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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 |
- Abstract
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Under a Creative Commons license
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.
Keywords:
Particle method,
diffusion,
aggregation,
gradient flow,
discrete gradient flow,
JKO scheme.
Mathematics Subject Classification:
Primary: 65M12; Secondary: 35K05.
Citation:
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic and 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. |
| [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. |
| [3] |
H. Attouch,
Variational Convergence for Functions and Operators, Applicable Mathematics. Pitman Advanced Publishing Program, 1984. |
| [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. |
| [5] |
J.-D. Benamou, G. Carlier, Q. 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. |
| [6] |
D. Benedetto, E. Caglioti, J. 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. |
| [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. |
| [8] |
A. Blanchet, V. 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. |
| [9] |
H. Brézis,
Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies. Elsevier Science, 1973. |
| [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. |
| [11] |
V. Calvez, B. 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. |
| [12] |
J. A. Carrillo, A. 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. |
| [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. |
| [14] |
J. A. Carrillo, M. 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. |
| [15] |
J. A. Carrillo, R. 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. |
| [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. |
| [17] |
J. A. Carrillo, F. S. Patacchini, P. 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. |
| [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. |
| [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. |
| [20] |
S. Graf and H. Luschgy,
Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000.
doi: 10.1007/BFb0103945. |
| [21] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [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. |
| [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. |
| [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. |
| [25] |
R. J. McCann,
A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.
doi: 10.1006/aima.1997.1634. |
| [26] |
R. J. McCann,
A Convexity Theory for Interacting Gases and Equilibrium Crystals, PhD thesis, Princeton University, 1994. |
| [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. |
| [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. |
| [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).
|
| [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. |
| [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. |
| [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. |
| [33] |
G. Russo,
Deterministic diffusion of particles, Comm. Pure Appl. Math., 43 (1990), 697-733.
doi: 10.1002/cpa.3160430602. |
| [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. |
| [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. |
| [36] |
C. M. Topaz, A. 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. |
| [37] |
J. L. Vázquez,
The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. |
| [38] |
C. Villani,
Topics in Optimal Transportation, Graduate studies in mathematics. American Mathematical Society, Providence (R. I. ), 2003.
doi: 10.1090/gsm/058. |
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. |
| [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. |
| [3] |
H. Attouch,
Variational Convergence for Functions and Operators, Applicable Mathematics. Pitman Advanced Publishing Program, 1984. |
| [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. |
| [5] |
J.-D. Benamou, G. Carlier, Q. 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. |
| [6] |
D. Benedetto, E. Caglioti, J. 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. |
| [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. |
| [8] |
A. Blanchet, V. 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. |
| [9] |
H. Brézis,
Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies. Elsevier Science, 1973. |
| [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. |
| [11] |
V. Calvez, B. 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. |
| [12] |
J. A. Carrillo, A. 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. |
| [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. |
| [14] |
J. A. Carrillo, M. 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. |
| [15] |
J. A. Carrillo, R. 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. |
| [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. |
| [17] |
J. A. Carrillo, F. S. Patacchini, P. 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. |
| [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. |
| [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. |
| [20] |
S. Graf and H. Luschgy,
Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000.
doi: 10.1007/BFb0103945. |
| [21] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [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. |
| [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. |
| [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. |
| [25] |
R. J. McCann,
A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.
doi: 10.1006/aima.1997.1634. |
| [26] |
R. J. McCann,
A Convexity Theory for Interacting Gases and Equilibrium Crystals, PhD thesis, Princeton University, 1994. |
| [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. |
| [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. |
| [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).
|
| [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. |
| [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. |
| [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. |
| [33] |
G. Russo,
Deterministic diffusion of particles, Comm. Pure Appl. Math., 43 (1990), 697-733.
doi: 10.1002/cpa.3160430602. |
| [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. |
| [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. |
| [36] |
C. M. Topaz, A. 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. |
| [37] |
J. L. Vázquez,
The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. |
| [38] |
C. Villani,
Topics in Optimal Transportation, Graduate studies in mathematics. American Mathematical Society, Providence (R. I. ), 2003.
doi: 10.1090/gsm/058. |
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 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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