Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 635-658.

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a Galerkin method, uniformly stable in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert–Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov–Galerkin setting to evaluate the dual residual norm.

DOI : 10.1051/m2an/2018073
Classification : 65M12, 65M22, 35K20
Mots-clés : Parabolic equations, tensor methods, proper generalized decomposition, greedy algorithm
Boiveau, Thomas 1 ; Ehrlacher, Virginie 1 ; Ern, Alexandre 1 ; Nouy, Anthony 1

1 
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     title = {Low-rank approximation of linear parabolic equations by space-time tensor {Galerkin} methods},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {635--658},
     publisher = {EDP-Sciences},
     volume = {53},
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     year = {2019},
     doi = {10.1051/m2an/2018073},
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Boiveau, Thomas; Ehrlacher, Virginie; Ern, Alexandre; Nouy, Anthony. Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 635-658. doi : 10.1051/m2an/2018073. http://www.numdam.org/articles/10.1051/m2an/2018073/

R. Andreev, Stability of space-time Petrov-Galerkin discretizations for parabolic evolution equations. Ph.D. thesis, ETH Zürich (2012).

R. Andreev, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33 (2013) 242–260. | DOI | MR | Zbl

R. Andreev, Space-time discretization of the heat equation. Numer. Algorithms 67 (2014) 713–731. | DOI | MR | Zbl

M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. | DOI | MR | Zbl

E. Cancès, V. Ehrlacher and T. Lelièvre, Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433–2467. | DOI | MR | Zbl

E. Cancès, V. Ehrlacher and T. Lelièvre, Greedy algorithms for high-dimensional eigenvalue problems. Constr. Approx. 40 (2014) 387–423. | DOI | MR | Zbl

F. Chinesta, R. Keunings and A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. Springer Briefs in Applied Sciences and Technology. Springer, Cham (2014). | DOI | MR | Zbl

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol 5 Evolution Problems I. Springer-Verlag, Berlin, Germany (1992). | MR | Zbl

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. In Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | DOI | MR | Zbl

A. Ern, I. Smears and M. Vohralk, Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems. SIAM J. Numer. Anal. 55 (2017) 2811–2834. | DOI | MR | Zbl

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. | DOI | MR | Zbl

A. Falcó and A. Nouy, Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. 121 (2012) 503–530. | DOI | MR | Zbl

M.J. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29 (2007) 556–578. | DOI | MR | Zbl

M.J. Gander and H. Zhao, Overlapping Schwarz waveform relaxation for the heat equation in n dimensions. BIT 42 (2002) 779–795. | DOI | MR | Zbl

E. Giladi and H.B. Keller, Space-time domain decomposition for parabolic problems. Numer. Math. 93 (2002) 279–313. | DOI | MR | Zbl

M. Griebel, D. Oeltz, A sparse grid space-time discretization scheme for parabolic problems. Computing 81 (2007) 1–34. | DOI | MR | Zbl

M.D. Gunzburger and A. Kunoth, Space-time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations. SIAM J. Control Optim. 49 (2011) 1150–1170. | DOI | MR | Zbl

V.H. Hoang and C. Schwab, Sparse tensor Galerkin discretization of parametric and random parabolic PDEs – analytic regularity and generalized polynomial chaos approximation. SIAM J. Math. Anal. 45 (2013) 3050–3083. | DOI | MR | Zbl

J. Janssen and S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: the continuous-time case. SIAM J. Numer. Anal. 33 (1996) 456–474. | DOI | MR | Zbl

E. Kieri, C. Lubich and H. Walach, Discretized dynamical low-rank approximation in the presence of small singular values. SIAM J. Numer. Anal. 54 (2016) 1020–1038. | DOI | MR | Zbl

O. Koch and C. Lubich, Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29 (2007) 434–454. | DOI | MR | Zbl

P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation. Springer Science & Business Media, New York, NY (2012). | Zbl

C. Le Bris, T. Lelièvre and Y. Maday, Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. Constr. Approx. 30 (2009) 621–651. | DOI | MR | Zbl

J.-L. Lions, Y. Maday and G. Turinici, Résolution d’EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 661–668. | DOI | MR | Zbl

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, II. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181–182. Springer-Verlag, New York-Heidelberg (1972). | MR | Zbl

C. Lubich and I.V. Oseledets, A projector-splitting integrator for dynamical low-rank approximation. BIT 54 (2014) 171–188. | DOI | MR | Zbl

A. Mantzaflaris, F. Scholz and I. Toulopoulos, Low-rank space-time isogeometric analysis for parabolic problems with varying coefficients. Comp. Methods Appl. Math. 19 (2019) 123–136. | DOI | MR | Zbl

M. Neumüller and I. Smears, Time-parallel iterative solvers for parabolic evolution equations. SIAM J. Sci. Comput. 41 (2019) C28–C51. | DOI | MR | Zbl

A. Nouy, Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Methods Eng. 16 (2009) 251–285. | DOI | MR | Zbl

A. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 199 (2010) 1603–1626. | DOI | MR | Zbl

C.C. Paige and M.A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8 (1982) 43–71. | DOI | MR | Zbl

C. Schwab and R. Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comput. 78 (2009) 1293–1318. | DOI | MR | Zbl

F. Tantardini and A. Veeser, The L2-projection and quasi-optimality of Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 54 (2016) 317–340. | DOI | MR | Zbl

V.N. Temlyakov, Greedy approximation. Acta Numer. 17 (2008) 235–409. | DOI | MR | Zbl

V. Thomée, Galerkin finite element methods for parabolic problems, 2nd edition. In Vol. 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). | MR | Zbl

K. Urban and A.T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations. C. R. Math. Acad. Sci. Paris 350 (2012) 203–207. | DOI | MR | Zbl

A. Uschmajew, Local convergence of the alternating least squares algorithm for canonical tensor approximation. SIAM J. Matrix Anal. Appl. 33 (2012) 639–652. | DOI | MR | Zbl

J. Wloka, Partial Differential Equations. Translated from the German by C.B. Thomas and M.J. Thomas. Cambridge University Press, Cambridge (1987). | MR | Zbl

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