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1. Nonsmooth d'Alembertian for Lorentz distance functions

Author

Braun, Mathias

Subjects
Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, Mathematical Physics, Mathematics - Metric Geometry, 28A50, 51K10 (Primary), 35J92, 49Q22, 51F99, 53C21, 53C50, 83C75 (Secondary)
Abstract

We refine a recent distributional notion of d'Alembertian of a signed Lorentz distance function to an achronal set in a metric measure spacetime obeying the timelike measure contraction property. We show precise representation formulas and comparison estimates (both upper and lower bounds). Under a condition we call "infinitesimally strict concavity" (known for infinitesimally Minkowskian structures and established here for Finsler spacetimes), we prove the associated distribution is a signed measure certifying the integration by parts formula. This treatment of the d'Alembertian using techniques from metric geometry expands upon its recent elliptic interpretation; even in the smooth case, our formulas seem to pioneer its exact shape across the timelike cut locus. Two central ingredients our contribution unifies are the localization paradigm of Cavalletti-Mondino and the Lorentzian Sobolev calculus of Beran et al. In the second part of our work, we present several applications of these insights. First, we show the equivalence of the timelike curvature-dimension condition with a Bochner-type inequality. Second, we set up synthetic mean curvature (and barriers for CMC sets) exactly. Third, we prove volume and area estimates of Heintze-Karcher-type, which enable us to show several synthetic volume singularity theorems., Comment: 84 pages. Comments welcome

Published
2024

2. A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes

Author

Beran, Tobias, Braun, Mathias, Calisti, Matteo, Gigli, Nicola, McCann, Robert J., Ohanyan, Argam, Rott, Felix, and Sämann, Clemens

Subjects
Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, Mathematical Physics, Mathematics - Metric Geometry, 51K10, 83C75 (Primary), 35J92, 35Q75, 49Q22, 53C21, 53C50 (Secondary)
Abstract

We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus., Comment: 132 pages. Comments welcome

Published
2024

3. Causal convergence conditions through variable timelike Ricci curvature bounds

Author

Braun, Mathias and McCann, Robert J.

Subjects
Mathematical Physics, Mathematics - Differential Geometry, Mathematics - Metric Geometry, 51K10 (Primary), 28A50, 49Q22, 51F99, 53Z99, 83C75 (Secondary)
Abstract

We describe a nonsmooth notion of globally hyperbolic, regular length metric spacetimes $(\mathrm{M},l)$. It is based on ideas of Kunzinger-S\"amann, but does not require Lipschitz continuity of causal curves. We study geodesics on $\mathrm{M}$ and the space of probability measures over $\mathrm{M}$ in detail. Furthermore, for such a spacetime endowed with a reference measure $\mathfrak{m}$, a lower semicontinuous function $k\colon \mathrm{M} \to \textbf{R}$, and constants $0

Published
2023

4. Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes

Author

Braun, Mathias and Ohta, Shin-ichi

Subjects
Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, Mathematics - Metric Geometry, 53C50, 53C60, 58E10, 83C99
Abstract

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from below by a real number $K$ in every timelike direction satisfies the timelike curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also shown. Our discussion is based on the solvability of the Monge problem with respect to the $q$-Lorentz-Wasserstein distance as well as the characterization of $q$-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case., Comment: 56 pages. Comments welcome

Published
2023

5. Timelike Ricci bounds for low regularity spacetimes by optimal transport

Author

Braun, Mathias and Calisti, Matteo

Subjects
General Relativity and Quantum Cosmology, Mathematical Physics, Mathematics - Differential Geometry, Mathematics - Probability, 49J52, 53C50, 58E10, 83C99
Abstract

We prove that a globally hyperbolic smooth spacetime endowed with a $\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $\smash{\mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics., Comment: 23 pages. Final version

Published
2022

6. R\'enyi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

Author

Braun, Mathias

Subjects
Mathematical Physics, Mathematics - Differential Geometry, Mathematics - Metric Geometry, Mathematics - Probability, 49J52, 53C50, 58E10, 58Z05, 83C99
Abstract

For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger, S\"amann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by $K\in\boldsymbol{\mathrm{R}}$ and upper dimension bounds by $N\in[1,\infty)$, namely the timelike curvature-dimension conditions $\smash{\mathrm{TCD}_p(K,N)}$ and $\smash{\mathrm{TCD}_p^*(K,N)}$ in weak and strong forms, where $p\in (0,1)$, and the timelike measure-contraction properties $\smash{\mathrm{TMCP}(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$. These are formulated by convexity properties of the R\'enyi entropy with respect to $\mathfrak{m}$ along $\smash{\ell_p}$-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological $\smash{\ell_p}$-optimal couplings and chronological $\smash{\ell_p}$-geodesics. We also prove the equivalence of $\smash{\mathrm{TCD}_p^*(K,N)}$ and $\smash{\mathrm{TMCP}^*(K,N)}$ to their respective entropic counterparts in the sense of Cavalletti and Mondino. Some of these results are obtained under timelike $p$-essential nonbranching, a concept which is a priori weaker than timelike nonbranching., Comment: 74 pages. Various typos have been corrected. Some details and the assumption of regularity have been added. Corrected proof of uniqueness of chronological couplings. Final version

Published
2022

7. Good geodesics satisfying the timelike curvature-dimension condition

Author

Braun, Mathias

Subjects
Mathematical Physics, Mathematics - Differential Geometry, Mathematics - Metric Geometry, 49J52, 53C50, 58E10, 58Z05, 83C99
Abstract

