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20 results on '"Balanced flow"'

1. Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent

Author

Gabriel Turinici, Imen Ayadi, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
FOS: Computer and information sciences, Computer Science::Machine Learning, Computer Science - Machine Learning, Computer science, neural network, adaptive stochastic gradient, Machine Learning (stat.ML), 010103 numerical & computational mathematics, 02 engineering and technology, 01 natural sciences, Machine Learning (cs.LG), [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Machine Learning, adaptive learning rate, [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], Statistics - Machine Learning, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Adaptive system, stochastic gradient descent, Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, deep learning optimization, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Artificial neural network, SGD, business.industry, deep learning, Function (mathematics), Numerical Analysis (math.NA), Runge–Kutta methods, Stochastic gradient descent, 020201 artificial intelligence & image processing, Artificial intelligence, Minification, Balanced flow, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], neural networks optimization
Abstract

International audience; The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by the numerical schemes used for general evolution equations we introduce a second order stochastic Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled, in an adaptive framework, with a Stochastic Gradient Descent (SGD) to adjust automatically the learning rate of the SGD, without the need of any additional information on the Hessian of the loss functional. The adaptive SGD, called SGD-G2, is successfully tested on standard datasets.

Published
2021

2. Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

Author

Virginie Ehrlacher, Greta Marino, Jan-Frederik Pietschmann, MATHematics for MatERIALS (MATHERIALS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC), Chemnitz University of Technology / Technische Universität Chemnitz, VE acknowledges support from the ANR JCJC project COMODO (ANR-19-CE46-0002) and from the PHC PROCOPE project Number 42632VA. GM and JFP were supported by the DAADvia the PPP Grant No. 57447206, ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Cross-diffusion, Structure (category theory), Type (model theory), 01 natural sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Global existence, Entropy (arrow of time), Mathematics, Cahn-Hilliard, Applied Mathematics, Weak solution, 010102 general mathematics, Degenerate energy levels, Zero (complex analysis), Degenerate Ginzburg-Landau, Extension (predicate logic), Weak solutions, 010101 applied mathematics, 35D30, 35G31, 35G50, 1991Mathematics Subject Classification.35D30, 35G31, 35G50, Balanced flow, Analysis, Analysis of PDEs (math.AP)
Abstract

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment., 33 pages, comments are welcome

Published
2021

3. A discretization algorithm for time-varying composite gradient flow dynamics

Author

Kanat Camlibel, Aneel Tanwani, Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, University of Groningen [Groningen], Centre National de la Recherche Scientifique (CNRS), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, ANR-17-CE40-0019,ConVan,Contrôle des Systèmes Interconnectés sous Contraintes en Utilisant l'Analyse Variationelle(2017), Systems, Control and Applied Analysis, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)

Subjects
0209 industrial biotechnology, Discretization, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], MathematicsofComputing_NUMERICALANALYSIS, Monotonic function, 02 engineering and technology, Maximal monotone operators, 01 natural sciences, 020901 industrial engineering & automation, Differential inclusion, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, [INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY], Applied mathematics, 0101 mathematics, Subgradient method, Mathematics, Gradient flow dynamics, Convex functions, 010102 general mathematics, Constrained optimization, Control and Systems Engineering, Time-stepping algorithm, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Balanced flow, Gradient descent, Convex function
Abstract

International audience; The problem of minimizing the sum, or composition, of two objective functions is a frequent sight in the field of optimization. In this article, we are interested in studying relations between the discrete-time gradient descent algorithms used for optimization of such functions and their corresponding gradient flow dynamics, when one of the functions is in particular time-dependent. It is seen that the subgradient of the underlying convex function results in differential inclusions with time-varying maximal monotone operator. We describe an algorithm for discretization of such systems which is suitable for numerical implementation. Using appropriate tools from convex and functional analysis, we study the convergence with respect to the size of the sampling interval. As an application, we study how the discretization algorithm relates to gradient descent algorithms used for constrained optimization.

Published
2020

4. TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Author

Andrea Natale, Gabriele Todeschi, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), The work of A. Natale was supported by the European Research Council (ERC project NORIA). Gabriele Todeschi acknowledges that this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 754362., CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Finite volume method, Discretization, Dynamical systems theory, Numerical Analysis (math.NA), 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Wasserstein gradient flows, Energy diminishing scheme, Rate of convergence, FOS: Mathematics, Applied mathematics, Point (geometry), Mathematics - Numerical Analysis, 0101 mathematics, Balanced flow, Interior point method, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such problems which preserve the variational structure and have second order accuracy in space. We propose an interior point method to solve the discrete variational problem, providing an efficient and robust algorithm. We present two applications to test the scheme and show its order of convergence.

