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We start with a variety of technical and physical problems by showing how discrete modeling under uncertainties is technically meaningful and often even problem inherent. Then we select the technical problem of contaminant spread in water grids, to which we apply the method of risk-level propagation in depth. Extensions of the method such as modeling of contaminant mixing at junctions in the discretized phase space and the applicability of conservation laws arise naturally along these lines and are discussed in the context of the general theory.<\/jats:p>","DOI":"10.1017\/s0890060417000063","type":"journal-article","created":{"date-parts":[[2017,5,4]],"date-time":"2017-05-04T10:04:18Z","timestamp":1493892258000},"page":"173-184","source":"Crossref","is-referenced-by-count":0,"title":["Efficient simulation of threshold behavior"],"prefix":"10.1017","volume":"31","author":[{"given":"Dominic","family":"Kohler","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Johannes","family":"M\u00fcller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Birgit","family":"Obst","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Utz","family":"Wever","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,5,4]]},"reference":[{"key":"S0890060417000063_ref28","volume-title":"Guidelines for Drinking Water Quality","year":"2008"},{"key":"S0890060417000063_ref26","volume-title":"Water Security Initiative\u2014Contamination Warning System Pilots for Water Distribution Systems","year":"2007"},{"key":"S0890060417000063_ref25","volume-title":"Quantum Mechanics of Atoms and Molecules","author":"Thirring","year":"1981"},{"key":"S0890060417000063_ref13","volume-title":"Introduction to Solid State Physics","author":"Kittel","year":"2005"},{"key":"S0890060417000063_ref6","first-page":"221","volume-title":"Handbook of Dynamical Systems","author":"Dellnitz","year":"2002"},{"key":"S0890060417000063_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996313002"},{"key":"S0890060417000063_ref19","volume-title":"Numerical Methods for Conservation Laws","author":"LeVeque","year":"1999"},{"key":"S0890060417000063_ref12","volume-title":"Str\u00f6mungsberechnung f\u00fcr Rohrsysteme","author":"Horlacher","year":"2012"},{"key":"S0890060417000063_ref21","volume-title":"Sicherheit in der Trinkwasserversorgung dvgw-hinweise w 1001 und w 1002","author":"Niehues","year":"2010"},{"key":"S0890060417000063_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/S0927-7757(01)00719-1"},{"key":"S0890060417000063_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4928"},{"key":"S0890060417000063_ref8","article-title":"Probabilistic cellular automata with conserved quantities","volume":"17","author":"Fuks","year":"2003","journal-title":"Nonlinearity"},{"key":"S0890060417000063_ref1","volume-title":"Chemical Principles: The Quest for Insight","author":"Atkins","year":"2008"},{"key":"S0890060417000063_ref23","volume-title":"EPANET Multi-Species Extension User's Manual","author":"Shang","year":"2011"},{"key":"S0890060417000063_ref7","volume-title":"Towards a More Secure Society and Increased Industrial Competitiveness. 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