On the approximation of high-dimensional differential equations in the hierarchical Tucker format

The hierarchical Tucker format is a way to decompose a high-dimensional tensor recursively into sums of products of lower-dimensional tensors. The number of degrees of freedom in such a representation is typically many orders of magnitude lower than the number of entries of the original tensor. This makes the hierarchical Tucker format a promising approach to solve ordinary differential equations for high-dimensional tensors. In order to propagate the approximation in time, differential equations for the parameters of the hierarchical Tucker format are derived from the Dirac-Frenkel variational principle. We prove an error bound for the dynamical approximation in the hierarchical Tucker format by extending previous results of Koch and Lubich for the non-hierarchical Tucker format.

[1]  Uwe Manthe,et al.  A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. , 2008, The Journal of chemical physics.

[2]  Daniel Kressner,et al.  Algorithm 941 , 2014 .

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[5]  W. Dörfler,et al.  Convergence of an adaptive hp finite element strategy in higher space-dimensions , 2010 .

[6]  H. Meyer,et al.  Multilayer multiconfiguration time-dependent Hartree method: implementation and applications to a Henon-Heiles hamiltonian and to pyrazine. , 2010, The Journal of chemical physics.

[7]  Bart Vandereycken,et al.  The geometry of algorithms using hierarchical tensors , 2013, Linear Algebra and its Applications.

[8]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[9]  Othmar Koch,et al.  Dynamical Tensor Approximation , 2010, SIAM J. Matrix Anal. Appl..

[10]  Haobin Wang,et al.  Multilayer formulation of the multiconfiguration time-dependent Hartree theory , 2003 .

[11]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[12]  Lars Grasedyck,et al.  Hierarchical Singular Value Decomposition of Tensors , 2010, SIAM J. Matrix Anal. Appl..

[13]  C. Lubich.  From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , 2008 .

[14]  Reinhold Schneider,et al.  On manifolds of tensors of fixed TT-rank , 2012, Numerische Mathematik.

[15]  Reinhold Schneider,et al.  Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors , 2013, SIAM J. Matrix Anal. Appl..

[16]  Ulrich W. Kulisch,et al.  The exact dot product as basic tool for long interval arithmetic , 2011, Computing.

[17]  Othmar Koch,et al.  Dynamical Low-Rank Approximation , 2007, SIAM J. Matrix Anal. Appl..

[18]  W. Hackbusch,et al.  A New Scheme for the Tensor Representation , 2009 .