A new minimizing-movements scheme for curves of maximal slope

Stefanelli, Ulisse

doi:10.1051/cocv/2022028

Stefanelli, Ulisse

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 59

Résumé

Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on minimizing-movements schemes, most notably on the Euler scheme. We present here an alternative minimizing-movements approach, yielding more regular discretizations, serving as a-posteriori convergence estimator, and allowing for a simple convergence proof.

MR Zbl

DOI : 10.1051/cocv/2022028

Classification : 35K55
Keywords: Curves of maximal slope, minimizing movements, generalized geodesic convexity, nonlinear diffusion, Wasser stein spaces

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     author = {Stefanelli, Ulisse},
     title = {A new minimizing-movements scheme for curves of maximal slope},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {28},
     doi = {10.1051/cocv/2022028},
     mrnumber = {4469490},
     zbl = {07574255},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2022028/}
}

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Stefanelli, Ulisse. A new minimizing-movements scheme for curves of maximal slope. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 59. doi: 10.1051/cocv/2022028

Bibliographie
Cité par

[1] M. Agueh, Asymptotic behavior for doubly degenerate parabolic equations. C. R. Math. Acad. Sci. Paris 337 (2003) 331–336. | MR | Zbl | Numdam | DOI

[2] M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Adv. Differ. Equ. 10 (2005) 309–360. | MR | Zbl

[3] A. D. Alexandrov, A theorem on triangles in a metric space and some applications. Trudy Math. Inst. Steklov 38 (1951) 5–23. | MR | Zbl

[4] F. Alvarez, H. Attouch, J. Bolte and P. Redont, A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. J. Math. Pures Appl. (9) 81 (2002) 747–779. | MR | Zbl | DOI

[5] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995) 191–246. | MR | Zbl

[6] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, second ed., Birkhäuser Verlag, Basel (2008). | MR | Zbl

[7] E. Asplund, Averaged norms. Israel J. Math. 5 (1967) 227–233. | MR | Zbl | DOI

[8] E. Asplund, Topics in the theory of convex functions. In Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968). Oderisi, Gubbio (1969), pp. 1–33. | MR | Zbl

[9] H. Attouch and A. Cabot. Asymptotic stabilization of inertial gradient dynamics with time dependent viscosity. J. Differ. Equ. 263 (2017) 5412–5458. | MR | Zbl | DOI

[10] H. Attouch, Z. Chbani, J. Peypouquet and P. Redont. Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. Ser. B 168 (2018) 123–175. | MR | Zbl | DOI

[11] H. Attouch, J. Peypouquet and P. Redont. Fast convex optimization via inertial dynamics with Hessian driven damping. J. Differ. Equ. 261 (2016) 5734–5783. | MR | Zbl | DOI

[12] I. Ayadi and G. Turinici, Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent. Preprint (2020). | arXiv

[13] A. Bacho, E. Emmrich and A. Mielke, An existence result and evolutionary $Γ$ -convergence for perturbed gradient systems. J. Evol. Equ. 19 (2019) 479–522. | MR | Zbl | DOI

[14] H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). | MR | Zbl

[15] H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces. Second edition. CMS Books in Mathematics/Ouvrages de Mathéematiques de la SMC. Springer, Cham (2017). | MR | Zbl

[16] D. P. Bertsekas, Convex optimization algorithms. Athena Scientific, Belmont, MA (2015). | MR | Zbl

[17] R. I. Boţ, E. R. Csetnek and C. S. László, Approaching nonsmooth nonconvex minimization through second order proximalgradient dynamical systems. J. Evol. Equ. 18 (2018) 1291–1318. | MR | Zbl | DOI

[18] R. I. Boţ and E. R. Csetnek, A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities. Optimization 68 (2019) 1265–1277. | MR | Zbl | DOI

[19] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Number 5 in North Holland Math. Studies. North-Holland, Amsterdam (1973). | MR | Zbl

[20] A. Cabot, H. Engler and S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation. Trans. Am. Math. Soc. 361 (2009) 5983–6017. | MR | Zbl | DOI

[21] 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. Iberoamericana 19 (2003) 971–1018. | MR | Zbl | DOI

[22] J. A. Carrillo, R. J. Mccann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179 (2006) 217–263. | MR | Zbl | DOI

[23] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428–517. | MR | Zbl | DOI

