Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits...

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Vydáno v:	Journal of scientific computing Ročník 88; číslo 2; s. 36
Hlavní autoři:	Dektor, Alec, Rodgers, Abram, Venturi, Daniele
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.08.2021 Springer Nature B.V
Témata:	Adaptive algorithms Algorithms Approximation Computational Mathematics and Numerical Analysis Covariance matrix Decomposition Energy Fokker-Planck equation Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear differential equations Numerical integration Operators (mathematics) Partial differential equations Robustness (mathematics) Series expansion Tensors Theoretical Time integration Dynamical low-rank approximation Tensor train decomposition Tensor manifolds
ISSN:	0885-7474, 1573-7691
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Abstract	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.
AbstractList	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.
ArticleNumber	36
Author	Venturi, Daniele Dektor, Alec Rodgers, Abram
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Cites_doi	10.1016/j.jcp.2020.109341 10.1016/j.jcp.2019.108940 10.1016/j.jcp.2009.10.043 10.1007/978-3-642-28027-6 10.1073/pnas.112329799 10.1007/s10208-016-9317-9 10.1007/s10910-007-9250-x 10.1017/CBO9780511618352 10.1016/B978-0-12-576350-9.50006-0 10.1098/rspa.2013.0754 10.1016/j.physrep.2017.12.003 10.1017/S0962492904000182 10.1007/s10957-019-01473-3 10.1016/j.jcp.2003.08.021 10.1098/rspa.2011.0186 10.1007/978-1-4612-1039-9 10.1137/120896001 10.1051/m2an/2013067 10.1073/pnas.1922204117 10.1080/00268976400100041 10.1016/j.matcom.2008.03.007 10.1007/s10208-018-9381-4 10.1137/09076578X 10.1137/17M1146889 10.1016/j.jcp.2017.04.057 10.1137/050639703 10.1016/j.jcp.2018.10.045 10.1007/978-3-540-71050-9 10.1098/rspa.2014.0932 10.1137/090752286 10.1007/978-3-642-61544-3 10.1016/j.jcp.2021.110295 10.1137/15M1026791 10.1137/15M1036919 10.1007/s10208-013-9154-z 10.1137/140976546 10.1007/s40687-021-00265-4 10.1137/140966368 10.1016/j.jcp.2018.08.057 10.1137/16M1095202 10.1051/proc/201448001 10.1016/j.jcp.2017.11.039 10.1016/j.jcp.2020.109744 10.1137/18M1192780 10.1016/j.jcp.2015.10.030 10.1515/cmam-2018-0029 10.1016/j.jcp.2019.05.024 10.1016/j.jcp.2019.109125 10.1016/S0009-2614(00)00200-1 10.1017/S0962492914000063
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References	Falcó, Hackbusch, Nouy (CR22) 2020; 2011 Bigoni, Engsig-Karup, Marzouk (CR6) 2016; 38 Cho, Venturi, Karniadakis (CR14) 2016; 315 Narayan, Jakeman (CR41) 2014; 36 CR31 CR30 Oseledets (CR44) 2011; 33 Boelens, Venturi, Tartakovsky (CR7) 2018; 375 Feppon, Lermusiaux (CR24) 2019; 40 Venturi, Sapsis, Cho, Karniadakis (CR58) 2012; 468 CR8 Foo, Karniadakis (CR25) 2010; 229 CR49 CR48 Bachmayr, Schneider, Uschmajew (CR2) 2016; 16 Li, Rabitz (CR37) 2008; 43 CR42 Ruthotto, Osher, Li, Nurbekyan, Fung (CR52) 2020; 117 Raissi, Perdikaris, Karniadakis (CR47) 2019; 378 Nonnenmacher, Lubich (CR43) 2008; 79 Han, Li (CR19) 2019; 6 Dektor, Venturi (CR16) 2020; 404 Barthelmann, Novak, Ritter (CR4) 2000; 12 Hackbusch (CR28) 2012 Raissi, Karniadakis (CR46) 2018; 357 Zhu, Zabaras, Koutsourelakis, Perdikaris (CR60) 2019; 394 Sirisup, Karniadakis (CR54) 2004; 194 CR17 CR15 Koch, Neuhauser, Thalhammer (CR36) 2013; 47 Lubich, Vandereycken, Walach (CR39) 2018; 56 Lubich, Oseledets, Vandereycken (CR38) 2015; 53 CR56 Falcó, Hackbusch, Nouy (CR21) 2019; 19 Frenkel (CR26) 1934 CR50 di Marco, Pareschi (CR18) 2014; 23 Kieri, Vandereycken (CR33) 