Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems. (English) Zbl 07593593
Summary: In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.
MSC:
| 65-XX | Numerical analysis |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 65K05 | Numerical mathematical programming methods |
| 90C25 | Convex programming |
Keywords:
Wasserstein gradient flows; degenerate diffusions; variational principle; operator-splitting methods; non-local partial differential equations; optimal transport; entropic regularisationReferences:
| [1] | S. Adams; N. Dirr; M. A. Peletier; J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: A new micro-macro passage, Comm. Math. Phys., 307, 791-815 (2011) · Zbl 1267.60106 · doi:10.1007/s00220-011-1328-4 |
| [2] | D. Adams, M. H. Duong and G. dos Reis, Entropic regularisation of non-gradient systems, to appear, SIAM Journal on Mathematical Analysis. |
| [3] | L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Birkhäuser Verlag, Basel, 2008. · Zbl 1145.35001 |
| [4] | A. Arnold, I. M. Gamba, M. P. Gualdani, S. Mischler, C. Mouhot and C. Sparber, The Wigner-Fokker-Planck equation: Stationary states and large time behavior, Math. Models Methods Appl. Sci., 22 (2012), 1250034, 31 pp. · Zbl 1253.82065 |
| [5] | Y. Benoist; P. Foulon; F. Labourie, Flots d’Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5, 33-74 (1992) · Zbl 0759.58035 · doi:10.2307/2152750 |
| [6] | E. Bernton, Langevin Monte Carlo and JKO splitting, Proceedings of the Conference On Learning Theory, Stockholm, Sweden, 2018, 1777-1798. |
| [7] | T. Bodineau; R. Lefevere, Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats, J. Stat. Phys., 133, 1-27 (2008) · Zbl 1157.82027 · doi:10.1007/s10955-008-9601-4 |
| [8] | M. Bowles; M. Agueh, Weak solutions to a fractional Fokker-Planck equation via splitting and Wasserstein gradient flow, Appl. Math. Lett., 42, 30-35 (2015) · Zbl 1319.35288 · doi:10.1016/j.aml.2014.10.008 |
| [9] | T. Bühler and D. A. Salamon, Functional Analysis, volume 191, American Mathematical Society, Providence, RI, 2018. · Zbl 1408.46001 |
| [10] | E. A. Carlen; W. Gangbo, Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric, Arch. Ration. Mech. Anal., 172, 21-64 (2004) · Zbl 1182.76944 · doi:10.1007/s00205-003-0296-z |
| [11] | E. A. Carlen; J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, Journal of Functional Analysis, 273, 1810-1869 (2017) · Zbl 1386.46057 · doi:10.1016/j.jfa.2017.05.003 |
| [12] | G. Carlier; V. Duval; G. Peyré; B. Schmitzer, Convergence of entropic schemes for optimal transport and gradient flows, SIAM Journal on Mathematical Analysis, 49, 1385-1418 (2017) · Zbl 1365.90197 · doi:10.1137/15M1050264 |
| [13] | G. Carlier; M. Laborde, A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts, Nonlinear Analysis: Theory, Methods & Applications, 150, 1-18 (2017) · Zbl 1361.35077 · doi:10.1016/j.na.2016.10.026 |
| [14] | J. A. Carrillo; J. Soler, On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in \(L^p\) spaces, Math. Methods Appl. Sci., 18, 825-839 (1995) · Zbl 0829.35096 · doi:10.1002/mma.1670181006 |
| [15] | M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, Advances in Neural Information Processing Systems, 26, 2292-2300 (2013) |
| [16] | F. Delarue; S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, Journal of Functional Analysis, 259, 1577-1630 (2010) · Zbl 1223.60037 · doi:10.1016/j.jfa.2010.05.002 |
| [17] | S. Di Marino and L. Chizat, A tumor growth model of Hele-Shaw type as a gradient flow, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 103, 38 pp. · Zbl 1460.92050 |
| [18] | R. J. DiPerna; P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547 (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835 |
| [19] | M. H. Duong, Long time behaviour and particle approximation of a generalised Vlasov dynamic, Nonlinear Anal., 127, 1-16 (2015) · Zbl 1330.35456 · doi:10.1016/j.na.2015.06.018 |
| [20] | M. H. Duong; Y. Lu, An operator splitting scheme for the fractional kinetic Fokker-Planck equation, Discrete Contin. Dyn. Syst., 39, 5707-5727 (2019) · Zbl 1423.49045 · doi:10.3934/dcds.2019250 |
| [21] | M. H. Duong; M. A. Peletier; J. Zimmer, GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles, Nonlinearity, 26, 2951-2971 (2013) · Zbl 1288.60029 · doi:10.1088/0951-7715/26/11/2951 |
| [22] | M. H. Duong; M. A. Peletier; J. Zimmer, Conservative-dissipative approximation schemes for a generalized Kramers equation, Math. Methods Appl. Sci., 37, 2517-2540 (2014) · Zbl 1370.35165 · doi:10.1002/mma.2994 |
| [23] | M. H. Duong; H. M. Tran, Analysis of the mean squared derivative cost function, Mathematical Methods in the Applied Sciences, 40, 5222-5240 (2017) · Zbl 1384.49007 · doi:10.1002/mma.4382 |
| [24] | M. H. Duong; H. M. Tran, On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type, Discrete Contin. Dyn. Syst., 38, 3407-3438 (2018) · Zbl 1393.35074 · doi:10.3934/dcds.2018146 |
