Optimal control with stochastic PDE constraints and uncertain controls

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby forgoing the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented. © 2011 Elsevier B.V.

Model structure and uncertainty for stochastic non-point source modelling applications

The uncertainty associated with a rainfall-runoff and non-point source loading (NPS) model can be attributed to both the parameterization and model structure. An interesting implication of the areal nature of NPS models is the direct relationship between model structure (i.e. sub-watershed size) and sample size for the parameterization of spatial data. The approach of this research is to find structural limitations in scale for the use of the conceptual NPS model, then examine the scales at which suitable stochastic depictions of key parameter sets can be generated. The overlapping regions are optimal (and possibly the only suitable regions) for conducting meaningful stochastic analysis with a given NPS model. Previous work has sought to find optimal scales for deterministic analysis (where, in fact, calibration can be adjusted to compensate for sub-optimal scale selection); however, analysis of stochastic suitability and uncertainty associated with both the conceptual model and the parameter set, as presented here, is novel; as is the strategy of delineating a watershed based on the uncertainty distribution. The results of this paper demonstrate a narrow range of acceptable model structure for stochastic analysis in the chosen NPS model. In the case examined, the uncertainties associated with parameterization and parameter sensitivity are shown to be outweighed in significance by those resulting from structural and conceptual decisions. © 2011 Copyright IAHS Press.

Comparison of stochastic methods for control in air traffic management

This paper provides a direct comparison of two stochastic optimisation techniques (Markov Chain Monte Carlo and Sequential Monte Carlo) when applied to the problem of conflict resolution and aircraft trajectory control in air traffic management. The two methods are then also compared to another existing technique of Mixed-Integer Linear Programming which is also popular in distributed control. © 2011 IFAC.

A Stochastic Model for Electron Multiplication Charge-Coupled Devices - From Theory to Practice

Electron multiplication charge-coupled devices (EMCCD) are widely used for photon counting experiments and measurements of low intensity light sources, and are extensively employed in biological fluorescence imaging applications. These devices have a complex statistical behaviour that is often not fully considered in the analysis of EMCCD data. Robust and optimal analysis of EMCCD images requires an understanding of their noise properties, in particular to exploit fully the advantages of Bayesian and maximum-likelihood analysis techniques, whose value is increasingly recognised in biological imaging for obtaining robust quantitative measurements from challenging data. To improve our own EMCCD analysis and as an effort to aid that of the wider bioimaging community, we present, explain and discuss a detailed physical model for EMCCD noise properties, giving a likelihood function for image counts in each pixel for a given incident intensity, and we explain how to measure the parameters for this model from various calibration images. © 2013 Hirsch et al.

Visualising a stochastic model of California Water Resources Using Sankey Diagrams

This paper describes a novel approach to the analysis of supply and demand of water in California. A stochastic model is developed to assess the future supply of and demand for water resources in California. The results are presented in the form of a Sankey diagram where present and stochastically-varying future fluxes of water in California and its sub-regions are traced from source to services by mapping the various transformations of water from when it is first made available for use, through its treatment, recycling and reuse, to its eventual loss in a variety of sinks. This helps to highlight the connections of water with energy and land resources, including the amount of energy used to pump and treat water, the amount of water used for energy production, and the land resources that create a water demand to produce crops for food. By mapping water in this way, policy-makers can more easily understand the competing uses of water, through the identification of the services it delivers (e.g. sanitation, food production, landscaping), the potential opportunities for improving themanagement of the resource and the connections with other resources which are often overlooked in a traditional sector-based management strategy. This paper focuses on a Sankey diagram for water, but the ultimate aim is the visualisation of linked resource futures through inter-connected Sankey diagrams for energy, land and water, tracking changes from the basic resources for all three, their transformations, and the final services they provide.

The stochastic implicit Euler method - A stable coupling scheme for Monte Carlo burnup calculations

Existing Monte Carlo burnup codes use various schemes to solve the coupled criticality and burnup equations. Previous studies have shown that the coupling schemes of the existing Monte Carlo burnup codes can be numerically unstable. Here we develop the Stochastic Implicit Euler method - a stable and efficient new coupling scheme. The implicit solution is obtained by the stochastic approximation at each time step. Our test calculations demonstrate that the Stochastic Implicit Euler method can provide an accurate solution to problems where the methods in the existing Monte Carlo burnup codes fail. © 2013 Elsevier Ltd. All rights reserved.