Tony LELIEVRE
- Organization: Ecole des Ponts ParisTech
- Research Unit: CERMICS
- Project: Molecular
and Multiscale Simulation
- Phone: 33 (0)1 64 15 35 28
- Fax: 33 (0)1 64 15 35 86
- E-mail :
- Office : B304, Coriolis building
- Mail:
CERMICS - Ecole des Ponts ParisTech
6 et 8 avenue Blaise Pascal
Cité Descartes - Champs sur Marne
77455 Marne la Vallée Cedex 2 (FRANCE)
Version
française
I am currently researcher at CERMICS and professor at the Ecole des Ponts ParisTech.
I am a member of the Molecular
and
Multiscale
Modelling group at CERMICS and of the Matherials project-team at INRIA.
I am the coordinator of the ERC project MSMath.
Open PhD position related to the MSMath ERC project, click here.
Here is my cv (pdf).
I defended my habilitation thesis on 3rd June 2009 (abstract, manuscript). Title: Mathematical and numerical analysis of some models for materials, from the microscopic scale to the macroscopic scale.
During year 2004-2005, I had a post-doc position at the Centre
de Recherches Mathématiques at the University of
Montreal,
in the context of a special year about the
Mathematics of Stochastic and Multiscale Modeling.
I defended my PhD thesis on 21st June 2004 (abstract, manuscript). Title: Multiscale models for viscoelastic fluids. PhD supervisors: Benjamin
Jourdain and Claude
Le Bris.
Research interests
- Micro-macro models for viscoelastic fluids (dumbbell model,
CONNFFESSIT method, variance reduction methods, entropy estimates).
- Free surface flow and magnetohydrodynamics. Application to
the simulation of aluminium electrolysis cells (Arbitrary Lagrangian
Eulerian method, moving contact line problem).
- Molecular simulation (molecular dynamics, constrained dynamics,
adaptive methods, Quantum Monte Carlo methods).
Teaching activities
- Professor at the Ecole des Ponts ParisTech
- Applied mathematics and numerical methods: Cours MOPSI (Modéliser Programmer Simuler).
- Université Pierre et Marie Curie ( M2 Mathématiques et
Applications )
Lectures and videos
- Lectures:
- CEMRACS 2008: Multiscale modelling of complex fluids: a mathematical initiation slides, lecture notes.
- Ecole doctorale ECODOQUI 2008: Monte Carlo methods in molecular dynamics slides, lecture notes (in French).
- Workshop Stress tensor effects on fluid mechanics, China, 2010: Multiscale modelling of complex fluids: a mathematical initiation, slides, lecture notes.
- Cornell University: Free energy calculations, slides.
- IPAM tutorial, 2012: Monte Carlo methods in molecular dynamics part 1 part 2.
- ICMS tutorial, 2014: Numerical methods in molecular dynamics, or what is a metastable stochastic process ? pdf.
- Computational Toolbox Winterschool 2016, Engelberg: Numerical methods in molecular dynamics, part 1 part 2.
Events
- Workshop "Numerical methods in molecular simulation", Bonn, April 7-11th 2008.
- Workshop "Rheology of complex fluids: modeling and numerics", Ecole des Ponts, January 5-9th 2009.
- Workshop "Hybrid simulations of dynamical systems and applications to molecular dynamics", Institut Henri Poincaré, 27 septembre - 30 septembre 2010.
- Minisymposium "Numerical Methods for Molecular Dynamics", ENUMATH 2011 conference, September 5-9th 2011, slides.
- Workshop "Metastability and stochastic processes", Ecole des Ponts, September 21st-23rd 2011.
- CECAM workshop "Free energy calculations: From theory to applications", Ecole des Ponts, June 4th-8th 2012: CECAM webpage, local webpage.
- CEMRACS 2013: Modelling and simulation of complex systems: stochastic and deterministic approaches. Co-organized with Nicolas Champagnat and Anthony Nouy. The proceedings are now available here.
- NASPDE 2013 : Numerical Analysis of Stochastic PDEs, 10 et 11 septembre 2013. Co-organized with Jocelyne Erhel and Erwan Faou.
