Open Access
Issue |
ESAIM: M2AN
Volume 57, Number 6, November-December 2023
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Page(s) | 3335 - 3371 | |
DOI | https://doi.org/10.1051/m2an/2023083 | |
Published online | 29 November 2023 |
- C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide. 3rd edition. Springer, Berlin (2006). [Google Scholar]
- H. AlRachid, L. Mones and C. Ortner, Some remarks on preconditioning molecular dynamics. SMAI J. Comput. Math. 4 (2018) 57–80. [CrossRef] [MathSciNet] [Google Scholar]
- C. Andrieu and J. Thoms, A tutorial on adaptive MCMC. Stat. Comput. 18 (2008) 343–373. [Google Scholar]
- L. Angeli, D. Crisan and M. Ottobre, Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods. Split-Step and Tamed Schemes. Preprint arXiv:2303.15463 (2023). [Google Scholar]
- J. Baker, P. Fearnhead, E.B. Fox and C. Nemeth, Control variates for stochastic gradient MCMC. Stat. Comput. 29 (2019) 599–615. [CrossRef] [MathSciNet] [Google Scholar]
- D. Belomestny, L. Iosipoi, E. Moulines, A. Naumov and S. Samsonov, Variance reduction for Markov chains with application to MCMC. Stat. Comput. 30 (2020) 973–997. [CrossRef] [MathSciNet] [Google Scholar]
- A. Beskos and A. Stuart, MCMC methods for sampling function space, in ICIAM 07 – 6th International Congress on Industrial and Applied Mathematics. Eur. Math. Soc., Zürich (2009) 337–364. [Google Scholar]
- A. Beskos, G. Roberts, A. Stuart and J. Voss, MCMC methods for diffusion bridges. Stoch. Dyn. 8 (2008) 319–350. [CrossRef] [Google Scholar]
- R.N. Bhattacharya, On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete 60 (1982) 185–201. [CrossRef] [MathSciNet] [Google Scholar]
- F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation, M2AN Math. Model. Numer. Anal. 44 (2010) 867–884. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- G. Bussi and M. Parrinello, Accurate sampling using Langevin dynamics. Phys. Rev. E 75 (2007) 056707. [CrossRef] [PubMed] [Google Scholar]
- P. Cattiaux, D. Chafaï and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes. ALEA Lat. Am. J Probab. Math. Stat. 9 (2012) 337–382. [MathSciNet] [Google Scholar]
- M. Chak, N. Kantas and G.A. Pavliotis, On the generalised Langevin equation for simulated annealing. SIAM/ASA J. Uncertainty Quantif. 11 (2023) 139–167. [CrossRef] [MathSciNet] [Google Scholar]
- N.S. Chatterji, N. Flammarion, Y.-A. Ma, P.L. Bartlett and M.I. Jordan, On the theory of variance reduction for stochastic gradient Monte Carlo. PMLR 80 (2018) 764–773. [Google Scholar]
- X. Chen, S. Liu, R. Sun and M. Hong, On the convergence of a class of adam-type algorithms for non-convex optimization, in 2019. 7th International Conference on Learning Representations, ICLR 2019. Conference date: 06–05-2019 Through 09–05-2019 (2019). [Google Scholar]
- X. Cheng, N.S. Chatterji, P.L. Bartlett and M.I. Jordan, Underdamped Langevin MCMC: a non-asymptotic analysis, in Proceedings of the 31st Conference On Learning Theory. Vol. 75 of Proceedings of Machine Learning Research, edited by S. Bubeck, V. Perchet and P. Rigollet. 06–09 Jul 2018. PMLR (2018) 300–323. [Google Scholar]
- D. Crisan, P. Dobson and M. Ottobre, Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups. Trans. Am. Math. Soc. 374 (2021) 3289–3330. [CrossRef] [Google Scholar]
- D. Crisan, P. Dobson, B. Goddard, M. Ottobre and I. Souttar, Poisson equations with locally-Lipschitz coefficients and uniform in time averaging for stochastic differential equations via strong exponential stability. Preprint arXiv:2204.02679 (2022). [Google Scholar]
- A.S. Dalalyan and L. Riou-Durand, On sampling from a log-concave density using kinetic Langevin diffusions. Bernoulli 26 (2020) 1956–1988. [CrossRef] [MathSciNet] [Google Scholar]
- B. Delyon and Y. Hu, Simulation of conditioned diffusion and application to parameter estimation. Stochastic Process. Appl. 116 (2006) 1660–1675. [CrossRef] [MathSciNet] [Google Scholar]
- Z. Ding, Q. Li, J. Lu and S.J. Wright, Random coordinate underdamped Langevin Monte Carlo. Preprint arXiv:2010.11366 (2020). [Google Scholar]
- Z. Dong and X. Peng, Malliavin matrix of degenerate SDE and gradient estimate. Electron. J. Probab. 19 (2014) 26. [CrossRef] [Google Scholar]
