Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems

Abstract We consider dynamical low-rank approximation on the manifold of fixed-rank matrices and tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, under suitable approximability assumptions, we prove error estimates for the explicit Euler method equipped with quasi-optimal projections to the manifold. Then we discuss the possibilities and difficulties with higher-order explicit methods. In particular, we discuss ways for limiting rank growth in the increments, and robustness with respect to small singular values.

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

[2]  Fabio Nobile,et al.  Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs , 2015, SIAM J. Sci. Comput..

[3]  Christian Lubich,et al.  Dynamical low-rank approximation: applications and numerical experiments , 2008, Math. Comput. Simul..

[4]  Daniel Kressner,et al.  Recompression of Hadamard Products of Tensors in Tucker Format , 2017, SIAM J. Sci. Comput..

[5]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[6]  Dietrich Braess,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Approximation of 1/x by Exponential Sums in [1, ∞) , 2022 .

[7]  Mark Coppejans,et al.  Breaking the Curse of Dimensionality , 2000 .

[8]  U. Manthe,et al.  Wave‐packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl , 1992 .

[9]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[10]  M. Beck,et al.  The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propa , 1999 .

[11]  W. Huisinga,et al.  A Dynamical Low-Rank Approach to the Chemical Master Equation , 2008, Bulletin of mathematical biology.

[12]  Daniel Kressner,et al.  A literature survey of low‐rank tensor approximation techniques , 2013, 1302.7121.

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

[14]  Daniel Kressner,et al.  Matrices with Hierarchical Low-Rank Structures , 2016 .

[15]  Wolfgang Hackbusch.  New estimates for the recursive low-rank truncation of block-structured matrices , 2016, Numerische Mathematik.

[16]  U. Schollwoeck.  The density-matrix renormalization group in the age of matrix product states , 2010, 1008.3477.

[17]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[18]  U. Manthe,et al.  The multi-configurational time-dependent Hartree approach , 1990 .

[19]  C. Lubich,et al.  A projector-splitting integrator for dynamical low-rank approximation , 2013, BIT Numerical Mathematics.

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

[21]  André Uschmajew,et al.  On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format , 2013, SIAM J. Numer. Anal..

[22]  Hanna Walach,et al.  Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values , 2016, SIAM J. Numer. Anal..

[23]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[24]  J. H. Verner,et al.  Some Runge-Kutta formula pairs , 1991 .

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

[26]  P. Kramer,et al.  Geometry of the Time-Dependent Variational Principle in Quantum Mechanics , 1981 .

[27]  Eugene E. Tyrtyshnikov,et al.  Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions , 2009, SIAM J. Sci. Comput..

[28]  Ivan V. Oseledets,et al.  Time Integration of Tensor Trains , 2014, SIAM J. Numer. Anal..

[29]  Daniel Kressner,et al.  Preconditioned Low-rank Riemannian Optimization for Linear Systems with Tensor Product Structure , 2015, SIAM J. Sci. Comput..

[30]  F. Verstraete,et al.  Matrix product density operators: simulation of finite-temperature and dissipative systems. , 2004, Physical review letters.

[31]  Hans-Dieter Meyer,et al.  Multidimensional quantum dynamics : MCTDH theory and applications , 2009 .

[32]  Lars Grasedyck,et al.  Existence and Computation of Low Kronecker-Rank Approximations for Large Linear Systems of Tensor Product Structure , 2004, Computing.

[33]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[34]  Reinhold Schneider,et al.  The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format , 2012, SIAM J. Sci. Comput..

[35]  Pierre-Antoine Absil,et al.  Low-rank retractions: a survey and new results , 2015, Comput. Optim. Appl..

[36]  A Smerzi,et al.  Discrete solitons and breathers with dilute Bose-Einstein condensates. , 2001, Physical review letters.

[37]  P. Dirac.  Note on Exchange Phenomena in the Thomas Atom , 1930, Mathematical Proceedings of the Cambridge Philosophical Society.

[38]  Ernst Hairer,et al.  Solving Differential Equations on Manifolds , 2011 .