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A tensor decomposition algorithm for large ODEs with conservation laws
Numerical Analysis · Submitted 31 Mar 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.
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The abstract, sentence by sentence
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes the TT approximation of the solution directly, without ever solving the original large problem, and encapsulates the Galerkin model reduction of the ODE. This allows an efficient estimation of the time discretization error, and hence provides a way to adapt the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. | contribution 0.99 | 0.85 | 0.12 | asserted 0.99 |
| s1 | A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. | background 0.76 | 0.29 | 0.02 | asserted 1.00 |
| s2 | The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. | method 0.98 | 0.57 | 0.08 | asserted 1.00 |
| s3 | This method computes the TT approximation of the solution directly, without ever solving the original large problem, and encapsulates the Galerkin model reduction of the ODE. | method 0.66 | 0.83 | 0.23 | asserted 1.00 |
| s4 | This allows an efficient estimation of the time discretization error, and hence provides a way to adapt the time steps. | interpretation 0.45 | 0.74 | 0.06 | asserted 0.96 |
| s5 | Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the euclidean norm. | method 0.65 | 0.75 | 0.10 | asserted 0.81 |
| s6 | In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy. | result 1.00 | 0.96 | 0.70 | asserted 1.00 |
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Original abstract
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes the TT approximation of the solution directly, without ever solving the original large problem, and encapsulates the Galerkin model reduction of the ODE. This allows an efficient estimation of the time discretization error, and hence provides a way to adapt the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it
Uses its method or tool 1 paper
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The two most important instances are the following: from the TT-Toolbox, the block cross interpolation Algorithm [ref] for the TT approximation of $κ̃_α$ [ref], and from the companion package tAMEn [ref] , the linear system solver in the TT format (see [ref] ).
Computation of the Response Surface in the Tensor Train data format · Numerical Experiments · 1.00
Cites it as background 1 paper
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In literature, this map $ℳ$ can be referred to as Taylor maps and models [ref] , tensor decomposition [ref] , matrix Lie transform [ref] , exponential machines [ref] , and others.
Polynomial Neural Networks and Taylor maps for Dynamical Systems Simulation and Learning · Introduction and Related Works · 0.97
What it cites · 11 sentences without a confident label
Uses the method or tool of 1 paper
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In Fig. [ref], we investigate the TT ranks of the solution and the CPU times of the calculations with the following parameters: the tensor truncation threshold $=10^-5$, local residual gap $η=10$, number of Chebyshev points in time $=80$, the residual TT rank in tAMEn $ρ=3$, and the time grid is exp-uniform in accordance with [ref] , $t_j=exp(0.05·j)$, $j=1,…,200$, such that $=t_j-t_j-1$ for the step $j$.
Simultaneous state-time approximation of the chemical master equation using tensor product formats · Chemical master equation · 0.90
Compares with 1 paper
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We remind that the Crank-Nicolson calculations in [ref] required about one hour on the same computer.
Simultaneous state-time approximation of the chemical master equation using tensor product formats · Chemical master equation · 0.90
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