2019; 19 Villani (CR59) 2009 Bungartz, Griebel (CR9) 2004; 13 Rodgers, Venturi (CR51) 2020 Babaee, Choi, Sapsis, Karniadakis (CR1) 2017; 344 Cho, Venturi, Karniadakis (CR13) 2014; 2171 Koch, Lubich (CR35) 2010; 31 Venturi (CR55) 2018; 732 CR29 Salas, Thibault (CR53) 2019; 182 CR27 Kieri, Lubich, Walach (CR32) 2016; 54 McLachlan (CR40) 1964; 8 Baldeaux, Gnewuch (CR3) 2014; 52 Cercignani (CR11) 1988 CR20 Beylkin, Mohlenkamp (CR5) 2002; 99 Koch, Lubich (CR34) 2007; 29 Cao, Chen, Gunzbuger (CR10) 2009; 6 Feppon, Lermusiaux (CR23) 2018; 39 Raab (CR45) 2000; 319 Chkifa, Cohen, Schwab (CR12) 2014; 14 Venturi, Karniadakis (CR57) 2014; 470 AD McLachlan (1539_CR40) 1964; 8 J Baldeaux (1539_CR3) 2014; 52 J Frenkel (1539_CR26) 1934 D Salas (1539_CR53) 2019; 182 M Raissi (1539_CR46) 2018; 357 H Cho (1539_CR13) 2014; 2171 G Li (1539_CR37) 2008; 43 G Beylkin (1539_CR5) 2002; 99 1539_CR31 1539_CR30 H Babaee (1539_CR1) 2017; 344 A Falcó (1539_CR22) 2020; 2011 O Koch (1539_CR35) 2010; 31 HJ Bungartz (1539_CR9) 2004; 13 G di Marco (1539_CR18) 2014; 23 M Bachmayr (1539_CR2) 2016; 16 IV Oseledets (1539_CR44) 2011; 33 O Koch (1539_CR36) 2013; 47 A Raab (1539_CR45) 2000; 319 D Venturi (1539_CR57) 2014; 470 L Ruthotto (1539_CR52) 2020; 117 C Cercignani (1539_CR11) 1988 1539_CR20 W Hackbusch (1539_CR28) 2012 1539_CR29 F Feppon (1539_CR24) 2019; 40 1539_CR27 C Lubich (1539_CR38) 2015; 53 A Rodgers (1539_CR51) 2020 A Dektor (1539_CR16) 2020; 404 Y Zhu (1539_CR60) 2019; 394 A Nonnenmacher (1539_CR43) 2008; 79 A Narayan (1539_CR41) 2014; 36 H Cho (1539_CR14) 2016; 315 D Venturi (1539_CR55) 2018; 732 D Bigoni (1539_CR6) 2016; 38 S Sirisup (1539_CR54) 2004; 194 C Villani (1539_CR59) 2009 1539_CR56 V Barthelmann (1539_CR4) 2000; 12 M Raissi (1539_CR47) 2019; 378 A Falcó (1539_CR21) 2019; 19 1539_CR17 1539_CR15 A Chkifa (1539_CR12) 2014; 14 E Kieri (1539_CR33) 2019; 19 AMP Boelens (1539_CR7) 2018; 375 E Kieri (1539_CR32) 2016; 54 F Feppon (1539_CR23) 2018; 39 D Venturi (1539_CR58) 2012; 468 1539_CR50 C Lubich (1539_CR39) 2018; 56 1539_CR8 J Foo (1539_CR25) 2010; 229 1539_CR42 O Koch (1539_CR34) 2007; 29 Y Cao (1539_CR10) 2009; 6 1539_CR49 1539_CR48 W E,J Han (1539_CR19) 2019; 6
References_xml	– volume: 2171 start-page: 1 issue: 470 year: 2014 end-page: 21 ident: CR13 article-title: Statistical analysis and simulation of random shocks in Burgers equation publication-title: Proc. R. Soc. A – volume: 315 start-page: 817 year: 2016 end-page: 837 ident: CR14 article-title: Numerical methods for high-dimensional probability density function equations publication-title: J. Comput. Phys. – ident: CR49 – volume: 53 start-page: 917 issue: 2 year: 2015 end-page: 941 ident: CR38 article-title: Time integration of tensor trains publication-title: SIAM J. Numer. Anal. – volume: 52 start-page: 1128 issue: 3 year: 2014 end-page: 1155 ident: CR3 article-title: Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition publication-title: SIAM J. Numer. Anal. – volume: 39 start-page: 510 issue: 1 year: 2018 end-page: 538 ident: CR23 article-title: A geometric approach to dynamical model order reduction publication-title: SIAM J. Matrix Anal. Appl. – year: 1934 ident: CR26 publication-title: Wave Mechanics: Advanced General Theory – volume: 47 start-page: 1265 issue: 5 year: 2013 end-page: 1286 ident: CR36 article-title: Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics publication-title: ESAIM Math. Model. Numer. Anal. – year: 2009 ident: CR59 publication-title: Optimal Transport: Old and New – volume: 12 start-page: 273 year: 2000 