- Workshop "Numerical methods for high-dimensional problems". Ecole des Ponts, April 14-18th 2014. Co-organized with Virginie Ehrlacher, Yvon Maday and Anthony Nouy.
- Journées Inverse problems for multiscale and stochastic problems, Ecole des Ponts, October 2-3rd 2014. Co-organized with Virginie Ehrlacher, Frédéric Legoll and Karam Sab.
- MoMaS'14 conference. CIRM, November 17-20th. Co-organized with Grégoire Allaire, Clément Cancès, Alexandre Ern and Raphaèle Herbin.
- Free-energy calculations: A mathematical perspective, BIRS, Casa Matemática Oaxaca, Mexico, July 19-24th, 2015. Co-organized with Chris Chipot and Robert Skeel.
- Workshop "COmputational Statistics and MOlecular Simulation", Paris, February 2-5th, 2016. Co-organized with Arnaud Guyader and Gabriel Stoltz.
- Journées EDP/Probas. Co-organized with Florent Malrieu. Next events:
- IHP conference on "Recent developments in numerical methods for model reduction", November 7-10, 2016. Co-organized with Simona Perotto and Gianluigi Rozza.
- IPAM Long Program on "Complex High-Dimensional Energy Landscapes", September 11 - December 15 2017. Co-organized with Cecilia Clementi, Graeme Henkelman, Richard Hennig, Mitchell Luskin, Noa Marom, Petr Plechac and Christof Schuette.
- ICTS program on "Large deviation theory in statistical physics: Recent advances and future challenges", August 14 - October 13 2017. Co-organized with Arvind Ayyer, Frank den Hollander, Abhishek Dhar, Juan P. Garrahan, Christopher Jarzynski, Manjunath Krishnapur, Sanjib Sabhapandit and Hugo Touchette.
Scientific awards
Editorial activity
Books
- (with J-F. Gerbeau and C. Le Bris) Mathematical methods for
the Magnetohydrodynamics of liquid metals, 324 pp., Numerical Mathematics
and Scientific
Computation, Oxford University Press, 2006.
- (with M. Rousset and G. Stoltz) Free energy computations: A mathematical perspective, 472 pp., Imperial College Press, 2010.
Journal articles
- (with B. Jourdain and C. Le Bris) Numerical analysis of
micro-macro
simulations of polymeric fluid flows: a simple case, Mathematical
Models
and Methods in Applied Sciences, 12(9),
1205-1243, (2002).
- (with J-F. Gerbeau and C. Le Bris) Simulations of MHD flows
with
moving
interfaces, Journal of Computational Physics, 184, 163-191, (2003).
- (with J-F. Gerbeau and C. Le Bris) Modelling and simulation
of the
industrial
production of aluminium: the nonlinear approach, Computers and
Fluids, 33, 801-814, (2004).
- Optimal error estimate for the CONNFFESSIT approach in a
simple
case,
Computers and Fluids, 33,
815-820, (2004).
- (with B. Jourdain and C. Le Bris) Existence of solution for a
micro-macro
model of polymeric fluid : the FENE model, Journal of Functional
Analysis, 209, 162-193, (2004).
- (with B. Jourdain and C. Le Bris) On a variance reduction
technique
for micro-macro simulations of polymeric fluids, Journal of
Non-Newtonian
Fluid Mechanics, 122, 91-106,
(2004).
- (with F. Dubois) Efficient pricing of Asian options by the
PDE
approach,
Journal of Computational Finance, 8(2),
55-64, (2005). See also the
program in C, implemented in Premia,
and some first attempts on the subject (ps
in French).
- (with J.P. Croisille, A. Ern et J. Proft) Analysis and
simulation
of
a coupled hyperbolic/parabolic model problem, Journal of Numerical
Mathematics, 13(2), 81-156,
(2005). See
also the reference
solutions
files.
- (with B. Jourdain and C. Le Bris) An elementary argument
regarding
the
long-time behaviour of the solution to a stochastic differential
equation, Annals of Craiova University, Mathematics and Computer
Science
series, 32, 39-47, (2005).
