Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

doi:10.1007/s42967-023-00357-8

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1562-1579.doi: 10.1007/s42967-023-00357-8

• ORIGINAL PAPERS • 上一篇下一篇

Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

Alec Dektor¹, Daniele Venturi²

1. Applied Mathematics & Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA;
2. Department of Applied Mathematics, University of California Santa Cruz, Santa Cruz, CA, 95064, USA

收稿日期:2023-08-08 修回日期:2023-11-28 接受日期:2023-11-29 出版日期:2024-02-23 发布日期:2024-02-23
通讯作者: Daniele Venturi,E-mail:venturi@ucsc.edu E-mail:venturi@ucsc.edu
基金资助:
This research was supported by the U.S. Air Force Office of Scientific Research (AFOSR) grant FA9550-20-1-0174 and the U.S. Army Research Office (ARO) grant W911NF-18-1-0309.

Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

Alec Dektor¹, Daniele Venturi²

1. Applied Mathematics & Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA;
2. Department of Applied Mathematics, University of California Santa Cruz, Santa Cruz, CA, 95064, USA

Received:2023-08-08 Revised:2023-11-28 Accepted:2023-11-29 Online:2024-02-23 Published:2024-02-23
Supported by:
This research was supported by the U.S. Air Force Office of Scientific Research (AFOSR) grant FA9550-20-1-0174 and the U.S. Army Research Office (ARO) grant W911NF-18-1-0309.

摘要/Abstract

摘要： We introduce a new tensor integration method for time-dependent partial differential equations (PDEs) that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations. Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution. The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi (2023) which has no guarantee of producing globally optimal rank-reducing coordinate transformations. Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

关键词: Tensor train, Crvilinear coordinates, Step-truncation tensor methods, High-dimensional PDEs, Dynamic tensor approximation

Abstract: We introduce a new tensor integration method for time-dependent partial differential equations (PDEs) that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations. Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution. The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi (2023) which has no guarantee of producing globally optimal rank-reducing coordinate transformations. Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

Key words: Tensor train, Crvilinear coordinates, Step-truncation tensor methods, High-dimensional PDEs, Dynamic tensor approximation

中图分类号:

Alec Dektor, Daniele Venturi. Coordinate-Adaptive Integration of PDEs on Tensor Manifolds[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1562-1579.

