Hostname: page-component-5b777bbd6c-v47t2 Total loading time: 0 Render date: 2025-06-17T13:56:17.084Z Has data issue: false hasContentIssue false
  • English
  • Français

A tensor approximation method based on ideal minimal residualformulations for the solution of high-dimensional problems∗

Published online by Cambridge University Press:  03 October 2014

M. Billaud-Friess ,
A. Nouy  and
O. Zahm
M. Billaud-Friess
Affiliation:
Ecole Centrale Nantes, Université de Nantes, GeM, UMR CNRS 6183, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.. marie.billaud-friess@ec-nantes.fr; anthony.nouy@ec-nantes.fr; olivier.zahm@ec-nantes.fr
A. Nouy
Affiliation:
Ecole Centrale Nantes, Université de Nantes, GeM, UMR CNRS 6183, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.. marie.billaud-friess@ec-nantes.fr; anthony.nouy@ec-nantes.fr; olivier.zahm@ec-nantes.fr
O. Zahm
Affiliation:
Ecole Centrale Nantes, Université de Nantes, GeM, UMR CNRS 6183, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.. marie.billaud-friess@ec-nantes.fr; anthony.nouy@ec-nantes.fr; olivier.zahm@ec-nantes.fr
Get access

Abstract

In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

Keywords

High-dimensional problemsnonlinear approximationlow-rank approximationproper generalized decompositionminimal residualstochastic partial differential equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1777 - 1806
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ammar, A., Mokdad, B., Chinesta, F. and Keunings, R., 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. Google Scholar
Ammar, A., Chinesta, F. and Falco, A., On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Engrg. 17 (2010) 473–486. Google Scholar
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.
Ballani, J. and Grasedyck, L., A projection method to solve linear systems in tensor format. Numer. Linear Algebra Appl. 20 (2013) 27-43. Google Scholar
Beylkin, G. and Mohlenkamp, M.J., Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26 (2005) 2133–2159. Google Scholar
Cances, E., Ehrlacher, V. and Lelievre, T., Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433–2467. Google Scholar
E. Cances, V. Ehrlacher and T. Lelievre, Greedy algorithms for high-dimensional non-symmetric linear problems (2012). Preprint: arXiv:1210.6688v1.
Cohen, A., Dahmen, W. and Welper, G., Adaptivity and variational stabilization for convection-diffusion equations. ESAIM: M2AN 46 (2012) 1247–1273. Google Scholar
Chinesta, F., Ladeveze, P. and Cueto, E., A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Engrg. 18 (2011) 395–404. Google Scholar
Dahmen, W., Huang, C., Schwab, C. and Welper, G., Adaptive petrov–galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420–2445. Google Scholar
De Lathauwer, L., De Moor, B. and Vandewalle, J., A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21 (2000) 1253–1278. Google Scholar
Doostan, A. and Iaccarino, G., A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 228 (2009) 4332–4345. Google Scholar
A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Appl. Math. Sci. (2004).
Espig, M. and Hackbusch, W., A regularized newton method for the efficient approximation of tensors represented in the canonical tensor format. Numer. Math. 122 (2012) 489–525. Google Scholar
Falcó, A. and Nouy, A., 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. Google Scholar
Falcó, A. and Hackbusch, W., On minimal subspaces in tensor representations. Found. Comput. Math. 12 (2012) 765–803. Google Scholar
Falcó, A. and Nouy, A., Proper generalized decomposition for nonlinear convex problems in tensor banach spaces. Numer. Math. 121 (2012) 503–530. Google Scholar
A. Falcó, W. Hackbusch and A. Nouy, Geometric structures in tensor representations. Preprint 9/2013, MPI MIS.
Figueroa, L. and Suli, E., Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators. Found. Comput. Math. 12 (2012) 573–623. Google Scholar
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).
Giraldi, L., Nouy, A., Legrain, G. and Cartraud, P., Tensor-based methods for numerical homogenization from high-resolution images. Comput. Methods Appl. Mech. Engrg. 254 (2013) 154–169. Google Scholar
Grasedyck, L., Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31 (2010) 2029–2054. Google Scholar
Grasedyck, L., Kressner, D. and Tobler, C., A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36 (2013) 53–78. Google Scholar
W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus. In vol. 42 of Springer Series in Computational Mathematics (2012).
Hackbusch, W. and Kuhn, S., A New Scheme for the Tensor Representation. J. Fourier Anal. Appl. 15 (2009) 706–722. Google Scholar
Holtz, S., Rohwedder, T. and Schneider, R., The Alternating Linear Scheme for Tensor Optimisation in the TT format. SIAM J. Sci. Comput. 34 (2012) 683–713. Google Scholar
Holtz, S., Rohwedder, T. and Schneider, R., On manifolds of tensors with fixed TT rank. Numer. Math. 120 (2012) 701–731. Google Scholar
Khoromskij, B.N. and Schwab, C., Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33 (2011) 364–385. Google Scholar
Khoromskij, B.N., Tensors-structured numerical methods in scientific computing: Survey on recent advances. Chemometrics and Intelligent Laboratory Systems 110 (2012) 1–19. Google Scholar
Kolda, T.G. and Bader, B.W., Tensor decompositions and applications. SIAM Review 51 (2009) 455–500. Google Scholar
Kressner, D. and Tobler, C., Low-rank tensor krylov subspace methods for parametrized linear systems. SIAM J. Matrix Anal. Appl. 32 (2011) 1288–1316. Google Scholar
P. Ladevèze, Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation. Springer Verlag (1999).
Ladevèze, P., Passieux, J.C. and Néron, D., The LATIN multiscale computational method and the Proper Generalized Decomposition. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1287–1296. Google Scholar
H. G. Matthies and E. Zander, Solving stochastic systems with low-rank tensor compression. Linear Algebra Appl. 436 (2012).
Nouy, A., A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg. 196 (2007) 4521-4537. Google Scholar
Nouy, A., Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Arch. Comput. Methods Engrg. 16 (2009) 251–285. Google Scholar
Nouy, A., Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Engrg. 17 (2010) 403–434. Google Scholar
Nouy, A., A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1603–1626. Google Scholar
Oseledets, I.V. and Tyrtyshnikov, E.E., Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31 (2009) 3744–3759. Google Scholar
Oseledets, I.V., Tensor-train decomposition. SIAM J. Sci. Comput. 33 (2011) 2295–2317. Google Scholar
Rohwedder, T. and Uschmajew, A., On local convergence of alternating schemes for optimization of convex problems in the tensor train format. SIAM J. Numer. Anal. 51 (2013) 1134–1162. Google Scholar
V. Temlyakov, Greedy Approximation. Camb. Monogr. Appl. Comput. Math. Cambridge University Press (2011).
Temlyakov, V., Greedy approximation. Acta Numerica 17 (2008) 235–409. Google Scholar
A. Uschmajew and B. Vandereycken, The geometry of algorithms using hierarchical tensors. Technical report, ANCHP-MATHICSE, Mathematics Section, EPFL (2012).