- (with B. Jourdain, C. Le Bris and F. Otto) Long-time asymptotics of a multiscale
model for polymeric fluid flows, Archive for Rational Mechanics
and
Analysis, 181(1),
97-148, (2006).
- (with E. Cancès and B. Jourdain) Quantum Monte Carlo
simulations
of fermions. A mathematical analysis of the fixed-node approximation,
Mathematical
Models
and Methods in Applied Sciences, 16(9),
1403-1440, (2006).
- (with M. Rousset and G. Stoltz) Computation
of free energy differences through nonequilibrium stochastic dynamics:
the reaction coordinate case, Journal of Computational Physics, 222(2), 624-643, (2007).
- (with M. Caffarel, E. Cancès, A. Scemama and G. Stoltz) An efficient sampling algorithm for
Variational Monte Carlo, Journal of Chemical Physics, 125,
114105 (2006).
- (with G. Ciccotti and E. Vanden-Eijnden) Projection of diffusions on submanifolds:
Application to mean force computation, Communications on Pure
and
Applied Mathematics, 61(3), 371-408, (2008).
- (with M. El Makrini and B. Jourdain) Diffusion Monte Carlo method: numerical
analysis in a simple case, Mathematical Modelling and Numerical
Analysis, 41(2), 189-213,
(2007).
- (with M. Rousset and G. Stoltz) Computation
of free energy profiles with parallel adaptive dynamics, Journal
of Chemical Physics 126,
134111 (2007).
- (with A. Ern) Adaptive models
for polymeric fluid flow simulation, C. R. Acad. Sci. Paris, Ser. I, 344(7), 473-476 (2007).
- (with D. Hu) New entropy
estimates for the Oldroyd-B model, and related models,
Communications in Mathematical Sciences 5(4), 909-916 (2007).
- (with C. Le Bris and E. Vanden-Eijnden) Analysis of some discretization schemes
for constrained Stochastic Differential Equations,
C.
R. Acad.
Sci. Paris, Ser. I, 346(7-8),
471-476 (2008).
- (with M. Rousset and G. Stoltz) Long-time
convergence of an Adaptive
Biasing Force method, Nonlinearity, 21, 1155-1181 (2008).
- (with E. Faou) Conservative stochastic differential
equations: Mathematical and numerical analysis, Mathematics of Computation, 78, 2047-2074 (2009).
- (with J-F. Gerbeau) Generalized
Navier Boundary Condition and Geometric Conservation Law for surface
tension, Computer Methods in Applied Mechanics and Engineering, 198(5-8), 644-656, (2009).
- A general two-scale criteria for logarithmic Sobolev inequalities, Journal of Functional Analysis, 256(7), 2211-2221 (2009).
- (with S. Boyaval and C. Mangoubi) Free-energy-dissipative schemes for the
Oldroyd-B model, Mathematical Modelling and Numerical
Analysis, 43, 523-561, (2009).
- (with C. Le Bris and Y. Maday) Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constructive Approximation, 30(3), 621-651, (2009).
- (with S. Boyaval) A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm, Communications
in Mathematical Sciences, 8(3), 735-762, (2010).
- (with B. Jourdain and R. Roux) Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process, Mathematical Modelling and
Numerical Analysis, 44, 831-865, (2010).
- (with C. Chipot and K. Minoukadeh) Potential of mean force calculations: a multiple-walker adaptive biasing force approach, Journal of Chemical Theory and Computation, 6(4), 1008-1017, (2010).
- (with S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen and A.T. Patera) Reduced basis techniques for stochastic problems, Archives of Computational Methods in Engineering, 17(4), 435-454, (2010).
- (with
B. Dickson, F. Legoll, G. Stoltz and P. Fleurat-Lessard) Free energy
calculations: An efficient adaptive biasing potential method, Journal of Physical Chemistry B, 114, 5823-5830, (2010).
- (with X. Blanc, F. Legoll and C. Le Bris) Beyond multiscale and multiphysics: Multimaths for model coupling, Networks and Heterogeneous Media, 5(3), 423-460, (2010).
- (with V. Legat and G. Samaey) A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells, Computers and Fluids, 43, 119-133, (2011).
