September  2017, 10(3): 613-641. doi: 10.3934/krm.2017025

Numerical study of a particle method for gradient flows

1. 

Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

2. 

School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

3. 

Mathematics Department, Technion--Israel Institute of Technology, Haifa 32000, Israel

* Corresponding author: J. A. Carrillo

Received  June 2016 Revised  October 2016 Published  December 2016

Fund Project: The first, second and third authors are supported by Engineering and Physical Sciences Research Council grant EP/K008404/1. The first author is also supported by the Royal Society through a Wolfson Research Merit Award. The last author is supported by ISF grant 998/5

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Citation: José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2005. doi: 10.1007/978-3-7643-8722-8. Google Scholar

[2]

L. Ambrosio and G. Savaré, Gradient flows of probability measures, In Handbook of Differential Equations: Evolutionary Equations. North-Holland, 3 (2007), 1-136. doi: 10.1016/S1874-5717(07)80004-1. Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics. Pitman Advanced Publishing Program, 1984. Google Scholar

[4]

J. -P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[5]

J.-D. BenamouG. CarlierQ. Mérigot and E. Oudet, Discretization of functionals involving the Monge-Ampère operator, Numer. Math., 134 (2016), 611-636. doi: 10.1007/s00211-015-0781-y. Google Scholar

[6]

D. BenedettoE. CagliotiJ. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990. doi: 10.1023/A:1023032000560. Google Scholar

[7]

M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012), B559-B583. doi: 10.1137/110853807. Google Scholar

[8]

A. BlanchetV. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar

[9]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies. Elsevier Science, 1973. Google Scholar

[10]

V. Calvez and T. Gallouët, Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., 36 (2016), 1175-1208. doi: 10.3934/dcds.2016.36.1175. Google Scholar

[11]

V. CalvezB. Perthame and M. Sharifi-tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Contemp. Math., 429 (2007), 45-62. doi: 10.1090/conm/429/08229. Google Scholar

[12]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. Google Scholar

[13]

J. A. Carrillo, Y. -P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, In Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation. Springer Vienna, 553 (2014), 1-46. doi: 10.1007/978-3-7091-1785-9_1. Google Scholar

[14]

J. A. CarrilloM. Di Francesco and G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., 135 (2007), 353-363. doi: 10.1090/S0002-9939-06-08594-7. Google Scholar

[15]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar

[16]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009), 4305-4329. doi: 10.1137/080739574. Google Scholar

[17]

J. A. CarrilloF. S. PatacchiniP. Sternberg and G. Wolansky, Convergence of a particle method for diffusive gradient flows in one dimension, SIAM J. Math. Anal., 48 (2016), 3708-3741. doi: 10.1137/16M1077210. Google Scholar

[18]

P. Degond and F. J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Statist. Comput., 11 (1990), 293-310. doi: 10.1137/0911018. Google Scholar

[19]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227. doi: 10.1137/050628015. Google Scholar

[20]

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000. doi: 10.1007/BFb0103945. Google Scholar

[21]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar

[22]

O. Junge, H. Osberger and D. Matthes, A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions, Preprint, arXiv: 1509.07721.Google Scholar

[23]

P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 332 (2001), 369-376. doi: 10.1016/S0764-4442(00)01795-X. Google Scholar

[24]

S. Mas-Gallic, The diffusion velocity method: A deterministic way of moving the nodes for solving diffusion equations, Transp. Theory Stat. Phys., 31 (2002), 595-605. doi: 10.1081/TT-120015516. Google Scholar

[25]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar

[26]

R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals, PhD thesis, Princeton University, 1994. Google Scholar

[27]

A. Mielke, On evolutionary Gamma-convergence for gradient systems, In Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Springer, 3 (2016), 187-249. doi: 10.1007/978-3-319-26883-5_3. Google Scholar

[28]

H. Osberger and D. Matthes, A convergent Lagrangian discretization for a nonlinear fourth order equation, Found. Comput. Math., (2015), 1-54. doi: 10.1007/s10208-015-9284-6. Google Scholar

[29]

H. Osbergers and D. Matthes, Convergence of a fully discrete variational scheme for a thin-film equation, Accepted in Radon Ser. Comput. Appl. Math., (2015). Google Scholar

[30]

H. Osberger and D. Matthes, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726. doi: 10.1051/m2an/2013126. Google Scholar

[31]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. Google Scholar

[32]

G. Russo, A particle method for collisional kinetic equations. Ⅰ. Basic theory and one-dimensional results, J. Comput. Phys., 87 (1990), 270-300. doi: 10.1016/0021-9991(90)90254-X. Google Scholar

[33]

G. Russo, Deterministic diffusion of particles, Comm. Pure Appl. Math., 43 (1990), 697-733. doi: 10.1002/cpa.3160430602. Google Scholar

[34]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046. Google Scholar

[35]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. Google Scholar

[36]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6. Google Scholar

[37]

J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar

[38]

C. Villani, Topics in Optimal Transportation, Graduate studies in mathematics. American Mathematical Society, Providence (R. I. ), 2003. doi: 10.1090/gsm/058. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2005. doi: 10.1007/978-3-7643-8722-8. Google Scholar

[2]

L. Ambrosio and G. Savaré, Gradient flows of probability measures, In Handbook of Differential Equations: Evolutionary Equations. North-Holland, 3 (2007), 1-136. doi: 10.1016/S1874-5717(07)80004-1. Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics. Pitman Advanced Publishing Program, 1984. Google Scholar

[4]

