Abstract

Dynamical low-rank algorithms are a class of numerical methods that compute low-rank approximations of dynamical systems. This is accomplished by projecting the dynamics onto a low-dimensional manifold and writing the solution directly in terms of the low-rank factors. The approach has been successfully applied to many types of differential equations. Recently, efficient dynamical low-rank algorithms have been proposed in [L. Einkemmer, A Low-Rank Algorithm for Weakly Compressible Flow, arXiv:1804.04561, 2018; L. Einkemmer and C. Lubich, SIAM J. Sci. Comput., 40 (2018), pp. B1330--B1360] to treat kinetic equations, including the Vlasov--Poisson and the Boltzmann equation. There it was demonstrated that the methods are able to capture the low-rank structure of the solution and significantly reduce numerical cost, while often maintaining high accuracy. However, no numerical analysis is currently available. In this paper, we perform an error analysis for a dynamical low-rank algorithm applied to the multiscale linear Boltzmann equation (a classical model in kinetic theory) to showcase the validity of the application of dynamical low-rank algorithms to kinetic theory. The equation, in its parabolic regime, is known to be rank 1 theoretically, and we will prove that the scheme can dynamically and automatically capture this low-rank structure. This work thus serves as the first mathematical error analysis for a dynamical low-rank approximation applied to a kinetic problem.

Keywords

  1. dynamical low-rank approximation
  2. multiscale analysis
  3. linear Boltzmann equation
  4. low-rank structure