Let $(M,\mathsf{d},\mathfrak{m},\ll,\leq,\tau)$ be a causally closed, $\mathscr{K}$-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition $\smash{\mathrm{wTCD}_p^e(K,N)}$ in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on $M$ which satisfy the entropic semiconvexity inequality defining $\smash{\mathrm{wTCD}_p^e(K,N)}$ and whose densities with respect to $\mathfrak{m}$ are additionally uniformly $L^\infty$ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property., Comment: 29 pages. Minor expositionary changes. Assumption of regularity has been added

Published
2022

8. Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Author

Braun, Mathias and Rigoni, Chiara

Subjects
Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, Mathematics - Probability, Primary: 51F30, 58J35, Secondary: 46E36, 47D08, 53C23
Abstract

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a constant $C>0$ and a bounded Lipschitz function $R\colon M \to (0,\infty)$ such that for every $x\in M$ and every $t>0$, \begin{align*} \sup_{y\in M} \mathsf{p}(t,x,y) \leq C\min\{t, R^2(x)\}^{-d/2}. \end{align*} This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by $(M,g)$. In the case $\partial M \neq \emptyset$, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on $\partial M$. We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense., Comment: 28 pages. Comments welcome

Published
2021

9. Vector calculus for tamed Dirichlet spaces

Author

Braun, Mathias

Subjects
Mathematics - Differential Geometry, Mathematics - Functional Analysis, Primary: 53C21, 58J35, Secondary: 31C25, 35J10, 35K20, 58J32
Abstract

In the language of $L^\infty$-modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M,\mathfrak{m})$ carrying a quasi-regular, strongly local Dirichlet form $\mathscr{E}$. Furthermore, we develop a second order calculus if $(M,\mathscr{E},\mathfrak{m})$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form., Comment: 118 pages. Minor corrections

Published
2021

11. Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel

Author

Braun, Mathias

Subjects
Mathematics - Functional Analysis, Mathematics - Differential Geometry, Mathematics - Metric Geometry, Mathematics - Spectral Theory, Primary: 35K08, 58J35, Secondary: 35P15, 58A14, 58C40
Abstract

We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space $(M,\mathsf{d},\mathfrak{m})$, $K\in\boldsymbol{\mathrm{R}}$. We show Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's inequality for $(\mathsf{H}_t)_{t\geq 0}$ w.r.t. the heat flow $(\mathsf{P}_t)_{t\geq 0}$ on $L^2(M)$. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for $1$-forms, the previous inequalities yield various $L^p$-properties of $(\mathsf{H}_t)_{t\geq 0}$, $p\in [1,\infty]$. Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian $\smash{\vec{\Delta}}$, of the negative functional Laplacian $-\Delta$, and of the Schr\"odinger operator $-\Delta+K$. In the $\mathrm{RCD}^*(K,N)$ case, we prove compactness of $\smash{\vec{\Delta}^{-1}}$ if $M$ is compact, and the independence of the $L^p$-spectrum of $\smash{\vec{\Delta}}$ on $p \in [1,\infty]$ under a volume growth condition. We terminate by giving an appropriate interpretation of a heat kernel for $(\mathsf{H}_t)_{t\geq 0}$. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it., Comment: 50 pages. The bibliography has been extended. Minor changes have been performed

Published
2020

12. Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

Author

Braun, Mathias and Güneysu, Batu

Subjects
Mathematics - Probability, Mathematics - Differential Geometry, Mathematics - Functional Analysis
Abstract

We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < \zeta^x\}}\big] < \infty, \end{equation*} then the lifetime $\zeta^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in M$ is a.s. infinite. This assumption on $k$ holds if $k^-$ belongs to the Kato class of $M$. We also derive a Bismut-Elworthy-Li derivative formula for $\nabla \mathsf{P}_tf$ for every $f\in L^\infty(M)$ and $t>0$ along the heat flow $(\mathsf{P}_t)_{t\geq 0}$ with generator $\Delta/2$, yielding its $L^\infty$-$\mathrm{Lip}$-regularization as a corollary. Moreover, given the stochastic completeness of $M$, but without any assumption on $k$ except continuity, we prove the equivalence of lower boundedness of $\mathrm{Ric}$ by $k$ to the existence, given any $x,y\in M$, of a coupling $(\mathsf{b}^x,\mathsf{b}^y)$ of Brownian motions on $M$ starting in $(x,y)$ such that a.s., \begin{equation*} \mathsf{d}\big(\mathsf{b}_t^x,\mathsf{b}_t^y\big) \leq \mathrm{e}^{-\int_s^t \underline{k}(\mathsf{b}_r^x,\mathsf{b}_r^y)/2\,\mathrm{d} r}\,\mathsf{d}\big(\mathsf{b}_s^x,\mathsf{b}_s^y\big) \end{equation*} holds for every $s,t\geq 0$ with $s\leq t$, involving the "average" $\underline{k}(u,v) := \inf_\gamma \int_0^1 k(\gamma_r)\,\mathrm{d} r$ of $k$ along geodesics from $u$ to $v$. Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-\'Emery Ricci tensor., Comment: 32 pages. Title has been changed. The case of weighted Riemannian manifolds has been included. The assumption on Dynkin decomposability has been replaced by a weaker exponential integrability condition. Theorem 1.3 has been added. The bibliography has been extended

Published
2020

14. Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds

Author

Braun, Mathias, Habermann, Karen, and Sturm, Karl-Theodor

Subjects
Mathematics - Functional Analysis, Mathematics - Analysis of PDEs, Mathematics - Differential Geometry, Mathematics - Probability
Abstract