Published
2020

5. On Energy Cascades in General Flows: A Lagrangian Application

Author

William K. Dewar, Andrew McC. Hogg, Thierry Penduff, Quentin Jamet, Adekunle-Opeoluwa Ajayi, J. Le Sommer, Institut des Géosciences de l’Environnement (IGE), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement (IRD)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Laboratoire de physique des océans (LPO), Institut de Recherche pour le Développement (IRD)-Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche pour le Développement (IRD)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), University of Michigan System, ANR-18-MPGA-0002,CONTACTS,Turbulence homogène de l'océan pour les simulateurs climatiques(2018), and Australian National University (ANU)

Subjects
Length scale, balanced flow dynamics, 010504 meteorology & atmospheric sciences, Kinetic energy, 01 natural sciences, lcsh:Oceanography, cascades, Environmental Chemistry, Statistical physics, lcsh:GC1-1581, energy transfers, lcsh:Physical geography, ComputingMilieux_MISCELLANEOUS, [SDU.STU.OC]Sciences of the Universe [physics]/Earth Sciences/Oceanography, 0105 earth and related environmental sciences, energetics, Physics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Global and Planetary Change, 010505 oceanography, Turbulence, Dissipation, Cascade, Energy cascade, Dissipative system, General Earth and Planetary Sciences, lcsh:GB3-5030, Energy (signal processing), energy
Abstract

International audience; An important characteristic of geophysically turbulent flows is the transfer of energy between scales. Balanced flows pass energy from smaller to larger scales as part of the well‐known upscale cascade while submesoscale and smaller scale flows can transfer energy eventually to smaller, dissipative scales. Much effort has been put into quantifying these transfers, but a complicating factor in realistic settings is that the underlying flows are often strongly spatially heterogeneous and anisotropic. Furthermore, the flows may be embedded in irregularly shaped domains that can be multiply connected. As a result, straightforward approaches like computing Fourier spatial spectra of nonlinear terms suffer from a number of conceptual issues. In this paper, we develop a method to compute cross‐scale energy transfers in general settings, allowing for arbitrary flow structure, anisotropy and inhomogeneity. We employ a Green's function approach to the kinwetic energy equation to relate kinetic energy at a point to its Lagrangian history. A spatial filtering of the resulting equation naturally decomposes kinetic energy into length scale dependent contributions and describes how the transfer of energy between those scales takes place. The method is applied to a doubly periodic simulation of vortex merger, resulting in the demonstration of the expected upscale energy cascade. Somewhat novel results are that the energy transfers are dominated by pressure work, rather than kinetic energy exchange, and dissipation is a noticeable influence on the larger scale energy budgets. We also describe, but do not employ here, a technique for developing filters to use in complex domains.

Published
2020

6. Dynamical aspects of generalized Schrödinger problem via Otto calculus -- A heuristic point of view

Author

Ivan Gentil, Christian Léonard, Luigia Ripani, Équations aux dérivées partielles, analyse (EDPA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Modélisation aléatoire de Paris X ( MODAL'X ), Université Paris Nanterre ( UPN ), ANR-17-CE40-0030,EFI,Entropy, flows, inequalities, ANR-11-IDEX-0007-02/10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon ( 2011 ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, General Mathematics, 010102 general mathematics, Schrödinger problem, Riemannian manifold, 01 natural sciences, Upper and lower bounds, Omega, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols.namesake, Newton equation, Variational inequality, symbols, curvature-dimension conditions, Embedding, Otto calculus, Wasserstein distance, 0101 mathematics, Balanced flow, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Schrödinger's cat, Mathematics - Probability, Mathematics, Probability measure
Abstract

International audience; The defining equation $(\ast):\ \dot \omega_t=-F'(\omega_t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot \omega ^{ \varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t),$ where $\varepsilon>0$, and (ii) by considering the \emph{accelerations} $\ddot \omega ^{ \varepsilon}_t$. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying Schrödinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero. We show heuristically that the solutions satisfy a Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequality under curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

Published
2020

7. Éléments finis hp adaptatifs avec contraction d’erreur garantie et solveurs multi-niveaux inexacts

Author

Daniel, Patrik, Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Sorbonne Université, Martin Vohralík, Alexandre Ern, and STAR, ABES

