the main difficulties that
has reduced the applicability of the current results is the high variability in surgery duration, making schedule recommendations hard to implement. In this work, we propose a timeindexed scheduling formulation to solve the operational problem. Our main contribution is that we propose the use of chance constraints related to the surgery duration's probability distribution for each surgeon to improve the scheduling performance. We show how to implement these chance constraints as linear ones in our timeindexed formulation, enhancing the performance of the resulting schedules significantly. Through data analysis of real historical instances, we develop specific constraints that improve the schedule, reducing the need for overtime without affecting the utilization significantly. Furthermore, these constraints give the operating room manager the possibility of balancing overtime and utilization through a tunning parameter in our formulation. Finally, through simulations and the use of real instances, we report the performance for four different metrics, showing the importance of using historical data to get the right balance between the utilization and overtime. 
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.

Dang, C., Valdebenito, M. A., Faes, M. G. R., Wei, P. F., & Beer, M. (2022). Structural reliability analysis: A Bayesian perspective. Struct. Saf., 99, 102259.
Abstract: Numerical methods play a dominant role in structural reliability analysis, and the goal has long been to produce a failure probability estimate with a desired level of accuracy using a minimum number of performance function evaluations. In the present study, we attempt to offer a Bayesian perspective on the failure probability integral estimation, as opposed to the classical frequentist perspective. For this purpose, a principled Bayesian Failure Probability Inference (BFPI) framework is first developed, which allows to quantify, propagate and reduce numerical uncertainty behind the failure probability due to discretization error. Especially, the posterior variance of the failure probability is derived in a semianalytical form, and the Gaussianity of the posterior failure probability distribution is investigated numerically. Then, a Parallel AdaptiveBayesian Failure Probability Learning (PABFPL) method is proposed within the Bayesian framework. In the PABFPL method, a varianceamplified importance sampling technique is presented to evaluate the posterior mean and variance of the failure probability, and an adaptive parallel active learning strategy is proposed to identify multiple updating points at each iteration. Thus, a novel advantage of PABFPL is that both prior knowledge and parallel computing can be used to make inference about the failure probability. Four numerical examples are investigated, indicating the potential benefits by advocating a Bayesian approach to failure probability estimation.

Dang, C., Wei, P. F., Faes, M. G. R., Valdebenito, M. A., & Beer, M. (2022). Interval uncertainty propagation by a parallel Bayesian global optimization method. Appl. Math. Model., 108, 220–235.
Abstract: This paper is concerned with approximating the scalar response of a complex computational model subjected to multiple input interval variables. Such task is formulated as finding both the global minimum and maximum of a computationally expensive blackbox function over a prescribed hyperrectangle. On this basis, a novel nonintrusive method, called `tripleengine parallel Bayesian global optimization', is proposed. The method begins by assuming a Gaussian process prior (which can also be interpreted as a surrogate model) over the response function. The main contribution lies in developing a novel infill sampling criterion, i.e., tripleengine pseudo expected improvement strategy, to identify multiple promising points for minimization and/or maximization based on the past observations at each iteration. By doing so, these identified points can be evaluated on the real response function in parallel. Besides, another potential benefit is that both the lower and upper bounds of the model response can be obtained with a single run of the developed method. Four numerical examples with varying complexity are investigated to demonstrate the proposed method against some existing techniques, and results indicate that significant computational savings can be achieved by making full use of prior knowledge and parallel computing.