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k\colon X\to\mathbb{R}$, we prove the equivalence of the synthetic approaches to Ricci curvature at $x\in X$ being bounded from below by $k(x)$ in terms of $\bullet$ the Bakry-\'Emery estimate $\Delta\Gamma(f)/2 - \Gamma(f,\Delta f) \geq k\,\Gamma(f)$ in an appropriate weak formulation, and $\bullet$ the curvature-dimension condition $\mathrm{CD}(k,\infty)$ in the sense Lott-Sturm-Villani with variable $k$. Moreover, for all $p\in(1,\infty)$, these properties hold if and only if the perturbed $p$-transport cost \begin{equation*} W_p^{\underline{k}}(\mu_1,\mu_2,t):=\inf_{(\mathsf{b}^1,\mathsf{b}^2)} \mathbb{E}\Big[\mathrm{e}^{\int_0^{2t} p \underline{k}\left(\mathsf{b}^1_{r}, \mathsf{b}^2_{r}\right)/2\,\mathrm{d} r} \mathsf{d}^p\!\left(\mathsf{b}^1_{2t},\mathsf{b}^2_{2t} \right)\!\Big]^{1/p} \end{equation*} is nonincreasing in $t$. The infimum here is taken over pairs of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ on $X$ with given initial distributions $\mu_1$ and $\mu_2$, respectively, and $\underline{k}(x,y) := \inf_\gamma \int_0^1 k(\gamma_s)\,\mathrm{d} s$ denotes the "average" of $k$ along geodesics $\gamma$ connecting $x$ and $y$. Furthermore, for any pair of initial distributions $\mu_1$ and $\mu_2$ on $X$, we prove the existence of a pair of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ such that a.s. for every $s,t\in[0,\infty)$ with $s\leq t$, we have \begin{equation*} \mathsf{d}\!\left(\mathsf{b}_t^1,\mathsf{b}_t^2\right)\leq \mathrm{e}^{-\int_s^t \underline{k}\left(\mathsf{b}_r^1,\mathsf{b}_r^2\right)/2\,\mathrm{d} r} \mathsf{d}\!\left(\mathsf{b}_s^1,\mathsf{b}_s^2\right)\!. \end{equation*}, Comment: 38 pages

Published
2019
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17. Timelike Ricci bounds for low regularity spacetimes by optimal transport.

Author

Braun, Mathias and Calisti, Matteo

Subjects
*RENYI'S entropy, *GEODESICS, *SPACETIME, *SENSES, *EQUATIONS
Abstract

We prove that a globally hyperbolic smooth spacetime endowed with a C1-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even C1,1, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics. [ABSTRACT FROM AUTHOR]

Published
2024
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18. Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes.

Author

Braun, Mathias and Ohta, Shin-ichi

Subjects
*CURVATURE, *REAL numbers, *PROBABILITY measures, *SPACETIME
Abstract

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature \mathrm {Ric}_N is bounded from below by a real number K in every timelike direction satisfies the timelike curvature-dimension condition \mathrm {TCD}_q(K,N) for all q\in (0,1). The converse and a nonpositive-dimensional version (N \le 0) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q-Lorentz–Wasserstein distance as well as the characterization of q-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case. [ABSTRACT FROM AUTHOR]

Published
2024
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27. Reduced Order Modelling and Uncertainty Propagation Applied to Water Distribution Networks

Author

Braun, Mathias, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), and Docteur spécialité mathématiques appliquées et calcul scientifique, université de Bordeaux

Subjects
[SDE]Environmental Sciences, thesis, thèse
Abstract

Water distribution systems are large, spatially distributed infrastructures that ensure the distribution of potable water of sufficient quantity and quality. Mathematical models of these systems are characterized by a large number of state variables and parameter. Two major challenges are given by the time constraints for the solution and the uncertain character of the model parameters. The main objectives of this thesis are thus the investigation of projection based reduced order modelling techniques for the time efficient solution of the hydraulic system as well as the spectral propagation of parameter uncertainties for the improved quantification of uncertainties. The thesis gives an overview of the mathematical methods that are being used. This is followed by the definition and discussion of the hydraulic network model, for which a new method for the derivation of the sensitivities is presented based on the adjoint method. The specific objectives for the development of reduced order models are the application of projection based methods, the development of more efficient adaptive sampling strategies and the use of hyper-reduction methods for the fast evaluation of non-linear residual terms. For the propagation of uncertainties spectral methods are introduced to the hydraulic model and an intrusive hydraulic model is formulated. With the objective of a more efficient analysis of the parameter uncertainties, the spectral propagation is then evaluated on the basis of the reduced model. The results show that projection based reduced order models give a considerable benefit with respect to the computational effort. While the use of adaptive sampling resulted in a more efficient use of pre-calculated system states, the use of hyper-reduction methods could not improve the computational burden and has to be explored further. The propagation of the parameter uncertainties on the basis of the spectral methods is shown to be comparable to Monte Carlo simulations in accuracy, while significantly reducing the computational effort.; Les réseaux de distribution d'eau consistent en de grandes infrastructures réparties dans l'espace qui assurent la distribution d'eau potable en quantité et en qualité suffisantes. Les modèles mathématiques de ces systèmes sont caractérisés par un grand nombre de variables d'état et de paramètres dont la plupart sont incertains. Les temps de calcul peuvent s'avérer conséquents pour les réseaux de taille importante et la propagation d'incertitude par des méthodes de Monte Carlo. Par conséquent, les deux principaux objectifs de cette thèse sont l'étude des techniques de modélisation à ordre réduit par projection ainsi que la propagation spectrale des incertitudes des paramètres. La thèse donne tout d'abord un aperçu des méthodes mathématiques utilisées. Ensuite, les équations permanentes des réseaux hydrauliques sont présentées et une nouvelle méthode de calcul des sensibilités est dérivée sur la base de la méthode adjointe. Les objectifs spécifiques du développement de modèles d'ordre réduit sont l'application de méthodes basées sur la projection, le développement de stratégies d'échantillonnage adaptatives plus efficaces et l'utilisation de méthodes d'hyper-réduction pour l'évaluation rapide des termes résiduels non linéaires. Pour la propagation des incertitudes, des méthodes spectrales sont introduites dans le modèle hydraulique et un modèle hydraulique intrusif est formulé. Dans le but d'une analyse plus efficace des incertitudes des paramètres, la propagation spectrale est ensuite évaluée sur la base du modèle réduit. Les résultats montrent que les modèles d'ordre réduit basés sur des projections offrent un avantage considérable par rapport à l'effort de calcul. Bien que l'utilisation de l'échantillonnage adaptatif permette une utilisation plus efficace des états système pré-calculés, l'utilisation de méthodes d'hyper-réduction n'a pas permis d'améliorer la charge de calcul. La propagation des incertitudes des paramètres sur la base des méthodes spectrales est comparable aux simulations de Monte Carlo en termes de précision, tout en réduisant considérablement l'effort de calcul.