- (with F. Legoll) Effective dynamics using conditional expectations, Nonlinearity, 23, 2131-2163, (2010).
- (with F. Cérou, A. Guyader and D. Pommier) A multiple replica approach to simulate reactive trajectories, Journal of Chemical Physics 134, 054108, (2011).
- (with K. Minoukadeh) Long-time convergence of an Adaptive Biasing Force method : the bi-channel case, Archive for Rational
Mechanics and Analysis, 202(1), 1-34, (2011).
- (with E. Cancès and V. Ehrlacher) Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, Mathematical
Models
and Methods in Applied Sciences, 21(12), 2433-2467, (2011).
- (with N. Chopin and G. Stoltz) Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors, Statistics and Computing, 22(4), 897-916, (2012).
- (with C. Chipot) Enhanced sampling of multidimensional free-energy landscapes using adaptive biasing forces, SIAM Journal of Applied Mathematics, 71(5), 1673-1695, (2011).
- (with A. Ern and R. Joubaud) Numerical study of a thin liquid film flowing down an inclined wavy plane, Physica D, 240(21), 1714-1723, (2011).
- (with M. Rousset and G. Stoltz) Langevin dynamics with constraints and computation of free energy differences, Mathematics of Computation, 81(280), 2071-2125, (2012).
- (with L. He and C. Le Bris) Periodic long-time behaviour for an approximate model of nematic polymers, Kinetic and Related Models, 5(2), 357-382, (2012).
- (with A. Ern and R. Joubaud) Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes, Nonlinearity 25, 1635-1652, (2012).
- (with C. Le Bris, M. Luskin and D. Perez) A mathematical formalization of the parallel replica dynamics, Monte Carlo Methods and Applications, 18(2), 119–146, (2012).
- (with D. Benoit, L. He and C. Le Bris) Mathematical analysis of a one-dimensional model for an aging fluid, Mathematical
Models
and Methods in Applied Sciences, 23(9), 1561-1602, (2013).
- (with M. Dobson, F. Legoll and G. Stoltz) Derivation of Langevin Dynamics in a nonzero Background Flow Field, Mathematical Modelling and
Numerical Analysis, 47(6), 1583-1626, (2013).
- (with F. Legoll and G. Samaey) A micro-macro parareal algorithm:
application to singularly perturbed ordinary differential equations, SIAM Journal on Scientific Computing, 35(4), A1951–A1986, (2013).
- (with F. Cérou, A. Guyader and F. Malrieu) On the length of one-dimensional reactive paths, ALEA Latin American Journal of Probability and Mathematical Statistics, 10(2), 359-389, (2013).
- (with F. Nier and G. Pavliotis) Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, Journal of Statistical Physics, 152(2), 237-274, (2013).
- (with F. Casenave and A. Ern) Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method, Mathematical Modelling and
Numerical Analysis, 48(1), 207-229, (2014).
- (with B. Jourdain and B. Miasojedow) Optimal scaling for the transient phase of Metropolis Hastings algorithms: the longtime behavior, Bernoulli, 20(4), 1930-1978, (2014).
- (with G. Fort, B. Jourdain, E. Kuhn and G. Stoltz) Convergence of the Wang-Landau algorithm, Mathematics of Computation, 84(295), 2297-2327, (2015).
- (avec G. Fort, B. Jourdain, E. Kuhn et G. Stoltz) Efficiency of the Wang-Landau algorithm: a simple test case, AMRX, 2, 275-311, (2014).
- (with M. Athènes, L. Cao, M.-C. Marinica and G. Stoltz) Using Bayes formula to average the biasing forces and recover free energies in adaptive Monte Carlo simulations, Journal of Chemical Physics, 140, 104108, (2014).
- (with D. Aristoff) Mathematical analysis of Temperature Accelerated Dynamics, SIAM Multiscale Modeling and Simulation, 12(1), 290–317, (2014).
- (with B. Jourdain and B. Miasojedow) Optimal scaling for the transient phase of the random walk Metropolis algorithm: the mean-field limit, Annals of Applied Probability, 25(4), 2263-2300, (2015).