Subjects
Méthode des éléments finis, Finite element method, Hp adaptability, Estimation d'erreur a posteriori, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Elliptical problem, Réduction d'erreur, Erreur algébrique, Adaptativité hp, A posteriori error estimation, Balanced flow, Flux équilibré, Algebraic error, Error reduction, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Problème elliptique
Abstract

We propose new practical adaptive refinement algorrithms for conforming hp-finite element approximations of elliptic problems. We consider the use of both exact and inexact solevsr within the established framework of adaptive methods consisting of four concatenated modules : SOLVE, ESTIMATE, MARK, REFINE. The strategies are driven by guaranteed equilibrated flux a posteriori error estimators. Namely, for an inexact approximation obtained by an (arbitrary) iterative algebraic solver, the bounds for the total, the algebraic, and the discretization errors are provided. Our hp-refinement criterion hinges on from solving two local residual problems posed on patches of elements around marked vertices selected by a bulk-chasing criterion. They respectively emulate h-refinement and p-refinement. One particular feature of energy error in the next adaptative loop step with respect to the present one. Numerical experiments are presented to turns out to be excellent, with effectivity indices close to the optimal value of one. In practice, we observe asymptomatic exponential convergence rates, in both the exact and inexact algebraic solver settings. Finally, we also provide a theoretical analysis of the proposed strategies., Nous proposons de nouveaux algorithmes de raffinement adaptatif pour l'approximation des problèmes elliptiques par la méthode des éléments finis hp. Nous considérons des solveurs algébriques exacts puis inexacts au sein du cadre générique des méthodes adaptatives consistant en quatre modules concaténés: RESOLUTION, ESTIMATION, MARQUAGE, RAFFINEMENT. Les stratégies reposent sur la construction d'estimateurs d'erreur a posteriori par flux équilibrés. Notamment, pour une approximation inexacte obtenue par un solveur algébrique itératif (arbitraire), nous prouvons une borne sur l'erreur totale ainsi que sur l'erreur algébrique et l'erreur de discrétisation. La structure hiérarchique des espaces d'éléments finis hp est cruciale pour obtenir la borne supérieure sur l'erreur algébrique, ce qui nous permet de formuler des critères d'arrêt précis pour le solveur algébrique. Notre critère de raffinement hp repose sur la résolution de deux problèmes résiduels locaux, posés sur les macro-éléments autour des sommets du maillage qui ont été marqués. Ces derniers sont sélectionnés par un critère de type bulk-chasing. Ceux deux problèmes résiduels imitent l'effet du raffinement h et p. Une caractéristique de notre approche est que nous obtenons une quantité calculable qui donne une borne garantie sur le rapport entre l'erreur d'énergie (inconnue) à la prochaine étape de la boucle adaptative et l'erreur actuelle (i.e. sur le facteur de réduction d'erreur). Des simulations numériques sont présentées afin de valider les stratégies adaptatives. Nous examinons la précision de notre borne sur le facteur de réduction d'erreur qui s'avère être excellente, avec des indices d'efficacité proches de 1. En pratique, nous observons des taux de convergence asymptotiquement exponentiels, aussi bien dans le cadre de la résolution algébrique exacte que dans celui de la résolution inexacte.Enfin, nous menons une analyse théorique des stratégies proposées.

Published
2019

8. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations

Author

Federica Bubba, Luis Almeida, Benoît Perthame, Camille Pouchol, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), and European Project: 740623,H2020 Pilier ERC,ADORA(2017)

Subjects
Statistics and Probability, Discretization, Upwind schemes, Upwind scheme, 01 natural sciences, Maximum principle, Gradient flow, FOS: Mathematics, Applied mathematics, Scharfetter-Gummel method, Mathematics - Numerical Analysis, 0101 mathematics, Keller-Segel system, Mathematics, Mathematical and theoretical biology, Finite volume method, Applied Mathematics, Numerical analysis, Energy dissipation, General Engineering, Numerical Analysis (math.NA), 16. Peace & justice, Computer Science Applications, 010101 applied mathematics, Mathematical biology, 2010 Mathematics Subject Classification. Primary: 35K55, 35Q84, 65M08, 65M22, 92C17, Fokker–Planck equation, Numerical methods, Balanced flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes whose goals are to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach. These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting feature is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotony properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare., pp:1-19

Published
2019

9. A degenerate Cahn‐Hilliard model as constrained Wasserstein gradient flow

Author

Clément Cancès, Flore Nabet, Daniel Matthes, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), International Association for Applied Mathematics and Mechanics, Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe

Subjects
Physics, 010304 chemical physics, 010102 general mathematics, 0103 physical sciences, Mathematical analysis, Degenerate energy levels, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Balanced flow, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2019

10. Gradient stability of high-order BDF methods and some applications

Author

Morgan Pierre, Anass Bouchriti, Nour Eddine Alaa, Laboratoire de Mathématiques et Applications (LMA-Poitiers), and Université de Poitiers-Centre National de la Recherche Scientifique (CNRS)

Subjects
Algebra and Number Theory, Discretization, Applied Mathematics, 010102 general mathematics, 01 natural sciences, Stability (probability), Allen-Cahn equation, Mathematics::Numerical Analysis, 010101 applied mathematics, gradient flow, BDF methods, Order (group theory), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Balanced flow, High order, Lojasiewicz-Simon inequality, Analysis, Allen–Cahn equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

It is well-known that the backward differentation formulae (BDF) of order 1, 2 and 3 are gradient stable. This means that when such a method is used for the time discretization of a gradient flow, the associated discrete dynamical system exhibit properties similar to the continuous case, such as the existence of a Lyapunov functional. By means of a Lojasiewicz-Simon inequality, we prove convergence to equilibrium for the 3-step BDF scheme applied to the Allen-Cahn equation with an analytic nonlinearity. By introducing a notion of quadratic-stability, we also show that the BDF methods of order 4 and 5 are gradient stable, and that the k-step BDF schemes are not gradient stable for k ≥ 7. Some numerical simulations illustrate the theoretical results.

Published
2018

11. GRADIENT FLOWS, SECOND-ORDER GRADIENT SYSTEMS AND CONVEXITY

Author

Tahar Zamène Boulmezaoud, Aris Daniilidis, Philippe Cieutat, Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), MINECO of Spain [MTM2014-59179-C2-1-P], ERDF of EU [MTM2014-59179-C2-1-P], [BASAL AFB-170001], and [FONDECYT 1171854]

Subjects
021103 operations research, Dynamical systems theory, convexity, 010102 general mathematics, Mathematical analysis, 0211 other engineering and technologies, Order (ring theory), 02 engineering and technology, Function (mathematics), dynamical systems, 01 natural sciences, Convexity, Square (algebra), Theoretical Computer Science, gradient flows, Gradient system, second-order gradient equations, High Energy Physics::Experiment, 0101 mathematics, Balanced flow, Connection (algebraic framework), [MATH]Mathematics [math], Software, Mathematics
Abstract

We disclose an interesting connection between the gradient flow of a e(2)-smooth function psi and strongly evanescent orbits of the second-order gradient system defined by the square-norm of del psi under an adequate convexity assumption. As a consequence, we obtain the following surprising result for two C-2, convex, and bounded from below functions psi(1), psi(2) if parallel to del psi(1)parallel to = parallel to del psi(2)parallel to, then psi(1) = psi(2) + k for some k is an element of R.

Published
2018

12. Acceleration of the imaginary time method for spectrally computing the stationary states of Gross-Pitaevskii equations

Author

Xavier Antoine, Christophe Besse, Romain Duboscq, Vittorio Rispoli, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)

Subjects
accelerated gradient, General Physics and Astronomy, Acceleration (differential geometry), 010103 numerical & computational mathematics, Bose-Einstein condensation, 01 natural sciences, law.invention, rotating Gross-Pitaevskii equation, imaginary time formulation, Simple (abstract algebra), law, 0103 physical sciences, stationary states, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, pseudospectral approximation, 010306 general physics, Physics, Condensed Matter::Quantum Gases, Condensed Matter::Other, Mathematical analysis, Dipole, Hardware and Architecture, Minimization algorithm, Balanced flow, Rotation (mathematics), Stationary state, Bose–Einstein condensate, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; The aim of this paper is to propose a simple accelerated spectral gradient flow formulation for solving the Gross-Pitaevskii Equation (GPE) when computing the stationary states of Bose-Einstein Condensates. The new algorithm, based on the recent iPiano minimization algorithm [35], converges three to four times faster than the standard implicit gradient scheme. To support the method, we provide a complete numerical study for 1d-2d-3d GPEs, including rotation and dipolar terms.