Published
2019

29. Augmented Resilience of Water Distribution Systems following Severe Abnormal Events

Author

Piller, Olivier, Sedehizade, Fereshte, Bernard, Thomas, Braun, Mathias, CHEIFETZ, Nicolas, Deuerlein, Jochen, Wagner, Martin, Lapébie, Emmanuel, Trick, Iris, Weber, Jean-Marc, Werey, Cathy, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Berliner Wasserbertriebe, Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), Veolia Eau - Ile de France, 3S Consult GmbH, DVGW-Technologiezentrum Wasser (TZW), Deutsche Verein des Gas- und Wasserfaches (DVGW), GRAMAT (DAM/GRAMAT), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Fraunhofer Institute for Interfacial Engineering and Biotechnology (Fraunhofer IGB), Ville et Eurométropole de Strasbourg - Service de l'Eau, Gestion Territoriale de l'Eau et de l'environnement (UMR GESTE), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), ANR, ANR-14-PICS-0003,ResiWater,Outils, modèles et réseaux sécurisés et innovants de capteurs pour une résilience augmentée des infrastructures liées à l'eau(2014), aucun, Berliner Wasserbetriebe, Fraunhofer-Institut für Optronik, Systemtechnik und Bildauswertung (IOSB), Veolia Environnement (FRANCE), Service de l'eau, Eurométropole, BWB BERLINER WASSER BETRIEBE BERLIN DEU, Partenaires IRSTEA, Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES), Piller, Olivier, and APPEL FRANCO-ALLEMAND SUR LA PROTECTION DES INFRASTRUCTURES CRITIQUES - Outils, modèles et réseaux sécurisés et innovants de capteurs pour une résilience augmentée des infrastructures liées à l'eau - - ResiWater2014 - ANR-14-PICS-0003 - PICS - VALID

Subjects
[SPI] Engineering Sciences [physics], [MATH] Mathematics [math], sensors, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], [SPI]Engineering Sciences [physics], [INFO.EIAH] Computer Science [cs]/Technology for Human Learning, [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], Training Simulatorq, [SPI.GCIV.RISQ]Engineering Sciences [physics]/Civil Engineering/Risques, Optimisation, RESEAU DE DISTRIBUTION D'EAU, CAPTEUR, [MATH]Mathematics [math], [SPI.GCIV.RISQ] Engineering Sciences [physics]/Civil Engineering/Risques, water distribution system, ComputingMilieux_MISCELLANEOUS, Resilience, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, [STAT.ML] Statistics [stat]/Machine Learning [stat.ML], SIMULATION NUMERIQUE, numerical simulation, [SDE]Environmental Sciences, [SHS.GESTION]Humanities and Social Sciences/Business administration, [INFO.EIAH]Computer Science [cs]/Technology for Human Learning, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Water Distribution Networks, [SHS.GESTION] Humanities and Social Sciences/Business administration, Simulation
Abstract

[Departement_IRSTEA]Eaux [TR1_IRSTEA]GEUSICCWI 2017 - Computing and Control for the Water Industry , Sheffield, , 05-/09/2017 - 07/09/2017; International audience; The detection of faults and the capacity to return quickly to a normal state after failures and interruption of services are essential for water utilities. The ResiWater project aims to improve the following three aspects for better network security and enhanced resilience: prevention, surveillance and response of water distribution systems facing the major threats. In this paper, we present the ResiWater project main results after two years. A focus is made on the resilience framework and the development of high-performance sensors for fast detection of water quality deterioration or system breakdown.