[24] P. Clément and W. Desch, Some remarks on the equivalence between metric formulations of gradient flows. Boll. Unione Mat. Ital. (9) 3 (2010) 583–588. | MR | Zbl

[25] P. Clément and W. Desch, A Crandall-Liggett approach to gradient flows in metric spaces. J. Abstr. Differ. Equ. Appl. 1 (2010) 46–60. | MR | Zbl

[26] P. Clément and J. Maas, A Trotter product formula for gradient flows in metric spaces. J. Evol. Equ. 11 (2011) 405–427. | MR | Zbl | DOI

[27] E. De Giorgi, New problems on minimizing movements. In: Boundary value problems for PDE and applications, edited by C. Baiocchi and J.-L. Lions. Masson, Paris (1993), pp. 81–98. | MR | Zbl

[28] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68 (1980) 180–187. | MR | Zbl

[29] M. H. Duong, M. A. Peletier and J. Zimmer, GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles. Nonlinearity 26 (2013) 2951–2971. | MR | Zbl | DOI

[30] F. Fleißner, $Γ$ -convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach. ESAIM: COCV 25 (2019) Paper No. 28, 29 pp. | MR | Zbl | Numdam

[31] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, second edition. Springer, Berlin (2006). | MR | Zbl

[32] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. | MR | Zbl | DOI

[33] J. Jost, Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zuärich. Birkhäauser Verlag, Basel (1997). | MR | Zbl

[34] A. Jüngel, U. Stefanelli and L. Trussardi, Two structure-preserving time discretizations for gradient flows. Appl. Math. Optim. 80 (2020) 733–764. | MR | Zbl | DOI

[35] G. Legendre and G. Turinici, Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces. C. R. Math. Acad. Sci. Paris 355 (2017) 345–353. | MR | Zbl | Numdam | DOI

[36] D. Matthes and S. Plazotta, A variational formulation of the BDF2 method for metric gradient flows. ESAIM: Math. Model. Numer. Anal. 53 (2019) 145–172. | MR | Zbl | Numdam | DOI

[37] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV 18 (2012) 36–80. | MR | Zbl | Numdam

[38] M. Muratori and G. Savaré, Gradient flows and evolution variational inequalities in metric spaces. I: Structural properties. J. Funct. Anal. 278 (2020) 108347, 67 pp. | MR | Zbl | DOI

[39] Y. E. Nesterov, A method for solving the convex programming problem with convergence rate $0 (1 / k^{2})$ . Dokl. Akad. Nauk SSSR 269 (1983) 543–547. | MR | Zbl

[40] S. Ohta, Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Am. J. Math. 131 (2009) 475–516. | MR | Zbl | DOI

[41] F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26 (2001) 101–174. | MR | Zbl | DOI

[42] L. Portinale and U. Stefanelli, Penalization via global functionals of optimal-control problems for dissipative evolution. Adv. Math. Sci. Appl. 28 (2019) 425–447. | MR | Zbl

[43] T. Roche, R. Rossi and U. Stefanelli, Stability results for doubly nonlinear differential inclusions by variational convergence. SIAM J. Control Optim. 52 (2014) 1071–1107. | Zbl | MR | DOI

[44] R. T. Rockafellar, Convex analysis. Princeton University Press (1970). | MR | Zbl | DOI

[45] R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008) 97–169. | MR | Zbl | Numdam

[46] R. Rossi, A. Segatti and U. Stefanelli, Global attractors for gradient flows in metric spaces. J. Math. Pures Appl. 95 (2011) 204–244. | MR | Zbl | DOI

[47] C. Ryll-Nardzewski, On Borel measurability of orbits. Fund. Math. 56 (1964) 129–130. | MR | Zbl | DOI

[48] F. Santambrogio, (Euclidean, metric, and Wasserstein) gradient flows: an overview. Bull. Math. Sci. 7 (2017) 87–154. | MR | Zbl | DOI

[49] B. Shi, S. S. Du, M. I. Jordan and W. J. Su, Understanding the acceleration phenomenon via high-resolution differential equations. Math. Program. (2021). | DOI | MR | Zbl

[50] A. Tribuzio, Perturbations of minimizing movements and curves of maximal slope. Netw. Heterog. Media 13 (2018) 423–448. | MR | Zbl | DOI

[51] D. H. Wagner, Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977) 859–903. | MR | Zbl | DOI

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