Dang, C., Wei, P. F., Faes, M. G. R., Valdebenito, M. A., & Beer, M. (2022). Parallel adaptive Bayesian quadrature for rare event estimation. Reliab. Eng. Syst. Saf., 225, 108621.
Abstract: Various numerical methods have been extensively studied and used for reliability analysis over the past several decades. However, how to understand the effect of numerical uncertainty (i.e., numerical error due to the discretization of the performance function) on the failure probability is still a challenging issue. The active learning probabilistic integration (ALPI) method offers a principled approach to quantify, propagate and reduce the numerical uncertainty via computation within a Bayesian framework, which has not been fully investigated in context of probabilistic reliability analysis. In this study, a novel method termed `Parallel Adaptive Bayesian Quadrature' (PABQ) is proposed on the theoretical basis of ALPI, and is aimed at broadening its scope of application. First, the Monte Carlo method used in ALPI is replaced with an importance ball sampling technique so as to reduce the sample size that is needed for rare failure event estimation. Second, a multipoint selection criterion is proposed to enable parallel distributed processing. Four numerical examples are studied to demonstrate the effectiveness and efficiency of the proposed method. It is shown that PABQ can effectively assess small failure probabilities (e.g., as low as 10(7)) with a minimum number of iterations by taking advantage of parallel computing.

Dölz, J., Harbrecht, H., JerezHanckes, C., & Multerer M. (2022). Isogeometric multilevel quadrature for forward and inverse random acoustic scattering. Comput. Methods in Appl. Mech. Eng., 388, 114242.
Abstract: We study the numerical solution of forward and inverse timeharmonic acoustic scattering problems by randomly shaped obstacles in threedimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.

EscapilInchauspe, P., & JerezHanckes, C. (2020). Helmholtz Scattering by Random Domains: FirstOrder Sparse Boundary Elements Approximation. SIAM J. Sci. Comput., 42(5), A2561–A2592.
Abstract: We consider the numerical solution of timeharmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance, and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using firstorder shape Taylor expansions, we derive tensor deterministic firstkind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique [M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and P. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128141]. We supply extensive numerical experiments confirming the predicted error convergence rates with polylogarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.

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.

FustosToribio, I., ManqueRoa, N., Vasquez Antipan, D., Hermosilla Sotomayor, M., & Gonzalez, V. L. (2022). Rainfallinduced landslide early warning system based on corrected mesoscale numerical models: an application for the southern Andes. Nat. Hazards Earth Syst. Sci., 22(6), 2169–2183.
Abstract: Rainfallinduced landslides (RILs) are an issue in the southern Andes nowadays. RILs cause loss of life and damage to critical infrastructure. Rainfallinduced landslide early warning systems (RILEWSs) can reduce and mitigate economic and social damages related to RIL events. The southern Andes do not have an operationalscale RILEWS yet. In this contribution, we present a preoperational RILEWS based on the Weather and Research Forecast (WRF) model and geomorphological features coupled to logistic models in the southern Andes. The models have been forced using precipitation simulations. We correct the precipitation derived from WRF using 12 weather stations through a bias correction approach. The models were trained using 57 wellcharacterized RILs and validated by ROC analysis. We show that WRF has strong limitations in representing the spatial variability in the precipitation. Therefore, accurate precipitation needs a bias correction in the study zone. We used accurate precipitation simulation and slope, demonstrating a high predicting capacity (area under the curve, AUC, of 0.80). We conclude that our proposal could be suitable at an operational level under determined conditions. A reliable RIL database and operational weather networks that allow realtime correction of the mesoscale model in the implemented zone are needed. The RILEWSs could become a support to decisionmakers during extremeprecipitation events related to climate change in the south of the Andes.

Gordon, M. A., Vargas, F. J., & Peters, A. A. (2021). Comparison of Simple Strategies for Vehicular Platooning With Lossy Communication. IEEE Access, 9, 103996–104010.
Abstract: This paper studies vehicle platooning with communication channels subject to random data loss. We focus on homogeneous discretetime platoons in a predecessorfollowing topology with a constant time headway policy. We assume that each agent in the platoon sends its current position to the immediate follower through a lossy channel modeled as a Bernoulli process. To reduce the negative effects of data loss over the string stability and performance of the platoon, we use simple strategies that modify the measurement, error, and control signals of the feedback control loop, in each vehicle, when a dropout occurs. Such strategies are based on holding the previous value, dropping to zero, or replacing with a prediction based on a simple linear extrapolation. We performed a simulationbased comparison among a set of different strategies, and found that some strategies are favorable in terms of performance, while some others present improvements for string stabilization. These results strongly suggest that proper design of compensation schemes for the communications of interconnected multiagent systems plays an important role in their performance and their scalability properties.

