A. Arsenev, Existence in the large of a weak solution to the Vlasov system of equations, Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, pp.136-147, 1975.

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Annales de l'IHP Analyse non linéaire, pp.101-118, 1985.
DOI : 10.1016/S0294-1449(16)30405-X

J. Brackbill, On energy and momentum conservation in particle-in-cell plasma simulation, Journal of Computational Physics, vol.317, pp.405-427, 2016.
DOI : 10.1016/j.jcp.2016.04.050

E. Cances, V. Ehrlacher, and T. Lelievre, CONVERGENCE OF A GREEDY ALGORITHM FOR HIGH-DIMENSIONAL CONVEX NONLINEAR PROBLEMS, Mathematical Models and Methods in Applied Sciences, vol.21, issue.12, pp.2433-2467, 2011.
DOI : 10.1142/S0218202511005799
URL : https://hal.archives-ouvertes.fr/hal-00469622

E. Cances, V. Ehrlacher, and T. Lelievre, Greedy algorithms for highdimensional non-symmetric linear problems, ESAIM: Proceedings, pp.95-131, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00745611

F. Casas, N. Crouseilles, E. Faou, and M. Mehrenberger, High-order Hamiltonian splitting for the Vlasov???Poisson equations, Numerische Mathematik, vol.150, issue.3, 2015.
DOI : 10.1007/s00211-016-0816-z

P. Cazeaux, S. Jan, and . Hesthaven, Multiscale time-integration for particle-in-cell methods, 2014.

F. Charles, B. Després, and M. Mehrenberger, Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation, SIAM Journal on Numerical Analysis, vol.51, issue.2, pp.840-863, 2013.
DOI : 10.1137/110851511
URL : https://hal.archives-ouvertes.fr/hal-01437778

F. Chinesta, A. Ammar, and E. Cueto, Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models, Archives of Computational Methods in Engineering, vol.190, issue.1, pp.327-350, 2010.
DOI : 10.1007/s11831-010-9049-y
URL : https://hal.archives-ouvertes.fr/hal-01007235

F. Chinesta, P. Ladeveze, and E. Cueto, A Short Review on Model Order Reduction Based on Proper Generalized Decomposition, Archives of Computational Methods in Engineering, vol.69, issue.9, pp.395-404, 2011.
DOI : 10.1007/s11831-011-9064-7
URL : https://hal.archives-ouvertes.fr/hal-01004940

H. Cho, G. Venturi, and . Karniadakis, Numerical methods for high-dimensional probability density function equations, Journal of Computational Physics, vol.305, pp.817-837, 2016.
DOI : 10.1016/j.jcp.2015.10.030

N. Crouseilles, G. Latu, and E. Sonnendrücker, A parallel Vlasov solver based on local cubic spline interpolation on patches, Journal of Computational Physics, vol.228, issue.5, pp.1429-1446, 2009.
DOI : 10.1016/j.jcp.2008.10.041
URL : https://hal.archives-ouvertes.fr/hal-00771580

N. Crouseilles, M. Mehrenberger, and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations, Journal of Computational Physics, vol.229, issue.6, pp.1927-1953, 2010.
DOI : 10.1016/j.jcp.2009.11.007
URL : https://hal.archives-ouvertes.fr/hal-00363643

P. Degond, L. Pareschi, and G. Russo, Modeling and computational methods for kinetic equations, 2004.
DOI : 10.1007/978-0-8176-8200-2

L. Desvillettes and J. Dolbeault, On long time asymptotics of the VlasovPoissonBoltzmann equation, Communications in partial differential equations, pp.451-489, 1991.

M. Espig and A. Khachatryan, Convergence of alternating least squares optimisation for rank-one approximation to high order tensors, 2015.

A. Falco and A. Nouy, A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart???Young approach, Journal of Mathematical Analysis and Applications, vol.376, issue.2, pp.469-480, 2011.
DOI : 10.1016/j.jmaa.2010.12.003
URL : https://hal.archives-ouvertes.fr/hal-00461094

A. Falcó and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numerische Mathematik, vol.115, issue.45???48, pp.503-530, 2012.
DOI : 10.1007/s00211-011-0437-5

E. Leonardo, E. Figueroa, and . Süli, Greedy approximation of high-dimensional Ornstein?Uhlenbeck operators, Foundations of Computational Mathematics, vol.12, issue.5, pp.573-623, 2012.

