Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-24T16:19:55.960Z Has data issue: false hasContentIssue false

Low-rank tensor methods for partial differential equations

Published online by Cambridge University Press:  11 May 2023

Markus Bachmayr*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany E-mail: bachmayr@igpm.rwth-aachen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

References

Absil, P.-A. and Oseledets, I. V. (2015), Low-rank retractions: A survey and new results, Comput. Optim. Appl. 62, 529.CrossRefGoogle Scholar
Absil, P.-A., Mahony, R. and Sepulchre, R. (2008), Optimization Algorithms on Matrix Manifolds, Princeton University Press.CrossRefGoogle Scholar
Ali, M. and Nouy, A. (2020a), Singular value decomposition in Sobolev spaces, Part I, Z. Anal. Anwend. 39, 349369.CrossRefGoogle Scholar
Ali, M. and Nouy, A. (2020b), Singular value decomposition in Sobolev spaces, Part II, Z. Anal. Anwend. 39, 371394.CrossRefGoogle Scholar
Ali, M. and Nouy, A. (2021), Approximation theory of tree tensor networks: Tensorized multivariate functions. Available at arXiv:2101.11932.Google Scholar
Ali, M. and Nouy, A. (2023), Approximation theory of tree tensor networks: Tensorized univariate functions, Constr. Approx. Available at doi:10.1007/s00365-023-09620-w.CrossRefGoogle Scholar
Ali, M. and Urban, K. (2020), HT-AWGM: A hierarchical Tucker-adaptive wavelet Galerkin method for high-dimensional elliptic problems, Adv. Comput. Math. 46, 59.CrossRefGoogle Scholar
Ammar, A., Chinesta, F. and Falcó, A. (2010), On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Engrg 17, 473486.CrossRefGoogle Scholar
Andreev, R. (2013), Stability of sparse space–time finite element discretizations of linear parabolic evolution equations, IMA J. Numer. Anal. 33, 242260.CrossRefGoogle Scholar
Andreev, R. and Tobler, C. (2015), Multilevel preconditioning and low-rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs, Numer. Linear Algebra Appl 22, 317337.CrossRefGoogle Scholar
Arnold, A. and Jahnke, T. (2014), On the approximation of high-dimensional differential equations in the hierarchical Tucker format, BIT Numer. Math. 54, 305341.CrossRefGoogle Scholar
Bachmayr, M. (2012a), Adaptive low-rank wavelet methods and applications to two-electron Schrödinger equations. PhD thesis, RWTH Aachen.Google Scholar
Bachmayr, M. (2012b), Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation, ESAIM Math. Model. Numer. Anal. 46, 13371362.CrossRefGoogle Scholar
Bachmayr, M. and Cohen, A. (2017), Kolmogorov widths and low-rank approximations of parametric elliptic PDEs, Math. Comp. 86, 701724.CrossRefGoogle Scholar
Bachmayr, M. and Dahmen, W. (2015), Adaptive near-optimal rank tensor approximation for high-dimensional operator equations, Found. Comput. Math. 15, 839898.CrossRefGoogle Scholar
Bachmayr, M. and Dahmen, W. (2016a), Adaptive low-rank methods for problems on Sobolev spaces with error control in ${\mathrm{L}}_2$ , ESAIM Math. Model. Numer. Anal. 50, 11071136.CrossRefGoogle Scholar
Bachmayr, M. and Dahmen, W. (2016b), Adaptive low-rank methods: Problems on Sobolev spaces, SIAM J. Numer. Anal. 54, 744796.CrossRefGoogle Scholar
Bachmayr, M. and Dahmen, W. (2020), Adaptive low-rank approximations for operator equations: Accuracy control and computational complexity, in 75 Years of Mathematics of Computation (Brenner, S. C. et al., eds), Vol. 754 of Contemporary Mathematics, American Mathematical Society, pp. 144.CrossRefGoogle Scholar
Bachmayr, M. and Faldum, M. (2023), A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations. Available at arXiv:2302.01658.Google Scholar
Bachmayr, M. and Kazeev, V. (2020), Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs, Found. Comput. Math. 20, 11751236.CrossRefGoogle Scholar
Bachmayr, M. and Schneider, R. (2017), Iterative methods based on soft thresholding of hierarchical tensors, Found. Comput. Math. 17, 10371083.CrossRefGoogle Scholar
Bachmayr, M., Cohen, A. and Dahmen, W. (2018), Parametric PDEs: Sparse or low-rank approximations?, IMA J. Numer. Anal. 38, 16611708.CrossRefGoogle Scholar
Bachmayr, M., Cohen, A., Dũng, D. and Schwab, C. (2017), Fully discrete approximation of parametric and stochastic elliptic PDEs, SIAM J. Numer. Anal. 55, 21512186.CrossRefGoogle Scholar
Bachmayr, M., Eisenmann, H., Kieri, E. and Uschmajew, A. (2021a), Existence of dynamical low-rank approximations to parabolic problems, Math. Comp. 90, 17991830.CrossRefGoogle Scholar