Published
2017

30. CCWI2017: F92 'Spectral Propagation of Parameter Uncertainties in Water Distribution Networks'

Author

Braun, Mathias, Piller, Olivier, Deuerlein, Jochen, and Mortazavi, Iraj

Subjects
90599 Civil Engineering not elsewhere classified, FOS: Civil engineering
Abstract

The hydraulic state of a water distribution network is governed by a large number of uncertain parameters. These parameters may be given by uncertain consumer demand, valves states, the value of pipe diameters or the roughness of the pipes. In practice, the influence of parameter variations is important in the decision-making process of water utilities, which emphasizes the need for proper quantification of the resulting uncertainties in head and flow. The central step in uncertainty quantification is the propagation of uncertainties through the system. In the past, the influence of parameter uncertainties on the system state has been studied using perturbation methods, stochastic collocation and interval state estimation. This paper presents the results of an alternative spectral approach that has been examined as part of the French-German research project ResiWater. The generalized Polynomial Chaos Expansion is applied to a small looped water distribution network with multiple uncertain input parameters using a non-intrusive projection method. These results are compared to the Monte Carlo simulation as representative of stochastic collocation methods. It is demonstrated that the Polynomial Chaos Expansion is capable to capture a high order of non-linear effects like the Monte Carlo simulation for a considerably lower computational effort.

Published
2017
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31. ResiWater deliverable report D4.1: Stable and reliable pressure dependent modeling

Author

Piller, Olivier, Deuerlein, Jochen, Braun, Mathias, Parisini, Fabrizio, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), 3S Consult GmbH, National Recherche Autres (appel d'offres international), irstea, and Financement bilatéral franco-allemand ANR BMBF

Subjects
[SPI]Engineering Sciences [physics], SOUS-RÉSEAU DÉCONNECTE, [MATH]Mathematics [math], PRIMAL-DUAL, MODÉLISATION CONDUITE PAR LA PRESSION
Abstract

This report describes different methods for modelling of pressurized water supply networks with consideration of pressure dependent demands (PDM: Pressure Driven Modelling). The capability of a hydraulic network simulation software to correctly deal with pressure dependent demands is essential for the development in ResiWater. In case of system failures that might be caused by pipe bursts, natural disasters or malevolent attacks by humans, the pressure is reduced due to decreased capacity of the system. In such case, the common DDM solvers (DDM: Demand Driven Modelling) deliver unrealistic results with negative pressures while maintaining the defined outflows at the side of the customers. PDM modelling fixes this problem by automatically reducing the demand when the pressure drops under the minimum service pressure. The deliverable starts with an overview of existing approaches to PDM. Problems with convergence are well documented in literature. Two different methods for solution are proposed in the main part. The first includes a step size control based on Goldstein criterion. It has been published already in a journal that's why it is not presented in detail in the deliverable. The second approach contains the so-called Content Model formulation of the PDM problem that delivers also statements about existence and uniqueness of the solution. For numerical calculation, an active set method is proposed. The methods have been implemented in Matlab and successfully tested for several test networks with up to 20.000 nodes under strong PDM conditions (almost all nodes are in PDM mode). Small examples exist also in Excel. The approach will be also implemented in Porteau and SIR 3S as part of the reliable hydraulic system solver (task 4.4). Tests were made in SIR 3S for the BWB pilot networks. The deliverable ends with Conclusion.

Published
2017

32. CCWI2017: F96 'Augmented Resilience of Water Distribution Systems following Severe Abnormal Events'

Author

Piller, Olivier, Fereshte Sedehizade, Bernard, Thomas, Braun, Mathias, Cheifetz, Nicolas, Deuerlein, Jochen, Wagner, Martin, Lapébie, Emmanuel, Trick, Iris, Jean-Marc Weber, and Werey, Caty

Subjects
90599 Civil Engineering not elsewhere classified, 6. Clean water, FOS: Civil engineering
Abstract

The detection of faults and the capacity to return quickly to a normal state after failures and interruption of services are essential for water utilities. The ResiWater project aims to improve the following three aspects for better network security and enhanced resilience: prevention, surveillance and response of water distribution systems facing the major threats. In this paper, we present the ResiWater project main results after two years. A focus is made on the resilience framework and the development of high-performance sensors for fast detection of water quality deterioration or system breakdown.

Published
2017
Full Text
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33. ResiWater deliverable report D4.3: Addressing parameter and measurement uncertainties for water supply networks under abnormal operational conditions

Author

Braun, Mathias, Piller, Olivier, Deuerlein, Jochen, Parisini, Fabrizio, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), 3S Consult GmbH, National Recherche Autres (appel d'offres international), irstea, and Financement bilatéral franco-allemand ANR BMBF

Subjects
[SPI]Engineering Sciences [physics], PROPAGATION D'INCERTITUDE, CHAOS POLYNOMIAL, [MATH]Mathematics [math], MODÈLE D'ORDRE RÉDUIT
Abstract

The objective of work package 4.3 is the development of a concept for estimation of the parameter uncertainty impacts on the reliability of the simulation results. The major steps of work package 4.3 are: • Survey and classification of parameter/measurement uncertainties (sources, range, importance); • Development of a concept for estimation of the parameter uncertainty impacts on the reliability of the simulation results; • Estimation of confidence intervals for hydraulic and water quality model results. This deliverable describes the treatment and modelling of parameter uncertainties based on the available information, the propagation of these uncertainties using the mathematical model of water distribution networks and the evaluation with respect to the confidence intervals. After that the effectiveness will be illustrated in a number of example applications. The deliverable is structured as follows: First a review on uncertainty quantification applications to water distribution networks is given, then the modelling of parameter uncertainties will be described, followed by an overview of the different options for propagation. The application is demonstrated using two example networks and the results are evaluated. The deliverable closes with the conclusions that could be drawn and the next steps in the development.