- A.B. Duncan, T. Lelièvre and G.A. Pavliotis, Variance reduction using nonreversible Langevin samplers. J. Stat. Phys. 163 (2016) 457–491. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- A.B. Duncan, N. Nüsken and G.A. Pavliotis, Using perturbed underdamped Langevin dynamics to efficiently sample from probability distributions. J. Stat. Phys. 169 (2017) 1098–1131. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- A. Durmus and E. Moulines, High-dimensional Bayesian inference via the unadjusted Langevin algorithm. Preprint arXiv:1605.01559 (2018). [Google Scholar]
- A. Durmus, A. Enfroy, É. Moulines and G. Stoltz, Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics. Preprint arXiv:2107.14542 (2021). [Google Scholar]
- J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Comm. Math. Phys. 212 (2000) 105–164. [CrossRef] [MathSciNet] [Google Scholar]
- S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1986). [Google Scholar]
- J. Foster, T. Lyons and H. Oberhauser, The shifted ODE method for underdamped Langevin MCMC. Preprint arXiv:2101.03446 (2021). [Google Scholar]
- E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance. Finan. Stoch. 3 (1999) 391–412. [CrossRef] [Google Scholar]
- A. Friedman, Stochastic differential equations and applications. Vol. 1, in Probability and Mathematical Statistics, Vol. 28. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1975). [Google Scholar]
- E. Ghadimi, H.R. Feyzmahdavian and M. Johansson, Global convergence of the heavy-ball method for convex optimization, in 2015 European Control Conference (ECC). (2015) 310–315. [CrossRef] [Google Scholar]
- A. Guillin and P. Monmarché, Optimal linear drift for the speed of convergence of an hypoelliptic diffusion. Electron. Commun. Probab. 21 (2016) 14. [CrossRef] [Google Scholar]
- A. Guillin and F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality. J. Differ. Equ. 253 (2012) 20–40. [CrossRef] [Google Scholar]
- M. Hairer, A.M. Stuart, J. Voss and P. Wiberg, Analysis of SPDEs arising in path sampling. I. The Gaussian case. Commun. Math. Sci. 3 (2005) 587–603. [CrossRef] [MathSciNet] [Google Scholar]
- M. Hairer, A.M. Stuart and J. Voss, Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab. 17 (2007) 1657–1706. [CrossRef] [MathSciNet] [Google Scholar]
- M. Hairer, A.M. Stuart and J. Voss, Sampling conditioned hypoelliptic diffusions. Ann. Appl. Probab. 21 (2011) 669–698. [CrossRef] [MathSciNet] [Google Scholar]
- Y. He, K. Balasubramanian and M.A. Erdogdu, On the ergodicity, bias and asymptotic normality of randomized midpoint sampling method. Adv. Neural Inf. Process. Syst. 33 (2020) 7366–7376. [Google Scholar]
- B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians. Vol. 1862 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2005). [CrossRef] [Google Scholar]
- L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147–171. [CrossRef] [MathSciNet] [Google Scholar]
- A.M. Horowitz, The second order Langevin equation and numerical simulations. Nucl. Phys. B 280 (1987) 510–522. [CrossRef] [Google Scholar]
- A.M. Horowitz, A generalized guided Monte Carlo algorithm. Phys. Lett. B 268 (1991) 247–252. [CrossRef] [Google Scholar]
- S. Hottovy, A. McDaniel, G. Volpe and J. Wehr, The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Comm. Math. Phys. 336 (2015) 1259–1283. [CrossRef] [MathSciNet] [Google Scholar]
- A. Kavalur, V. Guduguntla and W.K. Kim, Effects of Langevin friction and time steps in the molecular dynamics simulation of nanoindentation. Mol. Simul. 46 (2020) 911–922. [CrossRef] [Google Scholar]
- R. Khasminskii, Stochastic Stability of Differential Equations. Vol. 66 of Stochastic Modelling and Applied Probability, 2nd edition. Springer, Heidelberg (2012). With contributions by G.N. Milstein and M.B. Nevelson. [CrossRef] [Google Scholar]
- T. Komorowski, C. Landim and S. Olla, Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Vol. 345 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2012). [CrossRef] [Google Scholar]
