First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

The significance of occupancy steadiness in residential consumer response to Time of Use pricing: Evidence from a stochastic adjustment model

The recent roll-out of smart metering technologies in several developed countries has intensified research on the impacts of Time-of-Use (TOU) pricing on consumption. This paper analyses a TOU dataset from the Province of Trento in Northern Italy using a stochastic adjustment model. Findings highlight the non-steadiness of the relationship between consumption and TOU price. Weather and active occupancy can partly explain future consumption in relation to price.

The stochastic component in choice and regression to the mean

In this article, we illustrate experimentally an important consequence of the stochastic component in choice behaviour which has not been acknowledged so far. Namely, its potential to produce ‘regression to the mean’ (RTM) effects. We employ a novel approach to individual choice under risk, based on repeated multiple-lottery choices (i.e. choices among many lotteries), to show how the high degree of stochastic variability present in individual decisions can distort crucially certain results through RTM effects. We demonstrate the point in the context of a social comparison experiment.

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models.

Advances in the stochastic modelling of satellite-derived rainfall estimates using a sparse calibration dataset

As satellite technology develops, satellite rainfall estimates are likely to become ever more important in the world of food security. It is therefore vital to be able to identify the uncertainty of such estimates and for end users to be able to use this information in a meaningful way. This paper presents new developments in the methodology of simulating satellite rainfall ensembles from thermal infrared satellite data. Although the basic sequential simulation methodology has been developed in previous studies, it was not suitable for use in regions with more complex terrain and limited calibration data. Developments in this work include the creation of a multithreshold, multizone calibration procedure, plus investigations into the causes of an overestimation of low rainfall amounts and the best way to take into account clustered calibration data. A case study of the Ethiopian highlands has been used as an illustration.

Optimal power management strategy for energy storage with stochastic loads

In this paper, a power management strategy (PMS) has been developed for the control of energy storage in a system subjected to loads of random duration. The PMS minimises the costs associated with the energy consumption of specific systems powered by a primary energy source and equipped with energy storage, under the assumption that the statistical distribution of load durations is known. By including the variability of the load in the cost function, it was possible to define the optimality criteria for the power flow of the storage. Numerical calculations have been performed obtaining the control strategies associated with the global minimum in energy costs, for a wide range of initial conditions of the system. The results of the calculations have been tested on a MATLAB/Simulink model of a rubber tyre gantry (RTG) crane equipped with a flywheel energy storage system (FESS) and subjected to a test cycle, which corresponds to the real operation of a crane in the Port of Felixstowe. The results of the model show increased energy savings and reduced peak power demand with respect to existing control strategies, indicating considerable potential savings for port operators in terms of energy and maintenance costs.

Evaluation of the Plant–Craig stochastic convection scheme (v2.0) in the ensemble forecasting system MOGREPS-R (24 km) based on the Unified Model (v7.3)

The Plant–Craig stochastic convection parameterization (version 2.0) is implemented in the Met Office Regional Ensemble Prediction System (MOGREPS-R) and is assessed in comparison with the standard convection scheme with a simple stochastic scheme only, from random parameter variation. A set of 34 ensemble forecasts, each with 24 members, is considered, over the month of July 2009. Deterministic and probabilistic measures of the precipitation forecasts are assessed. The Plant–Craig parameterization is found to improve probabilistic forecast measures, particularly the results for lower precipitation thresholds. The impact on deterministic forecasts at the grid scale is neutral, although the Plant–Craig scheme does deliver improvements when forecasts are made over larger areas. The improvements found are greater in conditions of relatively weak synoptic forcing, for which convective precipitation is likely to be less predictable.