Published
2017

13. Incompressible immiscible multiphase flows in porous media: a variational approach

Author

Thomas Gallouët, Léonard Monsaingeon, Clément Cancès, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Département de Mathématiques [Liège], Université de Liège, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-13-JS01-0007,GEOPOR,Approche géométrique pour les écoulements en milieux poreux: théorie et numérique(2013), ANR-12-MONU-0013,ISOTACE,Systemes d'Interactions, Transport Optimal, Applications a la simulation en Economie.(2012), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe

Subjects
Capillary pressure, Motion (geometry), Space (mathematics), 01 natural sciences, constrained par-abolic system, 35A15, Physics::Fluid Dynamics, Wasserstein gradient flows, Mathematics - Analysis of PDEs, 35K65, 35A15, 49K20, 76S05, Phase (matter), Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [SDU.STU.HY]Sciences of the Universe [physics]/Earth Sciences/Hydrology, Mathematics - Optimization and Control, Multiphase porous media flows, 49K20, Mathematics, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 35K65, minimizing movement scheme, 010101 applied mathematics, constrained parabolic system, Optimization and Control (math.OC), Compressibility, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Balanced flow, Porous medium, Analysis, Analysis of PDEs (math.AP), 76S05
Abstract

International audience; We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schemè a la [R. Jordan, D. Kinder-lehrer & F. Otto, SIAM J. Math. Anal, 29(1):1–17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.

Published
2017

14. Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two

Author

Claire Scheid, David Chiron, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media (NACHOS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
Kadomtsev–Petviashvili equation, 01 natural sciences, symbols.namesake, Nonlinear Schr ̈odinger Equation, Saddle point, Gradient flow, Travelling wave, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Nonlinear Schrödinger equation, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Kadomtsev-Petviashvili Equation, Continuation method, Action (physics), 010101 applied mathematics, Nonlinear system, Modeling and Simulation, symbols, Balanced flow, MSC (2010): 35B38, 35C07, 35J20, 35J61, 35Q40, 35Q55, 35J60, Transonic, Constrained minimization, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We investigate numerically the two dimensional travelling waves of the Nonlinear Schrödinger Equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional, that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified (KP-I) asymptotic in the transonic limit, various multiplicity results and ''one dimensional spreading'' phenomena.

Published
2016

15. The gradient flow approach to hydrodynamic limits for the simple exclusion process

Author

Marielle Simon, Max Fathi, Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Particle system, Mathematical optimization, 010102 general mathematics, Structure (category theory), Process (computing), Mechanics, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Simple (abstract algebra), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, Limit (mathematics), 0101 mathematics, Balanced flow, Mathematics
Abstract

We present a new approach to prove the macroscopic hydrodynamic behaviour for interacting particle systems, and as an example we treat the well-known case of the symmetric simple exclusion process (SSEP). More precisely, we characterize any possible limit of its empirical density measures as solutions to the heat equation by passing to the limit in the gradient flow structure of the particle system.

Published
2016

16. Morse–Novikov theory, Heegaard splittings, and closed orbits of gradient flows

Author

Andrei Pajitnov, Hiroshi Matsuda, Hiroshi Goda, UMR 6629 CNRS/Université de Nantes, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Pure mathematics, Algebra and Number Theory, Applied Mathematics, 010102 general mathematics, Morse code, Mathematics::Geometric Topology, 01 natural sciences, law.invention, law, Lefschetz zeta function, 0103 physical sciences, Torsion (algebra), Novikov self-consistency principle, Mathematics::Differential Geometry, 010307 mathematical physics, 0101 mathematics, Balanced flow, Invariant (mathematics), [MATH]Mathematics [math], Mathematics::Symplectic Geometry, Heegaard splitting, Analysis, ComputingMilieux_MISCELLANEOUS, Knot (mathematics), Mathematics
Abstract

The works of Donaldson (2) and Mark (14) make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle- valued Morse map on a 3-manifold. We study these invariants using the Morse- Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements.

