European Numerical Mathematics and
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   Diverse Applications (Part 2)
Chair: Fred Vermolen
 10:40 25 mins Asymptotic analysis on large-scale gas networks Dimitrios Zacharenakis, Jan Giesselmann Abstract: \begin{Abstract} In the past decade the use and production of natural gas has grown massively, due to the transition to more environmentally friendly energy resources, and will continue as the global energy demand is increasing. While the use of pipeline networks has become the most popular way for the transportation of gas, there is much emphasis given on the cost-effective design and operation of a gas network. A rather detailed modeling approach is to use the isothermal Euler equations on the pipes and to couple them at the nodes. Even though there are many coupling conditions that are proposed in the literature, we will focus on energy consistent coupling conditions \cite{RG}. For applications, such as optimization of network operations (e.g. switching on and off of compressors), simpler models are usually employed. Since operators are often interested in large space and time scales, the equations on the pipes may be written as \begin{flalign} \begin{split} \left(\rho^{\epsilon}\right)_{t} + \frac{1}{\epsilon}\left(\rho^{\epsilon} v^{\epsilon}\right)_{x} &= 0, \\ \left(v^{\epsilon} \right)_{t} + \frac{1}{\epsilon} v^{\epsilon} \left(v^{\epsilon} \right)_{x} +\frac{1}{\epsilon} \left(W'(\rho^{\epsilon} )\right)_{x} &= - \frac{1}{\epsilon^{2}} v^{\epsilon} , \end{split} \label{network1} \tag{$\Sigma^{\epsilon}$} \end{flalign} where $\epsilon >0$ is a small parameter and it is desirable to replace these equations by their parabolic limit \begin{flalign*} \begin{split} &\left(\bar{\rho} \right)_{t} + \frac{1}{\epsilon} \left(\bar{\rho} \bar{v} \right)_{x} =0,\\ &\bar{v} = -\epsilon \frac{1}{\bar{\rho} } p(\bar{\rho} )_{x} = -\epsilon W'(\bar{\rho} )_{x}, \end{split} \end{flalign*} which in fact allows us to extend the result of \cite{LTZ} to networks. The analysis is based on the relative entropy framework in order to compare weak entropic solutions to smooth solutions. However, the coupling conditions seem to make it impossible to handle weak entropy solutions of (\ref{network1}) Therefore, we consider a diffusion regularization of (\ref{network1}) and couple the limit for $\epsilon \rightarrow 0$ with a vanishing diffusion limit. \end{Abstract} % REFERENCES \begin{thebibliography}{99.} \bibitem{LTZ} C. Lattanzio, A.E. Tzavaras, \textit{Relative entropy in diffusive relaxation}. SIAM J. Math. Anal. \textbf{45}, 1563 - 1584, (2013) \par\vspace{2mm} \bibitem{RG} G. A. Reigstad, \textit{Numerical network models and entropy principles for isothermal junction flow}. Netw. Heterog. Media \textbf{9}(1), 65 - 95, (2014) \end{thebibliography} 11:05 25 mins Deep Learning Approaches for Metallographic Image Segmentation Franziska Riegger Abstract: Mechanical properties of materials is significantly determined by its microstructure. Based on that, the latter reveals diverse information about the investigated engine component, e.g. its application history or its further usability. Until now, microstructural analysis has been carried out manually by experts. This approach is not only time consuming and costly, but also suffers from subjectivity. Such weaknesses motivate the attempt to automatise the analysis. At the same time, these efforts are hampered by the fact that the microstructure of a material can have multiple and complex manifestations. Hence, powerful tools are required to provide the same accuracy as the well-established expert-based method. With Deep Convolutional Neural Networks, one such competitive approach is found. Several such models are defined, optimized and evaluated on two application examples. Further, we examine useful metrics including one that encodes the important characteristics of the considered network architecture, such as the effective receptive field. 11:30 25 mins Approximation method with stochastic local iterated function systems Somogyi Ildikó, Anna Soós Abstract: The methods of real data interpolation can be generalized with fractal interpolation. These fractal interpolation functions can be constructed with the so-called iterated function systems. Local iterated function systems are important generalization of the classical iterated function systems. In order to obtain new approximation methods this methods can be combine with the classical interpolation methods. In this paper we focus on the study of the stochastic local fractal interpolation function in the case when the vertical scaling parameter is a random variable. 11:55 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.