F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Computer Physics Communications, vol.150, issue.3, pp.247-266, 2003.
DOI : 10.1016/S0010-4655(02)00694-X
URL : https://hal.archives-ouvertes.fr/hal-00129663

K. Germaschewski, W. Fox, S. Abbott, N. Ahmadi, K. Maynard et al., The Plasma Simulation Code: A modern particle-in-cell code with patch-based load-balancing, Journal of Computational Physics, vol.318, pp.305-326, 2016.
DOI : 10.1016/j.jcp.2016.05.013

T. Robert and . Glassey, The Cauchy problem in kinetic theory, SIAM, 1996.

A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, pp.401-481, 1996.

W. Hackbusch, N. Boris, E. E. Khoromskij, and . Tyrtyshnikov, Approximate iterations for structured matrices, Numerische Mathematik, vol.64, issue.2, pp.365-383, 2008.
DOI : 10.1007/s00211-008-0143-0

E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration illustrated by the StrmerVerlet method, Acta Numerica, vol.12, pp.399-450, 2003.
DOI : 10.1017/S0962492902000144

E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2006.

J. Hyung and . Hwang, Regularity for the Vlasov?Poisson system in a convex domain, SIAM journal on mathematical analysis, vol.36, issue.1, pp.121-171, 2004.

K. Kormann, A Semi-Lagrangian Vlasov Solver in Tensor Train Format, SIAM Journal on Scientific Computing, vol.37, issue.4, pp.613-632, 2015.
DOI : 10.1137/140971270

P. Ladeveze, J. Passieux, and D. Néron, The LATIN multiscale computational method and the Proper Generalized Decomposition, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.21-22, pp.1991287-1296, 2010.
DOI : 10.1016/j.cma.2009.06.023
URL : https://hal.archives-ouvertes.fr/hal-00414447

C. Le-bris, T. Lelievre, and Y. Maday, Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations, Constructive Approximation, vol.17, issue.2, pp.621-651, 2009.
DOI : 10.1007/s00365-009-9071-1
URL : https://hal.archives-ouvertes.fr/inria-00336911

P. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Inventiones Mathematicae, vol.33, issue.no 1, pp.415-430, 1991.
DOI : 10.1007/BF01232273

M. Eric-madaule, E. Restelli, and . Sonnendrücker, Energy conserving discontinuous Galerkin spectral element method for the Vlasov???Poisson system, Journal of Computational Physics, vol.279, pp.261-288, 2014.
DOI : 10.1016/j.jcp.2014.09.010

J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D: Nonlinear Phenomena, vol.4, issue.3, pp.394-406, 1982.
DOI : 10.1016/0167-2789(82)90043-4

P. Morrison, Hamiltonian and action principle formulations of plasma physicsa) Physics of Plasmas, p.58102, 1994.

A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.23-24, pp.1603-1626, 2010.
DOI : 10.1016/j.cma.2010.01.009
URL : https://hal.archives-ouvertes.fr/hal-00455635

M. Campos, P. , and M. Mehrenberger, Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system, Numerische Mathematik, vol.149, issue.2, pp.407-444, 2008.
DOI : 10.1007/s00211-007-0120-z
URL : https://hal.archives-ouvertes.fr/hal-00320364

T. Rohwedder and A. Uschmajew, On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format, SIAM Journal on Numerical Analysis, vol.51, issue.2, pp.1134-1162, 2013.
DOI : 10.1137/110857520

N. Vladimir and . Temlyakov, Greedy approximation, Acta Numerica, vol.17, pp.235-409, 2008.

A. Uschmajew, Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation, SIAM Journal on Matrix Analysis and Applications, vol.33, issue.2, pp.639-652, 2012.
DOI : 10.1137/110843587

V. Vedenyapin, A. Sinitsyn, and E. Dulov, Kinetic Boltzmann, Vlasov and Related Equations, 2011.

X. Wang, R. Samulyak, X. Jiao, and K. Yu, AP-Cloud: Adaptive Particle-in-Cloud method for optimal solutions to Vlasov???Poisson equation, Journal of Computational Physics, vol.316, pp.682-699, 2016.
DOI : 10.1016/j.jcp.2016.04.037

J. Xu, N. Peter, B. Ostroumov, J. Mustapha, and . Nolen, Scalable Direct Vlasov Solver with Discontinuous Galerkin Method on Unstructured Mesh, SIAM Journal on Scientific Computing, vol.32, issue.6, pp.3476-3494, 2010.
DOI : 10.1137/10078904X