参考文献

[1] Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publications, New York (1990)
[2] Baalrud, S.D., Daligault, J.: Mean force kinetic theory: A convergent kinetic theory for weakly and strongly coupled plasmas. Phys. Plasmas 26, 082106 (2019)
[3] Beylkin, G., Mohlenkamp, M.J.: Numerical operator calculus in higher dimensions. PNAS 99(16), 10246-10251 (2002)
[4] Bigoni, D., Engsig-Karup, A.P., Marzouk, Y.M.: Spectral tensor-train decomposition. SIAM J. Sci. Comput. 38(4), A2405-A2439 (2016)
[5] Boelens, A.M.P., Venturi, D., Tartakovsky, D.M.: Parallel tensor methods for high-dimensional linear PDEs. J. Comput. Phys. 375, 519-539 (2018)
[6] Boelens, A.M.P., Venturi, D., Tartakovsky, D.M.: Tensor methods for the Boltzmann-BGK equation. J. Comput. Phys. 421, 109744 (2020)
[7] Brennan, C., Venturi, D.: Data-driven closures for stochastic dynamical systems. J. Comput. Phys. 372, 281-298 (2018)
[8] Cercignani, C.: The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988)
[9] Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equation. J. Comput. Phys. 315, 817-837 (2016)
[10] Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional kinetic equations. In: Jin, S., Pareschi, L. (eds.) Uncertainty Quantification for Kinetic and Hyperbolic Equations, pp. 93-125. Springer (2017)
[11] Dektor, A., Rodgers, A., Venturi, D.: Rank-adaptive tensor methods for high-dimensional nonlinear PDEs. J. Sci. Comput. 88(36), 1-27 (2021)
[12] Dektor, A., Venturi, D.: Dynamic tensor approximation of high-dimensional nonlinear PDEs. J. Comput. Phys. 437, 110295 (2021)
[13] Dektor, A., Venturi, D.: Tensor rank reduction via coordinate flows. J. Comput. Phys. 491, 112378 (2023)
[14] Etter, A.: Parallel ALS algorithm for solving linear systems in the hierarchical Tucker representation. SIAM J. Sci. Comput. 38(4), A2585-A2609 (2016)
[15] Falco, A., Hackbusch, W., Nouy, A.: Geometric structures in tensor representations (final release). arXiv:1505.03027v2 (2015)
[16] Friedric, R., Daitche, A., Kamps, O., Lülff, J., Voßkuhle, M., Wilczek, M.: The Lundgren-Monin-Novikov hierarchy: kinetic equations for turbulence. Comptes Rendus Physique 13(9/10), 929-953 (2012)
[17] Grasedyck, L., Löbbert, C.: Distributed hierarchical SVD in the hierarchical Tucker format. Numer. Linear Algebra Appl. 25(6), e2174 (2018)
[18] Griebel, M., Li, G.: On the decay rate of the singular values of bivariate functions. SIAM J. Numer. Anal. 56(2), 974-993 (2019)
[19] Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Volume 21 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2007)
[20] Hosokawa, I.: Monin-Lundgren hierarchy versus the Hopf equation in the statistical theory of turbulence. Phys. Rev. E 73(1/2/3/4), 067301 (2006)
[21] Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nat. Rev. Phys. 3, 422-440 (2021)
[22] Khoromskij, B.N.: Tensor numerical methods for multidimensional PDEs: theoretical analysis and initial applications. In: CEMRACS 2013—Modelling and Simulation of Complex Systems: Stochastic and Deterministic Approaches, Volume 48 of ESAIM Proc. Surveys, pp. 1-28. EDP Sci., Les Ulis (2015)
[23] Kieri, E., Lubich, C., Walach, H.: Discretized dynamical low-rank approximation in the presence of small singular values. SIAM J. Numer. Anal. 54(2), 1020-1038 (2016)
[24] Lundgren, T.S.: Distribution functions in the statistical theory of turbulence. Phys. Fluids 10(5), 969-975 (1967)
[25] Luo, H., Bewley, T.R.: On the contravariant form of the Navier-Stokes equations in time-dependent curvilinear coordinate systems. J. Comput. Phys. 199, 355-375 (2004)
[26] Montgomery, D.: A BBGKY framework for fluid turbulence. Phys. Fluids 19, 802-810 (1976)
[27] Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295-2317 (2011)
[28] Pinkus, A.: Ridge Functions. Cambridge University Press, Cambridge (2015)
[29] Raissi, M., Karniadakis, G.E.: Hidden physics models: machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125-141 (2018)
[30] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 606-707 (2019)
[31] Rodgers, A., Dektor, A., Venturi, D.: Adaptive integration of nonlinear evolution equations on tensor manifolds. J. Sci. Comput. 92(39), 1-31 (2022)
[32] Schneider, R., Uschmajew, A.: Approximation rates for the hierarchical tensor format in periodic Sobolev spaces. J. Complexity 30(2), 56-71 (2014)
[33] Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Langer, R.E. (ed) Nonlinear Problems, pp. 69-98. The University of Wisconsin Press, Madison (1963)
[34] Uschmajew, A., Vandereycken, B.: The geometry of algorithms using hierarchical tensors. Linear Algebra Appl. 439(1), 133-166 (2013)
[35] Venturi, D.: Convective derivatives and Reynolds transport in curvilinear time-dependent coordinate systems. J. Phys. A: Math. Theor. 42(12), 125203 (2009)
[36] Venturi, D.: Conjugate flow action functionals. J. Math. Phys. 54(11), 113502 (2013)
[37] Venturi, D.: The numerical approximation of nonlinear functionals and functional differential equations. Phys. Rep. 732, 1-102 (2018)
[38] Venturi, D., Choi, M., Karniadakis, G.E.: Supercritical quasi-conduction states in stochastic Rayleigh-Bénard convection. Int. J. Heat Mass Transf. 55(13), 3732-3743 (2012)

Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

编辑推荐

Metrics

本文评价