end-page: 288 ident: CR4 article-title: High dimensional polynomial interpolation on sparse grids publication-title: Adv. Comput. Mech. – ident: CR29 – volume: 229 start-page: 1536 year: 2010 end-page: 1557 ident: CR25 article-title: Multi-element probabilistic collocation method in high dimensions publication-title: J. Comput. Phys. – ident: CR8 – volume: 23 start-page: 369 year: 2014 end-page: 520 ident: CR18 article-title: Numerical methods for kinetic equations publication-title: Acta Numer. – volume: 36 start-page: A2952 issue: 6 year: 2014 end-page: A2983 ident: CR41 article-title: Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation publication-title: SIAM J. Sci. Comput. – volume: 394 start-page: 56 year: 2019 end-page: 81 ident: CR60 article-title: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data publication-title: J. Comput. Phys. – ident: CR42 – volume: 375 start-page: 519 year: 2018 end-page: 539 ident: CR7 article-title: Parallel tensor methods for high-dimensional linear PDEs publication-title: J. Comput. Phys. – volume: 54 start-page: 1020 issue: 2 year: 2016 end-page: 1038 ident: CR32 article-title: Discretized dynamical low-rank approximation in the presence of small singular values publication-title: SIAM J. Numer. Anal. – volume: 56 start-page: 1273 issue: 3 year: 2018 end-page: 1290 ident: CR39 article-title: Time integration of rank-constrained Tucker tensors publication-title: SIAM J. Numer. Anal. – volume: 29 start-page: 434 issue: 2 year: 2007 end-page: 454 ident: CR34 article-title: Dynamical low-rank approximation publication-title: SIAM J. Matrix Anal. Appl. – ident: CR15 – volume: 378 start-page: 606 year: 2019 end-page: 707 ident: CR47 article-title: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations publication-title: J. Comput. Phys. – ident: CR50 – volume: 117 start-page: 9183 issue: 17 year: 2020 end-page: 9193 ident: CR52 article-title: A machine learning framework for solving high-dimensional mean field game and mean field control problems publication-title: PNAS – volume: 194 start-page: 92 issue: 1 year: 2004 end-page: 116 ident: CR54 article-title: A spectral viscosity method for correcting the long-term behavior of pod models publication-title: J. Comput. Phys. – volume: 43 start-page: 1207 issue: 3 year: 2008 end-page: 1232 ident: CR37 article-title: Regularized random-sampling high dimensional model representation (RS-HDMR) publication-title: J. Math. Chem. – year: 2020 ident: CR51 article-title: Stability analysis of hierarchical tensor methods for time-dependent PDEs publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109341 – volume: 732 start-page: 1 year: 2018 end-page: 102 ident: CR55 article-title: The numerical approximation of nonlinear functionals and functional differential equations publication-title: Phys. Rep. – volume: 14 start-page: 601 year: 2014 end-page: 633 ident: CR12 article-title: High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs publication-title: Found. Comput. Math. – volume: 19 start-page: 159 issue: 1 year: 2019 end-page: 204 ident: CR21 article-title: On the Dirac–Frenkel variational principle on tensor Banach spaces publication-title: Found. Comput. Math. – volume: 38 start-page: A2405 issue: 4 year: 2016 end-page: A2439 ident: CR6 article-title: Spectral tensor-train decomposition publication-title: SIAM J. Sci. Comput. – year: 1988 ident: CR11 publication-title: The Boltzmann Equation and Its Applications – volume: 79 start-page: 1346 issue: 4 year: 2008 end-page: 1357 ident: CR43 