Bachmayr, M., Götte, M. and Pfeffer, M. (2022), Particle number conservation and block structures in matrix product states, Calcolo 59, 24.CrossRefGoogle Scholar
Bachmayr, M., Nouy, A. and Schneider, R. (2021b), Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions. Available at arXiv:2112.01474. To appear in Pure Appl. Funct. Anal. Google Scholar
Bachmayr, M., Schneider, R. and Uschmajew, A. (2016), Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations, Found. Comput. Math. 16, 14231472.CrossRefGoogle Scholar
Ballani, J. and Grasedyck, L. (2013), A projection method to solve linear systems in tensor format, Numer. Linear Algebra Appl 20, 2743.CrossRefGoogle Scholar
Ballani, J. and Grasedyck, L. (2015), Hierarchical tensor approximation of output quantities of parameter-dependent PDEs, SIAM/ASA J. Uncertain. Quantif. 3, 852872.CrossRefGoogle Scholar
Ballani, J., Grasedyck, L. and Kluge, M. (2013), Black box approximation of tensors in hierarchical Tucker format, Linear Algebra Appl. 438, 639657.CrossRefGoogle Scholar
Bardos, C., Catto, I., Mauser, N. and Trabelsi, S. (2010), Setting and analysis of the multi-configuration time-dependent Hartree–Fock equations, Arch. Ration. Mech. Anal. 198, 273330.CrossRefGoogle Scholar
Bardos, C., Catto, I., Mauser, N. J. and Trabelsi, S. (2009), Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree–Fock equations: A sufficient condition, Appl. Math. Lett. 22, 147152.CrossRefGoogle Scholar
Benedikt, U., Auer, A. A., Espig, M. and Hackbusch, W. (2011), Tensor decomposition in post-Hartree–Fock methods, I: Two-electron integrals and MP2, J. Chem. Phys. 134, 054118.CrossRefGoogle ScholarPubMed
Benedikt, U., Auer, H., Espig, M., Hackbusch, W. and Auer, A. A. (2013a), Tensor representation techniques in post-Hartree–Fock methods: Matrix product state tensor format, Molecular Phys. 111, 23982413.CrossRefGoogle Scholar
Benedikt, U., Böhm, K.-H. and Auer, A. A. (2013b), Tensor decomposition in post-Hartree–Fock methods, II: CCD implementation, J. Chem. Phys. 139, 224101.CrossRefGoogle ScholarPubMed
Benner, P., Cohen, A., Ohlberger, M. and Willcox, K., eds (2017), Model Reduction and Approximation: Theory and Algorithms, Vol. 15 of Computational Science & Engineering, SIAM.CrossRefGoogle Scholar
Beylkin, G. and Mohlenkamp, M. J. (2002), Numerical operator calculus in higher dimensions, Proc. Natl. Acad. Sci. USA 99, 1024610251.CrossRefGoogle ScholarPubMed
Beylkin, G. and Mohlenkamp, M. J. (2005), Algorithms for numerical analysis in high dimensions, SIAM J. Sci. Comput. 26, 21332159.CrossRefGoogle Scholar
Beylkin, G. and Monzón, L. (2005), On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19, 1748.CrossRefGoogle Scholar
Beylkin, G. and Monzón, L. (2010), Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal. 28, 131149.CrossRefGoogle Scholar
Beylkin, G., Mohlenkamp, M. J. and Pérez, F. (2008), Approximating a wavefunction as an unconstrained sum of Slater determinants, J. Math. Phys. 49, 032107.CrossRefGoogle Scholar
Bigoni, D., Engsig-Karup, A. P. and Marzouk, Y. M. (2016), Spectral tensor-train decomposition, SIAM J. Sci. Comput. 38, A2405A2439.CrossRefGoogle Scholar
Billaud-Friess, M., Nouy, A. and Zahm, O. (2014), A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems, ESAIM Math. Model. Numer. Anal. 48, 17771806.CrossRefGoogle Scholar
Bischoff, F. A. and Valeev, E. F. (2011), Low-order tensor approximations for electronic wave functions: Hartree–Fock method with guaranteed precision, J. Chem. Phys. 134, 104104.CrossRefGoogle ScholarPubMed
Boiveau, T., Ehrlacher, V., Ern, A. and Nouy, A. (2019), Low-rank approximation of linear parabolic equations by space–time tensor Galerkin methods, ESAIM Math. Model. Numer. Anal. 53, 635658.CrossRefGoogle Scholar
Braess, D. (1986), Nonlinear Approximation Theory, Vol. 7 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Braess, D. and Hackbusch, W. (2005), Approximation of $1/x$ by exponential sums in $\left[1,\infty \right)$ , IMA J. Numer. Anal. 25, 685697.Google Scholar
Braess, D. and Hackbusch, W. (2009), On the efficient computation of high-dimensional integrals and the approximation by exponential sums, in Multiscale, Nonlinear and Adaptive Approximation (DeVore, R. and Kunoth, A., eds), Springer, pp. 3974.CrossRefGoogle Scholar