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.

HomemdeMello, T., Kong, Q. X., & GodoyBarba, R. (2022). A Simulation Optimization Approach for the Appointment Scheduling Problem with DecisionDependent Uncertainties. INFORMS J. Comput., Early Access.
Abstract: The appointment scheduling problem (ASP) studies how to manage patient arrivals to a healthcare system to improve system performance. An important challenge occurs when some patients may not show up for an appointment. Although the ASP is well studied in the literature, the vast majority of the existing work does not consider the wellobserved phenomenon that patient noshow is influenced by the appointment time, the usual decision variable in the ASP. This paper studies the ASP with random service time (exogenous uncertainty) with known distribution and patient decisiondependent noshow behavior (endogenous uncertainty). This problem belongs to the class of stochastic optimization with decisiondependent uncertainties. Such problems are notoriously difficult as they are typically nonconvex. We propose a stochastic projected gradient path (SPGP) method to solve the problem, which requires the development of a gradient estimator of the objective functiona nontrivial task, as the literature on gradientbased optimization algorithms for problems with decisiondependent uncertainty is very scarce and unsuitable for our model. Our method can solve the ASP problem under arbitrarily smooth showup probability functions. We present solutions under different patterns of noshow behavior and demonstrate that breaking the assumption of constant showup probability substantially changes the scheduling solutions. We conduct numerical experiments in a variety of settings to compare our results with those obtained with a distributionally robust optimization method developed in the literature. The cost reduction obtained with our method, which we call the value of distribution information, can be interpreted as how much the system performance can be improved by knowing the distribution of the service times, compared to not knowing it. We observe that the value of distribution information is up to 31% of the baseline cost, and that such value is higher when the system is crowded or/and the waiting time cost is relatively high.
Summary of Contribution: This paper studies an appointment scheduling problem under timedependent patient noshow behavior, a situation commonly observed in practice. The problem belongs to the class of stochastic optimization problems with decisiondependent uncertainties in the operations research literature. Such problems are notoriously difficult to solve as a result of the lack of convexity, and the present case requires different techniques because of the presence of continuous distributions for the service times. A stochastic projected gradient path method, which includes the development of specialized techniques to estimate the gradient of the objective function, is proposed to solve the problem. For problems with a similar structure, the algorithm can be applied once the gradient estimator of the objective function is obtained. Extensive numerical studies are presented to demonstrate the quality of the solutions, the importance of modeling timedependent noshows in appointment scheduling, and the value of using distribution information about the service times. 
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.

Ni, P. H., Jerez, D. J., Fragkoulis, V. C., Faes, M. G. R., Valdebenito, M. A., & Beer, M. (2022). Operator NormBased Statistical Linearization to Bound the First Excursion Probability of Nonlinear Structures Subjected to Imprecise Stochastic Loading. ASCEASME J. Risk Uncertain. Eng. Syst. ACiv. Eng., 8(1), 04021086.
Abstract: This paper presents a highly efficient approach for bounding the responses and probability of failure of nonlinear models subjected to imprecisely defined stochastic Gaussian loads. Typically, such computations involve solving a nested doubleloop problem, where the propagation of the aleatory uncertainty has to be performed for each realization of the epistemic parameters. Apart from neartrivial cases, such computation is generally intractable without resorting to surrogate modeling schemes, especially in the context of performing nonlinear dynamical simulations. The recently introduced operator norm framework allows for breaking this double loop by determining those values of the epistemic uncertain parameters that produce bounds on the probability of failure a priori. However, the method in its current form is only applicable to linear models due to the adopted assumptions in the derivation of the involved operator norms. In this paper, the operator norm framework is extended and generalized by resorting to the statistical linearization methodology to

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%.