Published
2017

36. SMaRT-OnlineWDN deliverable 6.4: Identification of automatic countermeasures

Author

Deuerlein, Jochen, Guth, Nicolai, Montalvo Arango, Idel, Meyer Harries, Lea, Braun, Mathias, Piller, Olivier, 3S Consult GmbH, Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Irstea, and ANR-11-SECU-0006,SMaRT-OnlineWDN,Boîte à outils pour la gestion en temps réel de la sécurité et pour la fiabilité des réseaux de distribution d'eau(2011)

Subjects
Water quality model, [SPI]Engineering Sciences [physics], ComputingMethodologies_PATTERNRECOGNITION, Response time, [SPI.GCIV.RISQ]Engineering Sciences [physics]/Civil Engineering/Risques, Contaminant source, Sensor alarm, Forward tracking, Look ahead, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Isolation
Abstract

An effective and adequate response to a contamination event is one of the most important issues that are addressed by SMaRT-OnlineWDN and can save lives of the population in case of a contamination event. As soon as one of the sensors raises an alarm the online source identification algorithm delivers candidate nodes for the contamination source. The worst case node that explains the current alarms(s) is selected as source for an forward calculation of the spread of contaminant. The results shows how large the affected area is at present and how it will develop in the future. In order to make optimal decisions for countermeasures the future spread of contamination is one of the most important information for the decision maker an therefore the basis for all response actions. For that purpose look ahead simulations are carried out that are based on the assumed source locations as well as the prognosis of the future flow velocities that are driven by the demands in the network. With the so calculated contamination maps it is possible to select the population that must be informed first or the valves that have to be closed in order to isolate the contaminant and prohibit its further spreads through the network.

Published
2015

37. SMaRT-OnlineWDN deliverable 4.3&4: Adaptation of transport equations and CFD Simulation of the transport phenomena and comparison with measured data

Author

Gilbert, Denis, Piller, Olivier, Ung, Hervé, Bernard, Thomas, Braun, Mathias, Korth, Andreas, Nitsche, Reik, Sedehizade, Fereshte, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), DVGW-Technologiezentrum Wasser (TZW), Deutsche Verein des Gas- und Wasserfaches (DVGW), Berliner Wasserbetriebe, ANR, Irstea, and ANR-11-SECU-0006,SMaRT-OnlineWDN,Boîte à outils pour la gestion en temps réel de la sécurité et pour la fiabilité des réseaux de distribution d'eau(2011)

Subjects
Transport modelling, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [SPI]Engineering Sciences [physics], [SPI.GCIV]Engineering Sciences [physics]/Civil Engineering, Imperfect Mixing, Laminar, Dispersion, [MATH]Mathematics [math], CFD, Turbulent, Design of Experiment
Abstract

The main objective of the SMaRT-OnlineWDN project is the development of an online security management toolkit for water distribution networks that is based on sensor measurements of water quality as well as water quantity. Pseudo-real time modelling of water quantity and water quality variables is the cornerstone of the project. Existing transport model tools are not adapted for online modelling and ignore some important phenomena that may be dominant when lookingat the network in greater detail with an observation time of several minutes. The aim of this deliverable is to report investigations by the SMaRT-OnlineWDN partners about processes of contaminant mixing a T-junctions and transport in pipes.

Published
2015

38. SMaRT-OnlineWDN: A Franco-German Project for the Online Security Management of Water Distribution Networks

Author

Bernard, Thomas, Piller, O., Gildbert, D., Braun, Mathias, Kühnert, Christian, Deuerlein, J., Korth, A., Nitsche, R., Sedehizade, F., Weber, J.-M., Maurel, M., Sandraz, A.-C., Werey, C., Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), 3S Consult GmbH, Deutsche Verein des Gas- und Wasserfaches (DVGW), Berliner Wasserbertriebe, Ville et Eurométropole de Strasbourg - Service de l'Eau, Veolia Environnement (FRANCE), Gestion Territoriale de l'Eau et de l'environnement (UMR GESTE), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES), and Irstea Publications, Migration

Subjects
[SPI]Engineering Sciences [physics], [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], [SPI] Engineering Sciences [physics], [SPI.GCIV.RISQ]Engineering Sciences [physics]/Civil Engineering/Risques, [SHS.GESTION]Humanities and Social Sciences/Business administration, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH] Mathematics [math], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math], [SHS.GESTION] Humanities and Social Sciences/Business administration, [SPI.GCIV.RISQ] Engineering Sciences [physics]/Civil Engineering/Risques, [STAT.ML] Statistics [stat]/Machine Learning [stat.ML]
Abstract

International audience; The SMaRT-OnlineWDN project ́s main objective is to develop an early warning system and a decision support toolkit for emergencies and the deliberate contamination of water distribution networks. There are four parts in this Project. One is to find an optimal sensor network as a combination of different sensor types e.g.: quality and hydraulic sensors and an optimal sensor placement. Another one is developing an online running model, which is automatically calibrated to the measured sensor data and simulates the current hydraulic status of the WDN. The third part is developing a model with more accurate transport and mixing mechanisms. Once a contamination event has been detected, the source identification tool will locate the contamination source by using the inverse transport model. Then, the spread of contamination is found out by simulating the future hydraulic status and using a forecast on water demand. For validating the models, the methods and the data acquisition, experiments are done at a real world test track. The fourth part of the project is a Risk analysis by studying the likelihood and impacts of deliberate contamination and how to behave in this case and also how to inform customers. The decision support toolkit is completed by combining all parts together.

