A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1777-1806.

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces.

DOI : 10.1051/m2an/2014019
Classification : 15A69, 35J50, 41A63, 65D15, 65N12
Mots-clés : high-dimensional problems, nonlinear approximation, low-rank approximation, proper generalized decomposition, minimal residual, stochastic partial differential equation
@article{M2AN_2014__48_6_1777_0,
     author = {Billaud-Friess, M. and Nouy, A. and Zahm, O.},
     title = {A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1777--1806},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
     year = {2014},
     doi = {10.1051/m2an/2014019},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014019/}
}
TY  - JOUR
AU  - Billaud-Friess, M.
AU  - Nouy, A.
AU  - Zahm, O.
TI  - A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2014
SP  - 1777
EP  - 1806
VL  - 48
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014019/
DO  - 10.1051/m2an/2014019
LA  - en
ID  - M2AN_2014__48_6_1777_0
ER  - 
%0 Journal Article
%A Billaud-Friess, M.
%A Nouy, A.
%A Zahm, O.
%T A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2014
%P 1777-1806
%V 48
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2014019/
%R 10.1051/m2an/2014019
%G en
%F M2AN_2014__48_6_1777_0
Billaud-Friess, M.; Nouy, A.; Zahm, O. A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1777-1806. doi : 10.1051/m2an/2014019. http://archive.numdam.org/articles/10.1051/m2an/2014019/

[1] A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. J. Non-Newtonian Fluid Mech. 139 (2006) 153-176. | Zbl

[2] A. Ammar, F. Chinesta and A. Falco, On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Engrg. 17 (2010) 473-486. | MR | Zbl

[3] M. Bachmayr and W. Dahmen, Adaptive near-optimal rank tensor approximation for high-dimensional operator equations. Found. Comput. Math. (2014) DOI:10.1007/s10208-013-9187-3.

[4] J. Ballani and L. Grasedyck, A projection method to solve linear systems in tensor format. Numer. Linear Algebra Appl. 20 (2013) 27-43. | MR | Zbl

[5] G. Beylkin and M.J. Mohlenkamp, Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26 (2005) 2133-2159. | MR | Zbl

[6] E. Cances, V. Ehrlacher and T. Lelievre, Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433-2467. | MR | Zbl

[7] E. Cances, V. Ehrlacher and T. Lelievre, Greedy algorithms for high-dimensional non-symmetric linear problems (2012). Preprint: arXiv:1210.6688v1. | MR

[8] A. Cohen, W. Dahmen and G. Welper, Adaptivity and variational stabilization for convection-diffusion equations. ESAIM: M2AN 46 (2012) 1247-1273. | Numdam | MR | Zbl

[9] F. Chinesta, P. Ladeveze and E. Cueto, A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Engrg. 18 (2011) 395-404.

[10] W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive petrov-galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420-2445. | MR | Zbl

[11] L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21 (2000) 1253-1278. | MR | Zbl

[12] A. Doostan and G. Iaccarino, A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 228 (2009) 4332-4345. | MR | Zbl

[13] A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Appl. Math. Sci. (2004). | MR | Zbl

[14] M. Espig and W. Hackbusch, A regularized newton method for the efficient approximation of tensors represented in the canonical tensor format. Numer. Math. 122 (2012) 489-525. | MR | Zbl

[15] A. Falcó and A. Nouy, A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl. 376 (2011) 469-480. | MR | Zbl

[16] A. Falcó and W. Hackbusch, On minimal subspaces in tensor representations. Found. Comput. Math. 12 (2012) 765-803. | MR | Zbl

[17] A. Falcó and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor banach spaces. Numer. Math. 121 (2012) 503-530. | MR | Zbl

[18] A. Falcó, W. Hackbusch and A. Nouy, Geometric structures in tensor representations. Preprint 9/2013, MPI MIS.

[19] L. Figueroa and E. Suli, Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators. Found. Comput. Math. 12 (2012) 573-623. | MR | Zbl

[20] L. Giraldi, Contributions aux Méthodes de Calcul Basées sur l'Approximation de Tenseurs et Applications en Mécanique Numérique. Ph.D. thesis, École Centrale Nantes (2012).

[21] L. Giraldi, A. Nouy, G. Legrain and P. Cartraud, Tensor-based methods for numerical homogenization from high-resolution images. Comput. Methods Appl. Mech. Engrg. 254 (2013) 154-169. | MR | Zbl

[22] L. Grasedyck, Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31 (2010) 2029-2054. | MR | Zbl

[23] L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36 (2013) 53-78. | MR | Zbl

[24] W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus. In vol. 42 of Springer Series in Computational Mathematics (2012). | MR | Zbl

[25] W. Hackbusch and S. Kuhn, A New Scheme for the Tensor Representation. J. Fourier Anal. Appl. 15 (2009) 706-722. | MR | Zbl

[26] S. Holtz, T. Rohwedder and R. Schneider, The Alternating Linear Scheme for Tensor Optimisation in the TT format. SIAM J. Sci. Comput. 34 (2012) 683-713. | MR | Zbl

[27] S. Holtz, T. Rohwedder and R. Schneider, On manifolds of tensors with fixed TT rank. Numer. Math. 120 (2012) 701-731. | MR | Zbl

[28] B.N. Khoromskij and C. Schwab, Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33 (2011) 364-385. | MR | Zbl

[29] B.N. Khoromskij, Tensors-structured numerical methods in scientific computing: Survey on recent advances. Chemometrics and Intelligent Laboratory Systems 110 (2012) 1-19.

[30] T.G. Kolda and B.W. Bader, Tensor decompositions and applications. SIAM Review 51 (2009) 455-500. | MR | Zbl

[31] D. Kressner and C. Tobler, Low-rank tensor krylov subspace methods for parametrized linear systems. SIAM J. Matrix Anal. Appl. 32 (2011) 1288-1316. | MR | Zbl

[32] P. Ladevèze, Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation. Springer Verlag (1999). | Zbl

[33] P. Ladevèze, J.C. Passieux and D. Néron, The LATIN multiscale computational method and the Proper Generalized Decomposition. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1287-1296. | MR | Zbl

[34] H. G. Matthies and E. Zander, Solving stochastic systems with low-rank tensor compression. Linear Algebra Appl. 436 (2012). | MR | Zbl

[35] A. Nouy, A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg. 196 (2007) 4521-4537. | MR | Zbl

[36] A. Nouy, Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Arch. Comput. Methods Engrg. 16 (2009) 251-285. | MR

[37] A. Nouy, Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Engrg. 17 (2010) 403-434. | MR | Zbl

[38] A. Nouy, A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1603-1626. | MR | Zbl

[39] I.V. Oseledets and E.E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31 (2009) 3744-3759. | MR | Zbl

[40] I.V. Oseledets, Tensor-train decomposition. SIAM J. Sci. Comput. 33 (2011) 2295-2317. | MR | Zbl

[41] T. Rohwedder and A. Uschmajew, On local convergence of alternating schemes for optimization of convex problems in the tensor train format. SIAM J. Numer. Anal. 51 (2013) 1134-1162. | MR | Zbl

[42] V. Temlyakov, Greedy Approximation. Camb. Monogr. Appl. Comput. Math. Cambridge University Press (2011). | MR

[43] V. Temlyakov, Greedy approximation. Acta Numerica 17 (2008) 235-409. | MR | Zbl

[44] A. Uschmajew and B. Vandereycken, The geometry of algorithms using hierarchical tensors. Technical report, ANCHP-MATHICSE, Mathematics Section, EPFL (2012). | MR | Zbl

Cité par Sources :