| [25] | A. Esposito; F. S. Patacchini; A. Schlichting; D. Slepčev, Nonlocal-interaction equation on graphs: Gradient flow structure and continuum limit, Archive for Rational Mechanics and Analysis, 240, 699-760 (2021) · Zbl 1510.35353 · doi:10.1007/s00205-021-01631-w |
| [26] | R. Glowinski, S. J. Osher and W. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Scientific Computation. Springer, Cham, 2016. · Zbl 1362.65002 |
| [27] | C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl., 250, 333-367 (2000) · Zbl 0971.82031 · doi:10.1006/jmaa.2000.7109 |
| [28] | C. Huang; R. Jordan, Variational formulations for Vlasov-Poisson-Fokker-Planck systems, Math. Methods Appl. Sci., 23, 803-843 (2000) · Zbl 0962.35003 |
| [29] | H. J. Hwang; J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18, 681-691 (2013) · Zbl 1277.35327 · doi:10.3934/dcdsb.2013.18.681 |
| [30] | P.-E. Jabin; Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with \({W}^{-1, \infty}\) kernels, Inventiones Mathematicae, 214, 523-591 (2018) · Zbl 1402.35208 · doi:10.1007/s00222-018-0808-y |
| [31] | R. Jordan; D. Kinderlehrer; F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1-17 (1998) · Zbl 0915.35120 · doi:10.1137/S0036141096303359 |
| [32] | H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7, 284-304 (1940) · Zbl 0061.46405 · doi:10.1016/S0031-8914(40)90098-2 |
| [33] | R. Kupferman, Fractional kinetics in Kac-Zwanzig heat bath models, J. Statist. Phys., 114, 291-326 (2004) · Zbl 1060.82034 · doi:10.1023/B:JOSS.0000003113.22621.f0 |
| [34] | M. Laborde, On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows, Topological Optimization and Optimal Transport, (2017), 17 of Radon Ser. Comput. Appl. Math., 304-332. · Zbl 1379.35159 |
| [35] | S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces, ESAIM Control Optim. Calc. Var., 15, 712-740 (2009) · Zbl 1178.35201 · doi:10.1051/cocv:2008044 |
| [36] | C. Liu, C. Wang and Y. Wang, A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance, Journal of Computational Physics, 436 (2021), Paper No. 110253, 22 pp. · Zbl 07513840 |
| [37] | J. Maas, Gradient flows of the entropy for finite Markov chains, Journal of Functional Analysis, 261, 2250-2292 (2011) · Zbl 1237.60058 · doi:10.1016/j.jfa.2011.06.009 |
| [38] | A. Marcos and A. Soglo, Solutions of a class of degenerate kinetic equations using steepest descent in Wasserstein space, Journal of Mathematics, (2020), Art. ID 7489532, 30 pp. · Zbl 1450.35208 |
| [39] | P. A. Markowich, C. A. Ringhofer and C. Schmeiser., Semiconductor Equations, Springer-Verlag, Vienna, 1990. · Zbl 0765.35001 |
| [40] | D. Matthes; B. Söllner, Discretization of flux-limited gradient flows: \( \gamma \)-convergence and numerical schemes, Mathematics of Computation, 89, 1027-1057 (2020) · Zbl 1437.35423 · doi:10.1090/mcom/3492 |
| [41] | A. Mielke; M. A. Peletier; D. R. M. Renger., On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41, 1293-1327 (2014) · Zbl 1304.35692 · doi:10.1007/s11118-014-9418-5 |
| [42] | H. C. Öttinger, GENERIC: Review of successful applications and a challenge for the future, preprint, 2018, arXiv: 1810.08470. |
| [43] | M. Ottobre; G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24, 1629-1653 (2011) · Zbl 1221.82076 · doi:10.1088/0951-7715/24/5/013 |
| [44] | G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8, 2323-2351 (2015) · Zbl 1335.90068 · doi:10.1137/15M1010087 |
| [45] | G. Peyré; M. Cuturi, Computational optimal transport: With applications to data science, Foundations and Trends in Machine Learning, 11, 355-607 (2019) |
| [46] | L. Rey-Bellet, Open classical systems, Open Quantum Systems, II, Lecture Notes in Math, volume 1881, Springer, Berlin, 2006, 41-78. · Zbl 1188.82038 |
| [47] | H. Risken, The Fokker-Planck Equation, Methods of solution and applications, 1989. · Zbl 0665.60084 |
| [48] | F. Santambrogio, Optimal transport for applied mathematicians, Birkäuser, NY, 55 (2015), 94. · Zbl 1401.49002 |
| [49] | S. Serfaty, Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887-2935. With an appendix by Mitia Duerinckx and Serfaty. · Zbl 1475.35341 |
| [50] | R. Sinkhorn; P. Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific Journal of Mathematics, 21, 343-348 (1967) · Zbl 0152.01403 · doi:10.2140/pjm.1967.21.343 |
| [51] | J. Smoller, Shock Waves and Reaction-Diffusion Equations, \(2^{nd}\) edition, Springer-Verlag, New York, 1994. · Zbl 0807.35002 |
| [52] | C. Villani, Optimal Transport: Old and New, 338. Springer-Verlag, Berlin, 2009. · Zbl 1156.53003 |
| [53] | C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202, 2009 · Zbl 1197.35004 |
| [54] | C. Villani, Topics in Optimal Transportation, volume 58, American Mathematical Soc, 2003. · Zbl 1106.90001 |
| [55] | Y. Yao; A. L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion, Physica D: Nonlinear Phenomena, 260, 77-89 (2013) · Zbl 1286.35050 · doi:10.1016/j.physd.2013.01.009 |
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