Published
2015

17. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

Author

Vincent Calvez, Thomas Gallouët, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Numerical Medicine (NUMED), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Quantitative methods for stochastic models in physics (MEPHYSTO), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe

Subjects
Keller Segel, Logarithm, Particle number, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Lagrangian and Eulerian specification of the flow field, critical mass, Rigidity (electromagnetism), Mathematics - Analysis of PDEs, Deterministic approximation, FOS: Mathematics, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Physics, Particle system, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Functional Analysis (math.FA), 010101 applied mathematics, Mathematics - Functional Analysis, Homogeneous, Balanced flow, Analysis, blow-up, Analysis of PDEs (math.AP)
Abstract

International audience; We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.; Nous étudions un schéma particulaire qui est une approximation discrète et déterministe de l'équation de Keller-Segel en dimension un avec potentiel logarithmique. Le système de particules est obtenu comme le flot gradient de l'énergie libre écrite en coordonnées Lagrangiennes. On se concentre sur la description de l'explosion des particules : en particulier sur le nombre de particules impliquées dans la création de la première singularité. On exhibe des bassins d'attraction pour lesquelles le nombre de particules est critique. On montre un théorème de rigidité faible concernant la dynamique du système rescalé. Ce travail est complété par une analyse détaillée du cas ou seulement trois particules interagissent.

Published
2015

18. Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity

Author

Romain Duboscq, Xavier Antoine, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)

Subjects
Discretization, [PHYS.COND.GAS]Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], Computation, nucleation, 010103 numerical & computational mathematics, 01 natural sciences, law.invention, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], law, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, [PHYS.COND]Physics [physics]/Condensed Matter [cond-mat], 010306 general physics, Mathematics, Condensed Matter::Quantum Gases, stochasticity, Fractional Brownian motion, Condensed Matter::Other, Finite difference, Quantum vortex, dynamics, 16. Peace & justice, condensates: stationary states, Balanced flow, Bose–Einstein condensate, Stationary state, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; The aim of this chapter is first to give an introduction to the derivation of the Gross-Pitaevskii Equations (GPEs) that arise in the modeling of Bose-Einstein Condensates (BECs). In particular, we describe some physical problems related to stationary states, dynamics, multi-components BECs and the possibility of handling stochastic effects into the equation. Next, we explain how to compute the stationary (and ground) states of the GPEs through the imaginary time method (also called Conjugate Normalized Gradient Flow) and finite difference or pseudo-spectral dis-cretization techniques. Examples are provided by using GPELab which is a Mat-lab toolbox dedicated to the numerical solution of GPEs. Finally, we explain how to discretize correctly the time-dependent GPE so that the schemes are physically admissible. We again provide some examples by using GPELab. Furthermore, extensions of the discretization schemes to some classes of stochastic (in time) GPEs are described and analyzed.

Published
2015

19. Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics

Author

Yann Brenier, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), and ANR-12-MONU-0013,ISOTACE,Systemes d'Interactions, Transport Optimal, Applications a la simulation en Economie.(2012)

Subjects
Numerical Analysis, Quadratic assignment problem, Applied Mathematics, Connection (vector bundle), Fluid mechanics, Calculus of Variations, Computational Mathematics, Mathematics - Analysis of PDEs, Mathematics Subject Classification, Modeling and Simulation, Combinatorial Optimization, Dissipative system, FOS: Mathematics, Optimal transport, Applied mathematics, Combinatorial optimization, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], AMS 35Q35, Calculus of variations, Balanced flow, Analysis, Mathematics, Analysis of PDEs (math.AP), PDEs of Fluid Mechanics
Abstract

There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savar\'e (for the heat equation), we provide for the initial value problem a concept of generalized "dissipative" solutions which always exist globally in time and are unique whenever theyare smooth.

Published
2014

20. A parameterization of gravity waves emitted by fronts and jets

Author

A. de la Cámara, François Lott, Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Physics, spontaneous adjustment mechanism, source-oriented gravity waves in climate models, Gravitational wave, Astrophysics::High Energy Astrophysical Phenomena, Flow (psychology), gravity waves from fronts and jets, law.invention, Computational physics, Momentum, Atmosphere, Geophysics, Classical mechanics, Drag, law, [SDU]Sciences of the Universe [physics], Intermittency, parameterization of gravity waves, General Earth and Planetary Sciences, Earth Science, Gravity wave, Stratosphere
Abstract

International audience; Based on the theoretical and experimental facts that gravity waves (GWs) can be spontaneously emitted during the evolution of a near-balanced flow, a stochastic parameterization of GWs linked to fronts and jets is proposed. Although the spontaneous adjustment theory used predicts "exponentially" small GW fields, it is shown that it is sufficient to produce realistic GW drag at mesospheric levels. Off-line tests using reanalyzed meteorological fields are conducted and show that the GWs emitted present a strong annual cycle following that of the sources. Also, the GW momentum fluxes in the lower stratosphere are qualitatively realistic in terms of intermittency. Online tests in a middle atmosphere general circulation model show that the scheme can potentially perform as well as highly tuned existing GW schemes.

Published
2015

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