- (with D. Benoit and C. Le Bris) Analysis and macroscopic limit of a one-dimensional model for aging fluids, SIAM Multiscale Modeling and Simulation, 12(3), 1335–1378, (2014).
- (with E. Cancès and V. Ehrlacher) Greedy algorithms for high-dimensional eigenvalue problems, Constructive Approximation, 40(3), 387-423, (2014).
- (with F. Casenave and A. Ern) A nonintrusive Reduced Basis Method applied to aeroacoustic simulations, Advances in Computational Mathematics, 1-26, (2014).
- (with D. Aristoff and G. Simpson) The parallel replica method for simulating long trajectories of Markov chains, AMRX, 2, 332-352, (2014).
- (with J. Comer, J.C. Gumbart, J. Hénin, A. Pohorille and C. Chipot), The adaptive biasing force method: everything you always wanted to know, but were afraid to ask, The Journal of Physical Chemistry B, 119(3), 1129-1151, (2015).
- (with C.E. Bréhier and M. Rousset) Analysis of Adaptive Multilevel Splitting algorithms in an idealized case, ESAIM P&S, 19, 361 - 394, (2015).
- (with F. Nier) Low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain, Analysis & PDE, 8(3), 561-628, (2015).
- (with A. Binder and G. Simpson) A Generalized Parallel Replica Dynamics, Journal of Computational Physics, 284, 595–616, (2015).
- (with H. Alrachid) Long-time convergence of an adaptive biasing force method: Variance reduction by Helmholtz projection, SMAI Journal of Computational Mathematics, 1, 55-82, (2015).
- (with F. Casenave and A. Ern) Variants of the Empirical Interpolation Method: symmetric formulation, choice of norms and rectangular extension, Advances in Computational Mathematics, 41(5), 961-986, (2015).
- (with H. Alrachid and R. Talhouk) Local and global solution for a nonlocal Fokker-Planck equation related to the adaptive biasing force process, Journal of Differential Equations, 260, 7032-7058, (2016).
- (with G. Fort, B. Jourdain and G. Stoltz) Self-Healing Umbrella Sampling: Convergence and efficiency, , Statistics and computing, to appear, (2016).
- (with A.B. Duncan and G.A. Pavliotis) Variance reduction using nonreversible Langevin samplers, Journal of Statistical Physics, to appear, (2016).
- (with C.-E. Bréhier, M. Gazeau, L. Goudenège and M. Rousset) Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms, Annals of Applied Probability, to appear, (2016).
- (with C. Mayne, K. Schulten and I. Teo), Adaptive multilevel
splitting method for molecular dynamics calculation of
benzamidine-trypsin dissociation time, Journal of Chemical Theory and
Computation, to appear, (2016).
- (with G. Stoltz), Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, 25, 681-880, (2016).
Research reports / Proceedings papers
- (with J-F. Gerbeau, C. Le Bris and N. Ligonesche) Metal
pad roll instabilities, proceeding of the 2002 TMS Annual
Meeting
and Exhibition, Light Metals, p. 483-487, 2002.
- (with B. Jourdain) Mathematical analysis of a stochastic
differential
equation arising in the micro-macro modelling of polymeric fluids, CERMICS
2002-225 report, Probabilistic Methods in Fluids Proceedings of the
Swansea 2002 Workshop, Eds: I M Davies, N Jacob, A Truman, O Hassan, K
Morgan & N P Weatherill, p. 205-223.
- (with J-F. Gerbeau and C. Le Bris) Numerical simulations of
two-fluids
MHD flows, Fundamental and Applied MHD. Proceedings of the Fifth
international
PAMIR Conference, p. I.101-I.105, 2002.
- (with J-F. Gerbeau and C. Le Bris) Modeling and simulation of
MHD
phenomena
in aluminium reduction cells, Proceedings of the fourth
International
Conference on Electromagnetic Processing of Materials, EPM 2003,
C3-10.4,
p. 57-62, 2003.
- (with B. Jourdain) Convergence of a stochastic particle
approximation
of the stress tensor for the FENE-P model, CERMICS
2004-263 report .
