European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   Uncertainty Quantification and Stochastic Models (Part 1)
10:30
25 mins
Uncertainty Quantification in Wind Power Forecasting
Waleed Alhaddad
Abstract: Reliable wind power generation forecasting is crucial to meet energy demand, to trade and invest. We propose a model to simulate and quantify uncertainties in such forecasts. This model is based on Stochastic Differential Equations whose time-dependent parameters are inferred using continuous optimization of an approximate Likelihood function. The result is a skew stochastic process that simulates uncertainty of wind power forecasts accounting for maximum power production limit and other temporal effects. We apply the model to historical Uruguayan data and forecasts.
10:55
25 mins
SECOND DERIVATIVE NUMERICAL INTEGRATORS FOR THE SOLUTIONS OF RANDOM ORDINARY DIFFERENTIAL EQUATIONS
Joshua Chollom, Makrop Longshak
Abstract: Abstract This paper presents a class of Second Derivative linear multi-step methods with excellent stability properties based on the Backward Differentiation Formula and the Generalized Adams Moulton strategy developed through the multi-step collocation approach. The procedure yielded second derivative block Numerical integrators for the solutions of Random Ordinary Differential Equations .The stability of the new methods investigated shows that they are A- stable, consistent, zero stable and hence convergent. The new SEDNI methods used in block form and tested on Random Ordinary Differential Equations confirms that they are efficient ,suitable for these class of problems and compete favorably with the state of the art Matlab ODE solver ode45.
11:20
25 mins
Bayesian network PDEs for multiscale representations of porous materials
Eric Hall, Kimoon Um, Markos Katsoulakis, Daniel Tartakovsky
Abstract: Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (Darcy- or continuum-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media, and the design of novel materials, e.g., for energy storage. Most microscopic properties exhibit complex statistical correlations and geometric constraints, which presents challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian networks. The proposed framework allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that are informed by domain knowledge and emulate engineering processes, e.g., the design of hierarchical nanoporous materials. These PDFs also serve as input for the forward propagation of parametric uncertainty. To assess the impact of correlations and causal relationships between microscopic parameters on macroscopic material properties, we use a moment-independent GSA based on the differential mutual information that leverages the structure of the Bayesian network and accounts for both correlated inputs and complex non-Gaussian QoIs. The global sensitivity indices are used to rank the effect of uncertainty in microscopic parameters on macroscopic QoIs and to quantify the impact of causality on the multiscale model's predictions. Our findings from numerical experiments indicate two practical outcomes. Firstly, the inclusion of correlations through structured priors based on causal relationships impacts predictions of QoIs which has important implications for decisions support and engineering design. Secondly, we observe the inclusion of structured priors with non-trivial correlations yields different effect rankings than independent priors and moreover these rankings are more consistent with the anticipated physics of a model problem.