Published
2014

39. A Gradient-type Method for Real-Time State Estimation of Water Distribution Networks

Author

Piller, Olivier, Montalvo Arango, Idel, Deuerlein, Jochen, Gilbert, Denis, Braun, Mathias, Ung, Hervé, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), 3S Consult GmbH, Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), and Irstea Publications, Migration

Subjects
ESTIMATION D'ÉTAT, MOINDRES CARRÉS, [SPI]Engineering Sciences [physics], [SPI] Engineering Sciences [physics], LEVENBERG MARQUARDT ALGORITHME, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH] Mathematics [math], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math]
Abstract

International audience; Drinking water distribution networks risk exposure to malicious or accidental contamination. Several levels of responses are conceivable. One of them consists of installing a sensor network to monitor the system in real time. Once a contamination has been detected, it is also important to take appropriate counter-measures. The SMaRT-OnlineWDN project relies on modeling to predict both the hydraulics and water quality. An online model use makes identification of the contaminant source and simulation of the contaminated area possible. The objective of this paper is to present SMaRT-OnlineWDN experience and research results for hydraulic state estimations with sampling frequency of a few minutes. A least squares problem with bound constraints is formulated to adjust the demand class coefficient to best fit the observed values at a given time. The criterion is a Huber function to limit the influence of outliers. A Tikhonov regularization is introduced for consideration of prior information in the parameter vector. Then the Levenberg-Marquardt algorithm is applied that uses derivative information for limiting the number of iterations. Confidence intervals for the state prediction are also given. The results are presented and discussed on real networks in France and Germany.

Published
2014

40. SMaRT-OnlineWDN D4.2: Investigation about the processes of the specified phenomena

Author

Gilbert, Denis, Piller, Olivier, Ung, Hervé, Bernard, Thomas, Braun, Mathias, Korth, Andreas, Nitsche, Reik, Sedehizade, Fereshte, Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), DVGW-Technologiezentrum Wasser (TZW), Deutsche Verein des Gas- und Wasserfaches (DVGW), Berliner Wasserbetriebe, National Recherche (appel d'offres international), irstea, SMaRT-OnlineWDN ANR/BMBF CSOSG Project, deliverable 4.2, WP4 Transport modelling, and ANR-11-SECU-0006,SMaRT-OnlineWDN,Boîte à outils pour la gestion en temps réel de la sécurité et pour la fiabilité des réseaux de distribution d'eau(2011)

Subjects
[SPI]Engineering Sciences [physics], [SPI.GCIV]Engineering Sciences [physics]/Civil Engineering, [MATH]Mathematics [math], [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]
Abstract

The main objective of the SMaRT-OnlineWDN project is the development of an online security management toolkit for water distribution networks that is based on sensor measurements of water quality as well as water quantity. Pseudo-real time modelling of water quantity and water quality variables is the cornerstone of the project. Existing transport model tools are not adapted for online modelling and ignore some important phenomena that may be dominant when looking at the network in greater detail with an observation time of several minutes. The aim of this deliverable is to report investigations by the SMaRT-OnlineWDN partners regarding processes of contaminant mixing at junctions and transport in pipes. Firstly, investigations at the Berliner Wasserbetriebe (BWB) are presented (section 1). The test field is a simple loop with old cast iron pipes representative of old pipes in the BWB network. Chemicals can be injected into the pipe by a pump and three multi-parameter sensors, located at different distances, measure hydraulic and quality parameters (flow, pressure, conductivity pH-value, oxygen,…) during the flow. This network was calibrated for the roughness and the effective diameter of pipes which is reported here. Experiments with salt and its transport under different regimes was also studied. So the transport phenomena like advection, dispersion and absorption can be studied. Next, a statistic and hydraulic analysis of the Tee and cross-junctions is achieved on the two networks in France (Strasbourg CUS) and Vedif network (demand area of Villejuif) ) (section 2). It was found that there are a lot of cases where double tee-junctions are present with a distance inter-tee inferior to 10 diameters which may favour imperfect mixing. A hydraulic analysis was also performed which ensures that every hydraulic regime is well represented. For example, for a double-tee junction with equal 100mm diameter 40% of the Reynolds cases are for laminar flow and 80% are under 10,000. For higher diameters the statistics fall to 20% for laminar flow and 60% under 10,000. The results of this analysis serve to design the new test rig built in Dresden by TZW and supply CFD cases for the numerical simulation. Finally, in section 3, the new test rig at TZW (Dresden) and all the investigations that were necessary to know which product to inject, the injection system and the experimental setup are presented. First investigations at TZW have been performed applying several colour tracers with different densities under laminar and turbulent flow conditions The experiments were conducted in a straight pipe with velocities in a range of 0.004m/s to 0.5 m/s. The main results under laminar flow conditions are: 1) Dispersion is the main process for spreading and mixing, 2) The behaviour (moving up or down) of the tracer depends particularly on the density of the injected liquid, 3) An injected liquid with a higher or lower density than the water moves at the pipe wall with a lower velocity than the water body.

Published
2014

41. SMaRT-OnlineWDN: Online Security Management and Reliability Toolkit for Water Distribution Networks

Author

Piller, Olivier, Bernard, Thomas, Braun, Mathias, Gilbert, Denis, Deuerlein, Jochen, Korth, Andreas, Nitsche, Reik, Maurel, Marie, SANDRAZ, Anne-Claire, Sedehizade, Fereshte, Weber, Jean-Marc, Werey, Cathy, Réseaux épuration et qualité des eaux (UR REBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (Fraunhofer IOSB), Fraunhofer (Fraunhofer-Gesellschaft), 3S Consult GmbH, DVGW-Technologiezentrum Wasser (TZW), Deutsche Verein des Gas- und Wasserfaches (DVGW), Veolia Environnement (FRANCE), Berliner Wasserbetriebe, Ville et Eurométropole de Strasbourg - Service de l'Eau, and École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)