J. -P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[5]

J.-D. BenamouG. CarlierQ. Mérigot and E. Oudet, Discretization of functionals involving the Monge-Ampère operator, Numer. Math., 134 (2016), 611-636. doi: 10.1007/s00211-015-0781-y. Google Scholar

[6]

D. BenedettoE. CagliotiJ. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990. doi: 10.1023/A:1023032000560. Google Scholar

[7]

M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012), B559-B583. doi: 10.1137/110853807. Google Scholar

[8]

A. BlanchetV. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar

[9]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies. Elsevier Science, 1973. Google Scholar

[10]

V. Calvez and T. Gallouët, Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., 36 (2016), 1175-1208. doi: 10.3934/dcds.2016.36.1175. Google Scholar

[11]

V. CalvezB. Perthame and M. Sharifi-tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Contemp. Math., 429 (2007), 45-62. doi: 10.1090/conm/429/08229. Google Scholar

[12]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. Google Scholar

[13]

J. A. Carrillo, Y. -P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, In Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation. Springer Vienna, 553 (2014), 1-46. doi: 10.1007/978-3-7091-1785-9_1. Google Scholar

[14]

J. A. CarrilloM. Di Francesco and G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., 135 (2007), 353-363. doi: 10.1090/S0002-9939-06-08594-7. Google Scholar

[15]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar

[16]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009), 4305-4329. doi: 10.1137/080739574. Google Scholar

[17]

J. A. CarrilloF. S. PatacchiniP. Sternberg and G. Wolansky, Convergence of a particle method for diffusive gradient flows in one dimension, SIAM J. Math. Anal., 48 (2016), 3708-3741. doi: 10.1137/16M1077210. Google Scholar

[18]

P. Degond and F. J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Statist. Comput., 11 (1990), 293-310. doi: 10.1137/0911018. Google Scholar

[19]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227. doi: 10.1137/050628015. Google Scholar

[20]

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000. doi: 10.1007/BFb0103945. Google Scholar

[21]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar

[22]

O. Junge, H. Osberger and D. Matthes, A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions, Preprint, arXiv: 1509.07721.Google Scholar

[23]

P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 332 (2001), 369-376. doi: 10.1016/S0764-4442(00)01795-X. Google Scholar

[24]

S. Mas-Gallic, The diffusion velocity method: A deterministic way of moving the nodes for solving diffusion equations, Transp. Theory Stat. Phys., 31 (2002), 595-605. doi: 10.1081/TT-120015516. Google Scholar

[25]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. Google Scholar

[26]

R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals, PhD thesis, Princeton University, 1994. Google Scholar

[27]

A. Mielke, On evolutionary Gamma-convergence for gradient systems, In Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Springer, 3 (2016), 187-249. doi: 10.1007/978-3-319-26883-5_3. Google Scholar

[28]

H. Osberger and D. Matthes, A convergent Lagrangian discretization for a nonlinear fourth order equation, Found. Comput. Math., (2015), 1-54. doi: 10.1007/s10208-015-9284-6. Google Scholar

[29]

H. Osbergers and D. Matthes, Convergence of a fully discrete variational scheme for a thin-film equation, Accepted in Radon Ser. Comput. Appl. Math., (2015). Google Scholar

[30]

H. Osberger and D. Matthes, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726. doi: 10.1051/m2an/2013126. Google Scholar

[31]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. Google Scholar

[32]

G. Russo, A particle method for collisional kinetic equations. Ⅰ. Basic theory and one-dimensional results, J. Comput. Phys., 87 (1990), 270-300. doi: 10.1016/0021-9991(90)90254-X. Google Scholar

[33]

G. Russo, Deterministic diffusion of particles, Comm. Pure Appl. Math., 43 (1990), 697-733. doi: 10.1002/cpa.3160430602. Google Scholar

[34]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046. Google Scholar

[35]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. Google Scholar

[36]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6. Google Scholar

[37]

J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar

[38]

C. Villani, Topics in Optimal Transportation, Graduate studies in mathematics. American Mathematical Society, Providence (R. I. ), 2003. doi: 10.1090/gsm/058. Google Scholar

Figure 1.  The heat equation
Figure 2.  The porous medium equation with $m=2$
Figure 3.  The linear Fokker-Planck equation with $\Delta t = 10^{-5}$ --Stabilisation in time of the scheme (rate of convergence to the discrete steady state)
Figure 4.  The nonlinear Fokker-Planck equation with $m=2$ and $\Delta t = \frac{0.1}{N^2}$ -Stabilisation in time of the scheme (rate of convergence to the discrete steady state)
Figure 5.  The particles' positions for the two-dimensional heat equation for $N=100$ with $\Delta t = 10^{-4}$
Figure 6.  Accuracy for the two-dimensional heat equation with $\Delta t = 10^{-4}$
Figure 7.  The modified Keller-Segel equation with $\chi = 1.5$ for $N=50$
Figure 8.  The blow-up of the modified Keller-Segel equation with $\chi = 1.5$
Figure 9.  Blow-up formation for the modified Keller-Segel equation with two initial Gaussian bumps
Figure 10.  The modified nonlinear Keller-Segel equation with $\chi = 1.4$ for different choices of $m$, for $N=50$ at $T=4$ with $\Delta t = 10^{-5}$
Figure 11.  Compactly supported potential $W(x) = -c\max(1-|x|,0) + c$ with nonlinear diffusion with $c = 8$ and $m = 3$, for $N=80$ with $\Delta t = 10^{-5}$
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