We introduce a solution methodology for risk-neutral and risk-averse stochastic programs with deterministic constraints. Our approach relies on principles from projected gradient descent and sample average approximation algorithms. However, we adaptively control the sample size used in computing the reduced gradient approximation at each iteration. This leads to a significant reduction in cost. Numerical experiments from finance and engineering illustrate the performance and efficacy of the presented algorithms.

For oceanographic applications, probabilistic forecasts typically have to deal with i) high-dimensional complex models, and ii) very sparse spatial observations. In search-and-rescue operations at sea, for instance, the short-term predictions of drift trajectories are essential to efficiently define search areas, but in-situ buoy observations provide only very sparse point measurements, while the mission is ongoing. Statistically optimal forecasts, including consistent uncertainty statements, rely on Bayesian methods for data assimilation to make the best out of both the complex mathematical modeling and the sparse spatial data.To identify suitable approaches for data assimilation in this context, we discuss localisation strategies and compare two state-of-the-art ensemble-based methods for applications with spatially sparse observations. The first method is a version of the ensemble-transform Kalman filter, where we tailor a localisation scheme for sparse point data. The second method is the implicit equal-weights particle filter which has recently been tested for related oceanographic applications.First, we study a linear spatio-temporal model for contaminant advection and diffusion, where the analytical Kalman filter provides a reference. Next, we consider a simplified ocean model for sea currents, where we conduct state estimation and predict drift. Insight is gained by comparing ensemble-based methods on a number of skill scores including prediction bias and accuracy, distribution coverage, rank histograms, spatial connectivity and drift trajectory forecasts.

We introduce adaptive sampling methods for stochastic programs with deterministic constraints. First, we propose and analyze a variant of the stochastic projected gradient method, where the sample size used to approximate the reduced gradient is determined on-the-fly and updated adaptively. This method is applicable to a broad class of expectation-based risk measures, and leads to a significant reduction in the individual gradient evaluations used to estimate the objective function gradient. Numerical experiments with expected risk minimization and conditional value-at-risk minimization support this conclusion, and demonstrate practical performance and efficacy for both risk-neutral and risk-averse problems. Second, we propose an SQP-type method based on similar adaptive sampling principles. The benefits of this method are demonstrated in a simplified engineering design application, featuring risk-averse shape optimization of a steel shell structure subject to uncertain loading conditions and model uncertainty.