Subjects
[SPI]Engineering Sciences [physics], [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], [SPI.GCIV.RISQ]Engineering Sciences [physics]/Civil Engineering/Risques, [SHS.GESTION]Humanities and Social Sciences/Business administration, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math]
Abstract

International audience; Water distribution Networks (WDNs) are critical infrastructures that are exposed to deliberate or accidental contamination. Until now, no monitoring system is capable of protecting a WDN in real time. In the immediate future water service utilities that are installing water quantity and quality sensors in their networks will be producing a continuous and huge data stream for treating. The main objective of the project SMaRT-OnlineWDN is the development of an online security management toolkit for water distribution networks that is based on sensor measurements of water quality as well as water quantity and online simulation. Its field of application ranges from detection of deliberate contamination, including source identification and decision support for effective countermeasures, to improved operation and control of a WDN under normal and abnormal conditions.

Published
2013

48. Modélisation réduite et propagation d’incertitudes pour les réseaux d’alimentation en eau potable

Author

Braun, Mathias, Angelo Iollo, Iraj Mortazavi, Olivier P. Le Maître [Président], Irène Vignon-Clémentel [Rapporteur], Zoran Kapelan [Rapporteur], Olivier Piller, Jochen Deuerlein, Pierre-Olivier Malaterre, Modeling Enablers for Multi-PHysics and InteractionS (MEMPHIS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Environnement, territoires et infrastructures (UR ETBX), Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Université de Bordeaux, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Iollo, Angelo, Mortazavi, Iraj, Piller, Olivier, Deuerlein, Jochen, Malaterre, Pierre-Olivier, Le Maître, Olivier P., Vignon-Clémentel, Irène, Kapelan, Zoran, and STAR, ABES

Subjects
Reduced order model, Modèle d’ordre réduit, Modèles d’ingénierie, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Polynomial chaos expansion, Méthodes basées sur la projection, Projection based methods, Water distribution networks, Développement de chaos polynomial, Engineering models, Quantification d’incertitude, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, Réseaux de distribution d’eau, Uncertainty quantification
Abstract

Water distribution systems are large, spatially distributed infrastructures that ensure the distribution of potable water of sufficient quantity and quality. Mathematical models of these systems are characterized by a large number of state variables and parameter. Two major challenges are given by the time constraints for the solution and the uncertain character of the model parameters. The main objectives of this thesis are thus the investigation of projection based reduced order modelling techniques for the time efficient solution of the hydraulic system as well as the spectral propagation of parameter uncertainties for the improved quantification of uncertainties. The thesis gives an overview of the mathematical methods that are being used. This is followed by the definition and discussion of the hydraulic network model, for which a new method for the derivation of the sensitivities is presented based on the adjoint method. The specific objectives for the development of reduced order models are the application of projection based methods, the development of more efficient adaptive sampling strategies and the use of hyper-reduction methods for the fast evaluation of non-linear residual terms. For the propagation of uncertainties spectral methods are introduced to the hydraulic model and an intrusive hydraulic model is formulated. With the objective of a more efficient analysis of the parameter uncertainties, the spectral propagation is then evaluated on the basis of the reduced model. The results show that projection based reduced order models give a considerable benefit with respect to the computational effort. While the use of adaptive sampling resulted in a more efficient use of pre-calculated system states, the use of hyper-reduction methods could not improve the computational burden and has to be explored further. The propagation of the parameter uncertainties on the basis of the spectral methods is shown to be comparable to Monte Carlo simulations in accuracy, while significantly reducing the computational effort., Les réseaux de distribution d’eau consistent en de grandes infrastructures réparties dans l’espace qui assurent la distribution d’eau potable en quantité et en qualité suffisantes. Les modèles mathématiques de ces systèmes sont caractérisés par un grand nombre de variables d’état et de paramètres dont la plupart sont incertains. Les temps de calcul peuvent s’avérer conséquents pour les réseaux de taille importante et la propagation d’incertitude par des méthodes de Monte Carlo. Par conséquent, les deux principaux objectifs de cette thèse sont l’étude des techniques de modélisation à ordre réduit par projection ainsi que la propagation spectrale des incertitudes des paramètres. La thèse donne tout d’abord un aperçu des méthodes mathématiques utilisées. Ensuite, les équations permanentes des réseaux hydrauliques sont présentées et une nouvelle méthode de calcul des sensibilités est dérivée sur la base de la méthode adjointe. Les objectifs spécifiques du développement de modèles d’ordre réduit sont l’application de méthodes basées sur la projection, le développement de stratégies d’échantillonnage adaptatives plus efficaces et l’utilisation de méthodes d’hyper-réduction pour l’évaluation rapide des termes résiduels non linéaires. Pour la propagation des incertitudes, des méthodes spectrales sont introduites dans le modèle hydraulique et un modèle hydraulique intrusif est formulé. Dans le but d’une analyse plus efficace des incertitudes des paramètres, la propagation spectrale est ensuite évaluée sur la base du modèle réduit. Les résultats montrent que les modèles d’ordre réduit basés sur des projections offrent un avantage considérable par rapport à l’effort de calcul. Bien que l’utilisation de l’échantillonnage adaptatif permette une utilisation plus efficace des états système pré-calculés, l’utilisation de méthodes d’hyper-réduction n’a pas permis d’améliorer la charge de calcul. La propagation des incertitudes des paramètres sur la base des méthodes spectrales est comparable aux simulations de Monte Carlo en termes de précision, tout en réduisant considérablement l’effort de calcul.

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