- N.V. Krylov, On Kolmogorov’s equations for finite-dimensional diffusions, in Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998). Vol. 1715 of Lecture Notes in Math. Springer, Berlin (1998) 1–63. [Google Scholar]
- B. Leimkuhler and C. Matthews, Rational construction of stochastic numerical methods for molecular sampling. Appl. Math. Res. Express. AMRX 2013 (2013) 34–56. [Google Scholar]
- B. Leimkuhler and C. Matthews, Robust and efficient configurational molecular sampling via Langevin dynamics. J. Chem. Phys. 138 (2013) 174102. [CrossRef] [PubMed] [Google Scholar]
- B. Leimkuhler, C. Matthews and G. Stoltz, The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36 (2016) 13–79. [Google Scholar]
- T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25 (2016) 681–880. [CrossRef] [MathSciNet] [Google Scholar]
- T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations: A Mathematical Perspective. Imperial College Press, London (2010). [CrossRef] [Google Scholar]
- T. Lelièvre, F. Nier and G.A. Pavliotis, Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Stat. Phys. 152 (2013) 237–274. [CrossRef] [MathSciNet] [Google Scholar]
- P. Monmarché, High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion. Electron. J. Stat. 15 (2021) 4117–4166. [MathSciNet] [Google Scholar]
- P. Monmarché, Almost sure contraction for diffusions on Rd. Application to generalized Langevin diffusions. Stochastic Process. Appl. 161 (2023) 316–349. [CrossRef] [MathSciNet] [Google Scholar]
- W. Mou, Y.-A. Ma, M.J. Wainwright, P.L. Bartlett and M.I. Jordan, High-order Langevin diffusion yields an accelerated MCMC algorithm. J. Mach. Learn. Res. 22 (2021) 41. [Google Scholar]
- C. Nemeth and P. Fearnhead, Stochastic gradient Markov Chain Monte Carlo. J. Am. Stat. Assoc. 116 (2021) 433–450. [CrossRef] [Google Scholar]
- A.B. Owen, Statistically efficient thinning of a Markov chain sampler. J. Comput. Graph. Stat. 26 (2017) 738–744. [CrossRef] [Google Scholar]
- G.A. Pavliotis, Asymptotic analysis of the Green-Kubo formula. IMA J. Appl. Math. 75 (6) 951–967 [Google Scholar]
- G.A. Pavliotis, Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations, Vol. 60 of Texts in Applied Mathematics. Springer, New York (2014) [CrossRef] [Google Scholar]
- B. Polyak, Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4 (1964) 1–17. [CrossRef] [Google Scholar]
- P.E. Protter, Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability. Vol. 21 of Applications of Mathematics (New York), 2nd edition. Springer-Verlag, Berlin (2004). [Google Scholar]
- M. Sachs, B. Leimkuhler and V. Danos, Langevin dynamics with variable coefficients and nonconservative forces: from stationary states to numerical methods. Entropy 19 (2017) 647. [CrossRef] [Google Scholar]
- J.M. Sanz-Serna and K.C. Zygalakis, Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations. J. Mach. Learn. Res. 22 (2021) 37. [Google Scholar]
- A. Scemama, T. Lelièvre, G. Stoltz, E. Cancès and M. Caffarel, An efficient sampling algorithm for variational Monte Carlo. J. Chem. Phys. 125 (2006) 114105. [CrossRef] [PubMed] [Google Scholar]
- R. Shen and Y.T. Lee, The randomized midpoint method for log-concave sampling, in: Advances in Neural Information Processing Systems, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d’ Alché-Buc, E. Fox and R. Garnett. Vol. 32. Curran Associates, Inc. (2019). [Google Scholar]
- R.D. Skeel and C. Hartmann, Choice of damping coefficient in Langevin dynamics. Eur. Phys. J. B 94 (2021) 1–13. [CrossRef] [Google Scholar]
- L.F. South, C.J. Oates, A. Mira and C. Drovandi, Regularised zero-variance control variates for high-dimensional variance reduction. Bayesian Anal. 18 (2023) 865–888. [CrossRef] [MathSciNet] [Google Scholar]
- J. Teichmann, Calculating the Greeks by cubature formulae. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) 647–670. [MathSciNet] [Google Scholar]
- D. Zou and Q. Gu, On the convergence of Hamiltonian Monte Carlo with stochastic gradients. Proceedings of the 38th International Conference on Machine Learning. . Vol. 139 of Proceedings of Machine Learning Research, edited by M. Meila and T. Zhang. PMLR (2021) 13012–13022. [Google Scholar]
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