- (with B. Jourdain and C. Le Bris) Coupling PDEs and SDEs: the
illustrative
example of the multiscale simulation of viscoelastic flows,
in
Multiscale Methods in Science and Engineering, B. Engquist, P.
Lötstedt,
O. Runborg, eds., Lect. Notes Comput. Sci. Eng. 44,
Springer,
151-170, (2005).
- (with T. Tomasino, M. Le Hervet and O. Martin) Stability analysis of simplified
electrolysis cells with Mistral, proceeding of the 2006 TMS
Annual
Meeting
and Exhibition, Light Metals, p. 335-340, 2006.
- (with J.-F. Gerbeau, C. Le Bris, A. Orriols and T. Tomasino) Linear versus nonlinear approaches for the
stability analysis of aluminium production cells, Proceedings of
the 2006 ECCOMAS conference.
- (with J.-F. Gerbeau) Variational
formulation of the Generalized Navier Boundary Condition, http://hal.inria.fr/inria-00081751,
in Recent Progress in Scientific computing, SCPDE05 conference proceeding,
Eds: W Liu, M Ng and Z-C Shi, Science Press, Beijing, 2007.
- (with Y. Achdou and O. Bokanowski) Partial differential equations in finance,
CERMICS
2007-363 report, in Encyclopedia of Financial Models, Ed: F.J. Fabozzi, John Wiley and Sons Inc, 2012.
- (with F. Legoll and G. Stoltz) Some
remarks on sampling methods in molecular dynamics, ESAIM Proceedings, 22, 217-233, (2008) pdf.
- (with C. Le Bris) Multiscale
modelling of complex fluids: A mathematical initiation, in Multiscale Modeling and Simulation in Science Series, B.
Engquist, P. Lötstedt, O. Runborg, eds., Lect. Notes Comput. Sci. Eng. 66, Springer, p. 49-138, (2009).
To get the Matlab programs, contact me.
- (with F. Legoll) Some remarks on free energy and coarse-graining, in Numerical Analysis and Multiscale Computations, B.
Engquist, P. Lötstedt, O. Runborg, eds., Lect. Notes Comput. Sci. Eng., 82, Springer, p. 279-329, (2012).
- (with C. Le Bris) Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics, http://arxiv.org/abs/1102.0325, Science China Mathematics, 55(2), p. 353-384, (2012).
- Two mathematical tools to analyze metastable stochastic processes,
in Numerical Mathematics and Advanced Applications 2011, A. Cangiani,
R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley, M.V.
Tretyakov, eds., Springer, p. 791-810, (2013).
- (with E. Cancès and V. Ehrlacher) Greedy algorithms for high-dimensional non-symmetric linear problems, ESAIM Proceedings, 41, 95-131, (2013) pdf.
- (with F. Casenave, A. Ern and G. Sylvand) A nonintrusive method to approximate linear systems with nonlinear parameter dependence, http://arxiv.org/abs/1307.4330 .
- (with J. Infante Acevedo) A non linear approximation method for solving high dimensional partial differential equations: Application in Finance, http://arxiv.org/abs/1309.3731 .
- (with D. Aristoff, C.G. Mayne and I. Teo) Adaptive multilevel splitting in molecular dynamics simulations, ESAIM Proceedings and Surveys, 48, 215-225, (2015) pdf.
- (with I. G. Tejada, L. Brochard, G. Stoltz, F. Legoll and E. Cancès). Combining a reactive potential with a harmonic approximation for molecular dynamics simulation of failure. J. Phys. : Conf. Ser. 574 012041, (2015).
- Accelerated dynamics: Mathematical foundations and algorithmic improvements, Eur. Phys. J. Special Topics 224, 2429-2444, (2015).
- (with R. Assaraf, B. Jourdain and R. Roux) Computation of
sensitivities for the invariant measure of a parameter dependent
diffusion, http://arxiv.org/abs/1509.01348 .
- (with G. Di Gesù, D. Le Peutrec and B. Nectoux) Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach, https://arxiv.org/abs/1605.02643 .
- (with F. Legoll and S. Olla) Pathwise estimates for an effective dynamics, http://arxiv.org/abs/1605.02644 .
Last update: May, 10th, 2016.