article-title: Dynamical low-rank approximation: applications and numerical experiments publication-title: Math. Comput. Simul. – year: 2012 ident: CR28 publication-title: Tensor Spaces and Numerical Tensor Calculus – ident: CR30 – volume: 31 start-page: 2360 issue: 5 year: 2010 end-page: 2375 ident: CR35 article-title: Dynamical tensor approximation publication-title: SIAM J. Matrix Anal. Appl. – volume: 6 start-page: 256 year: 2009 end-page: 273 ident: CR10 article-title: ANOVA expansions and efficient sampling methods for parameter dependent nonlinear PDEs publication-title: Int. J. Numer. Anal. Model. – volume: 99 start-page: 10246 issue: 16 year: 2002 end-page: 10251 ident: CR5 article-title: Numerical operator calculus in higher dimensions publication-title: Proc. Natl. Acad. Sci. USA – volume: 470 start-page: 1 issue: 2166 year: 2014 end-page: 20 ident: CR57 article-title: Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems publication-title: Proc. R. Soc. A – volume: 6 start-page: 1 issue: 10 year: 2019 end-page: 41 ident: CR19 article-title: A mean-field optimal control formulation of deep learning publication-title: Res. Math. Sci. – ident: CR56 – volume: 2011 start-page: 1 issue: 08466 year: 2020 end-page: 14 ident: CR22 article-title: Geometry of tree-based tensor formats in tensor banach spaces publication-title: ArXiv – volume: 40 start-page: 814 issue: 2 year: 2019 end-page: 844 ident: CR24 article-title: The extrinsic geometry of dynamical systems tracking nonlinear matrix projections publication-title: SIAM J. Matrix Anal. Appl. – volume: 344 start-page: 303 year: 2017 end-page: 319 ident: CR1 article-title: A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems publication-title: J. Comput. Phys. – ident: CR27 – volume: 468 start-page: 759 issue: 2139 year: 2012 end-page: 783 ident: CR58 article-title: A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems publication-title: Proc. R. Soc. A – volume: 319 start-page: 674 year: 2000 end-page: 678 ident: CR45 article-title: On the Dirac–Frenkel/Mclachlan variational principle publication-title: Chem. Phys. Lett. – volume: 182 start-page: 189 issue: 1 year: 2019 end-page: 210 ident: CR53 article-title: On characterizations of submanifolds via smoothness of the distance function in Hilbert spaces publication-title: J. Optim. Theory Appl. – volume: 16 start-page: 1423 issue: 6 year: 2016 ident: CR2 article-title: Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations publication-title: Found. Comput. Math. – volume: 8 start-page: 39 year: 1964 end-page: 44 ident: CR40 article-title: A variational solution of the time-dependent Schrödinger equation publication-title: Mol. Phys. – ident: CR48 – volume: 19 start-page: 73 issue: 1 year: 2019 end-page: 92 ident: CR33 article-title: Projection methods for dynamical low-rank approximation of high-dimensional problems publication-title: Comput. Methods Appl. Math. – ident: CR17 – ident: CR31 – volume: 33 start-page: 2295 issue: 5 year: 2011 end-page: 2317 ident: CR44 article-title: Tensor-train decomposition publication-title: SIAM J. Sci. Comput. – volume: 357 start-page: 125 year: 2018 end-page: 141 ident: CR46 article-title: Hidden physics models: machine learning of nonlinear partial differential equations publication-title: J. Comput. Phys. – volume: 13 start-page: 147 year: 2004 end-page: 269 ident: CR9 article-title: Sparse grids publication-title: Acta Numer. – ident: CR20 – volume: 404 start-page: 109125 year: 2020 ident: CR16 article-title: Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs publication-title: J. Comput. Phys. – volume: 2011 start-page: 1 issue: 08466 year: 2020 ident: 1539_CR22 publication-title: ArXiv – ident: 1539_CR27 doi: 10.1016/j.jcp.2019.108940 – volume: 229 start-page: 1536 year: 2010 ident: 1539_CR25 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2009.10.043 – volume-title: Tensor Spaces and Numerical Tensor Calculus year: 2012 ident: 1539_CR28 doi: 10.1007/978-3-642-28027-6 – volume: 99 start-page: 10246 issue: 16 year: 2002 ident: 1539_CR5 publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.112329799 – volume: 16 start-page: 1423 issue: 6 year: 2016 ident: 1539_CR2 publication-title: Found. Comput. Math. doi: 10.1007/s10208-016-9317-9 – volume: 12 start-page: 273 year: 2000 ident: 1539_CR4 publication-title: Adv. Comput. Mech. – volume: 6 start-page: 1 issue: 10 year: 2019 ident: 1539_CR19 publication-title: Res. Math. Sci. – volume: 43 start-page: 1207 issue: 3 year: 2008 ident: 1539_CR37 publication-title: J. Math. Chem. doi: 10.1007/s10910-007-9250-x – ident: 1539_CR29 doi: 10.1017/CBO9780511618352 – ident: 1539_CR42 doi: 10.1016/B978-0-12-576350-9.50006-0 – volume: 470 start-page: 1 issue: 2166 year: 2014 ident: 1539_CR57 publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2013.0754 – volume: 6 start-page: 256 year: 2009 ident: 1539_CR10 publication-title: Int. J. Numer. Anal. Model. – volume: 732 start-page: 1 year: 2018 ident: 1539_CR55 publication-title: Phys. Rep. doi: 10.1016/j.physrep.2017.12.003 – volume: 13 start-page: 147 year: 2004 ident: 1539_CR9 publication-title: Acta Numer. doi: 10.1017/S0962492904000182 – volume: 182 start-page: 189 issue: 1 year: 2019 ident: 1539_CR53 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-019-01473-3 – volume: 194 start-page: 92 issue: 1 year: 2004 ident: 1539_CR54 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2003.08.021 – volume: 468 start-page: 759 issue: 2139 year: 2012 ident: 1539_CR58 publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2011.0186 – volume-title: The Boltzmann Equation and Its Applications year: 1988 ident: 1539_CR11 doi: 10.1007/978-1-4612-1039-9 – volume: 52 start-page: 1128 issue: 3 year: 2014 ident: 1539_CR3 publication-title: SIAM J. Numer. Anal. doi: 10.1137/120896001 – volume: 47 start-page: 1265 issue: 5 year: 2013 ident: 1539_CR36 publication-title: ESAIM Math. Model. Numer. Anal. doi: 10.1051/m2an/2013067 – volume: 117 start-page: 9183 issue: 17 year: 2020 ident: 1539_CR52 publication-title: PNAS doi: 10.1073/pnas.1922204117 – volume: 8 start-page: 39 year: 1964 ident: 1539_CR40 publication-title: Mol. Phys. doi: 10.1080/00268976400100041 – volume: 79 start-page: 1346 issue: 4 year: 2008 ident: 1539_CR43 publication-title: Math. Comput. Simul. doi: 10.1016/j.matcom.2008.03.007 – volume: 19 start-page: 159 issue: 1 year: 2019 ident: 1539_CR21 publication-title: Found. Comput. Math. doi: 10.1007/s10208-018-9381-4 – volume: 31 start-page: 2360 issue: 5 year: 2010 ident: 1539_CR35 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/09076578X – volume: 56 start-page: 1273 issue: 3 year: 2018 ident: 1539_CR39 publication-title: SIAM J. Numer. Anal. doi: 10.1137/17M1146889 – volume: 344 start-page: 303 year: 2017 ident: 1539_CR1 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.04.057 – volume: 29 start-page: 434 issue: 2 year: 2007 ident: 1539_CR34 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/050639703 – volume: 378 start-page: 606 year: 2019 ident: 1539_CR47 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.10.045 – ident: 1539_CR50 – volume-title: Optimal Transport: Old and New year: 2009 ident: 1539_CR59 doi: 10.1007/978-3-540-71050-9 – ident: 1539_CR15 – ident: 1539_CR30 doi: 10.1098/rspa.2014.0932 – volume: 33 start-page: 2295 issue: 5 year: 2011 ident: 1539_CR44 publication-title: SIAM J. Sci. Comput. doi: 10.1137/090752286 – ident: 1539_CR49 doi: 10.1007/978-3-642-61544-3 – ident: 1539_CR17 doi: 10.1016/j.jcp.2021.110295 – volume: 54 start-page: 1020 issue: 2 year: 2016 ident: 1539_CR32 publication-title: SIAM J. Numer. Anal. doi: 10.1137/15M1026791 – volume: 38 start-page: A2405 issue: 4 year: 2016 ident: 1539_CR6 publication-title: SIAM J. Sci. Comput. doi: 10.1137/15M1036919 – volume: 14 start-page: 601 year: 2014 ident: 1539_CR12 publication-title: Found. Comput. Math. doi: 10.1007/s10208-013-9154-z – volume: 53 start-page: 917 issue: 2 year: 2015 ident: 1539_CR38 publication-title: SIAM J. Numer. Anal. doi: 10.1137/140976546 – ident: 1539_CR56 doi: 10.1007/s40687-021-00265-4 – ident: 1539_CR48 – volume: 2171 start-page: 1 issue: 470 year: 2014 ident: 1539_CR13 publication-title: Proc. R. Soc. A – ident: 1539_CR20 – volume: 36 start-page: A2952 issue: 6 year: 2014 ident: 1539_CR41 publication-title: SIAM J. Sci. Comput. doi: 10.1137/140966368 – volume: 375 start-page: 519 year: 2018 ident: 1539_CR7 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.08.057 – volume: 39 start-page: 510 issue: 1 year: 2018 ident: 1539_CR23 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/16M1095202 – ident: 1539_CR31 doi: 10.1051/proc/201448001 – volume: 357 start-page: 125 year: 2018 ident: 1539_CR46 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.11.039 – ident: 1539_CR8 doi: 10.1016/j.jcp.2020.109744 – volume: 40 start-page: 814 issue: 2 year: 2019 ident: 1539_CR24 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/18M1192780 – volume: 315 start-page: 817 year: 2016 ident: 1539_CR14 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2015.10.030 – volume: 19 start-page: 73 issue: 1 year: 2019 ident: 1539_CR33 publication-title: Comput. Methods Appl. Math. doi: 10.1515/cmam-2018-0029 – year: 2020 ident: 1539_CR51 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109341 – volume: 394 start-page: 56 year: 2019 ident: 1539_CR60 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.05.024 – volume: 404 start-page: 109125 year: 2020 ident: 1539_CR16 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109125 – volume: 319 start-page: 674 year: 2000 ident: 1539_CR45 publication-title: Chem. Phys. Lett. doi: 10.1016/S0009-2614(00)00200-1 – volume-title: Wave Mechanics: Advanced General Theory year: 1934 ident: 1539_CR26 – volume: 23 start-page: 369 year: 2014 ident: 1539_CR18 publication-title: Acta Numer. doi: 10.1017/S0962492914000063
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Snippet	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train...
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SubjectTerms	Adaptive algorithms Algorithms Approximation Computational Mathematics and Numerical Analysis Covariance matrix Decomposition Energy Fokker-Planck equation Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear differential equations Numerical integration Operators (mathematics) Partial differential equations Robustness (mathematics) Series expansion Tensors Theoretical Time integration
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Title	Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs
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