MSC codes

  1. 65F55
  2. 35L02
  3. 65M06
  4. 80A21

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
A. Arnold and T. Jahnke, On the approximation of high-dimensional differential equations in the hierarchical Tucker format, BIT, 54 (2014), pp. 305--341.
2.
C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), pp. 617--649.
3.
M. Bennoune, M. Lemou, and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), pp. 3781--3803.
4.
D. Conte and C. Lubich, An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics, ESAIM Math. Model. Numer. Anal., 44 (2010), pp. 759--780.
5.
N. Crouseilles, M. Mehrenberger, and F. Vecil, Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson, in ESAIM Proc. 32 (2011), pp. 211--230.
6.
W. Dahmen, R. DeVore, L. Grasedyck, and E. Süli, Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations, Found. Comput. Math., 16 (2016), pp. 813--874.
7.
P. Degond, F. Deluzet, L. Navoret, A. Sun, and M. Vignal, Asymptotic-preserving particle-in-cell method for the Vlasov--Poisson system near quasineutrality, J. Comput. Phys., 229 (2010), pp. 5630--5652.
8.
P. Degond, S. Jin, and J. Liu, Mach-number uniform asymptotic-preserving gauge schemes for compressible flows, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007).
9.
G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J. Numer. Anal., 49 (2011), pp. 2057--2077.
10.
G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), pp. 369--520.
11.
L. Einkemmer, A Low-Rank Algorithm for Weakly Compressible Flow, arXiv:1804.04561, 2018.
12.
L. Einkemmer, High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code, Comput. Phys. Commun., 202 (2016), pp. 326--336.
13.
L. Einkemmer, A performance comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions, J. Comput. Phys., 376 (2019), pp. 937--951.
14.
L. Einkemmer and C. Lubich, A low-rank projector-splitting integrator for the Vlasov--Poisson equation, SIAM J. Sci. Comput., 40 (2018), pp. B1330--B1360.
15.
L. Einkemmer and C. Lubich, A quasi-conservative dynamical low-rank algorithm for the vlasov equation, SIAM J. Sci. Comput., 41 (2019), pp. B1061--B1081.
16.
L. Einkemmer, A. Ostermann, and C. Piazzola, A Low-Rank Projector-Splitting Integrator for the Vlasov--Maxwell Equations with Divergence Correction, arXiv:1902.00424, 2019.
17.
F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 150 (2003), pp. 247--266.
18.
V. Grandgirard, J. Abiteboul, J. Bigot, T. Cartier-Michaud, N. Crouseilles, G. Dif-Pradalier, C. Ehrlacher, D. Esteve, X. Garbet, P. Ghendrih, G. Latu, M. Mehrenberger, C. Norscini, C. Passeron, F. Rozar, Y. Sarazin, E. Sonnendrücker, A. Strugarek, and D. Zarzoso, A 5D gyrokinetic full-f global semi-Lagrangian code for flux-driven ion turbulence simulations, Comput. Phys. Commun., 207 (2016), pp. 35--68.
19.
J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B, 94 (2016), p. 165116.
20.
J. Hu, S. Jin, and Q. Li, Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations, in Handbook of Numerical Methods for Hyperbolic Problems, R. Abgrall and C. Shu, eds., Handb. Numer. Anal. 18, Elsevier, Amsterdam, 2017, pp. 103--129.
21.
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), pp. 441--454.
22.
S. Jin and C. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 126 (1996), pp. 449--467.
23.
S. Jin and L. Pareschi, Asymptotic-preserving (AP) schemes for multiscale kinetic equations: A unified approach, in Hyperbolic Problems: Theory, Numerics, Applications, H. Freistühler and G. Warnecke, eds., Birkhäuser Basel, Basel, 2001, pp. 573--582.
24.
S. Jin, L. Pareschi, and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), pp. 913--936.
25.
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), pp. 1020--1038.
26.
O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434--454.
27.
O. Koch and C. Lubich, Dynamical tensor approximation, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360--2375.
28.
K. Kormann, A semi-Lagrangian Vlasov solver in tensor train format, SIAM J. Sci. Comput., 37 (2015), pp. 613--632.
29.
E. Larsen, J. Morel, and W. Miller, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys., 69 (1987), pp. 283--324.
30.
M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), pp. 334--368.
31.
Q. Li, J. Lu, and W. Sun, A convergent method for linear half-space kinetic equations, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 1583--1615.
32.
Q. Li, J. Lu, and W. Sun, Half-space kinetic equations with general boundary conditions, Math. Comp., 86 (2017), pp. 1269--1301.
33.
Q. Li, J. Lu, and W. Sun, Validity and regularization of classical half-space equations, J. Stat. Phys., 166 (2017), pp. 398--433.
34.
Q. Li and L. Pareschi, Exponential Runge-Kutta for the inhomogeneous Boltzmann equations with high order of accuracy, J. Comput. Phys., 259 (2014), pp. 402--420.
35.
Q. Li and L. Wang, Implicit asymptotic preserving method for linear transport equations, Commun. Comput. Phys., 22 (2017), pp. 157--181.
36.
C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Society, Zürich, 2008.
37.
C. Lubich, Time integration in the multiconfiguration time-dependent Hartree method of molecular quantum dynamics, Appl. Math. Res. Express AMREX, 2015 (2015), pp. 311--328.
38.
C. Lubich and I. Oseledets, A projector-splitting integrator for dynamical low-rank approximation, BIT, 54 (2014), pp. 171--188.
39.
C. Lubich, I. V. Oseledets, and B. Vandereycken, Time integration of tensor trains, SIAM J. Numer. Anal., 53 (2015), pp. 917--941.
40.
C. Lubich, T. Rohwedder, R. Schneider, and B. Vandereycken, Dynamical approximation by hierarchical Tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 470--494.
41.
C. Lubich, B. Vandereycken, and H. Walach, Time integration of rank-constrained tucker tensors, SIAM J. Numer. Anal., 56 (2018), pp. 1273--1290.
42.
H. Meyer, F. Gatti, and G. A. Worth, Multidimensional Quantum Dynamics, John Wiley & Sons, New York, 2009.
43.
H. Meyer, U. Manthe, and L. S. Cederbaum, The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett., 165 (1990), pp. 73--78.
44.
C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), pp. 29--201.
45.
A. Nonnenmacher and C. Lubich, Dynamical low-rank approximation: Applications and numerical experiments, Math. Comput. Simul., 79 (2008), pp. 1346--1357.
46.
A. Ostermann, C. Piazzola, and H. Walach, Convergence of a low-rank Lie--Trotter splitting for stiff matrix differential equations, SIAM J. Numer. Anal., 57 (2019), pp. 1947--1966.
47.
Z. Peng, Y. Chen, Y. Cheng, and F. Li, A Reduced Basis Method for Radiative Transfer Equation, arXiv/2103.07574, 2021.
48.
N. J. Sircombe and T. D. Arber, VALIS: A split-conservative scheme for the relativistic 2D Vlasov--Maxwell system, J. Comput. Phys., 228 (2009), pp. 4773--4788.
49.
D. Ter Haar, Men of Physics: L.D. Landau, Vol. 2, Elsevier, Amsterdam, 1969.
50.
L. Wu and Y. Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Commun. Math. Phys., 336 (2015), pp. 1473--1553.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2254 - 2285
ISSN (online): 1095-7170

History

Submitted: 23 November 2020
Accepted: 20 May 2021
Published online: 19 August 2021

Keywords

  1. dynamical low-rank approximation
  2. multiscale analysis
  3. linear Boltzmann equation
  4. low-rank structure

MSC codes

  1. 65F55
  2. 35L02
  3. 65M06
  4. 80A21

Authors

Affiliations

Funding Information

RNMS KI-NET : 1107291
Wisconsin Data Science Initiative
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1619778, DMS-1750488, TRIPODS: 1740707

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

On May 28, 2024, our site will enter Read Only mode for a limited time in order to complete a platform upgrade. As a result, the following functions will be temporarily unavailable: registering new user accounts, any updates to existing user accounts, access token activations, and shopping cart transactions. Contact [email protected] with any questions.