Bramble, J. H., Pasciak, J. E. and Xu, J. (1990), Parallel multilevel preconditioners, Math. Comp. 55, 122.CrossRefGoogle Scholar
Cai, J.-F., Candès, E. J. and Shen, Z. (2010), A singular value thresholding algorithm for matrix completion, SIAM J. Optim. 20, 19561982.CrossRefGoogle Scholar
Cancès, E., Ehrlacher, V. and Lelièvre, T. (2011), Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, Math. Models Methods Appl. Sci. 21, 24332467.CrossRefGoogle Scholar
Cancès, E., Ehrlacher, V. and Lelièvre, T. (2013), Greedy algorithms for high-dimensional non-symmetric linear problems, ESAIM Proc. 41, 95131.CrossRefGoogle Scholar
Cancès, E., Ehrlacher, V. and Lelièvre, T. (2014), Greedy algorithms for high-dimensional eigenvalue problems, Constr. Approx. 40, 387423.CrossRefGoogle Scholar
Ceruti, G. and Lubich, C. (2020), Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors, BIT Numer. Math. 60, 591614.CrossRefGoogle Scholar
Ceruti, G. and Lubich, C. (2022), An unconventional robust integrator for dynamical low-rank approximation, BIT Numer. Math. 62, 2344.CrossRefGoogle Scholar
Ceruti, G., Kusch, J. and Lubich, C. (2022), A rank-adaptive robust integrator for dynamical low-rank approximation, BIT Numer. Math. 62, 11491174.CrossRefGoogle Scholar
Ceruti, G., Lubich, C. and Sulz, D. (2023), Rank-adaptive time integration of tree tensor networks, SIAM J. Numer. Anal. 61, 194222.CrossRefGoogle Scholar
Chen, Z., Batselier, K., Suykens, J. A. K. and Wong, N. (2017), Parallelized tensor train learning of polynomial classifiers, IEEE Trans. Neural Netw. Learn. Syst. 29, 46214632.CrossRefGoogle Scholar
Chertkov, A. and Oseledets, I. V. (2021), Solution of the Fokker–Planck equation by cross approximation method in the tensor train format, Front. Artif. Intell. 4, 668215.CrossRefGoogle ScholarPubMed
Chinnamsetty, S. R., Espig, M., Khoromskij, B. N., Hackbusch, W. and Flad, H.-J. (2007), Tensor product approximation with optimal rank in quantum chemistry, J. Chem. Phys. 127, 084110.CrossRefGoogle ScholarPubMed
Cho, H., Venturi, D. and Karniadakis, G. E. (2016), Numerical methods for high-dimensional probability density function equations, J. Comput. Phys. 305, 817837.CrossRefGoogle Scholar
Cichocki, A., Mandic, D., De Lathauwer, L., Zhou, G., Zhao, Q., Caiafa, C. and Phan, H. A. (2015), Tensor decompositions for signal processing applications: From two-way to multiway component analysis, IEEE Signal Process. Mag. 32, 145163.CrossRefGoogle Scholar
Cohen, A. (2003), Numerical Analysis of Wavelet Methods, Vol. 32 of Studies in Mathematics and its Applications, North-Holland.Google Scholar
Cohen, A. and DeVore, R. (2015), Approximation of high-dimensional parametric PDEs, Acta Numer. 24, 1159.CrossRefGoogle Scholar
Cohen, A., Dahmen, W. and DeVore, R. (2001), Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp. 70, 2775.CrossRefGoogle Scholar
Cohen, A., Dahmen, W. and DeVore, R. (2002), Adaptive wavelet methods, II: Beyond the elliptic case, Found. Comput. Math. 2, 203245.CrossRefGoogle Scholar
Cohen, A., Daubechies, I. and Feauveau, J.-C. (1992), Biorthogonal bases of compactly supported wavelets, Commun. Pure Appl. Math. 45, 485560.CrossRefGoogle Scholar
Cohen, A., DeVore, R. and Schwab, C. (2010), Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs, Found. Comput. Math. 10, 615646.Google Scholar
Cohen, A., DeVore, R. and Schwab, C. (2011), Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s, Anal. Appl. (Singap.) 9, 1147.CrossRefGoogle Scholar
Conte, D. (2020), Dynamical low-rank approximation to the solution of parabolic differential equations, Appl. Numer. Math. 156, 377384.CrossRefGoogle Scholar
Conte, D. and Lubich, C. (2010), An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics, M2AN Math. Model. Numer. Anal. 44, 759780.CrossRefGoogle Scholar
Crosswhite, G. M. and Bacon, D. (2008), Finite automata for caching in matrix product algorithms, Phys. Rev. A 78, 012356.CrossRefGoogle Scholar
Da Silva, C. and Herrmann, F. J. (2015), Optimization on the hierarchical Tucker manifold: Applications to tensor completion, Linear Algebra Appl. 481, 131173.CrossRefGoogle Scholar
Dahlke, S., Fornasier, M. and Raasch, T. (2012), Multilevel preconditioning and adaptive sparse solution of inverse problems, Math. Comp. 81, 419446.CrossRefGoogle Scholar
Dahmen, W. (1997), Wavelet and multiscale methods for operator equations, Acta Numer. 6, 55228.CrossRefGoogle Scholar
Dahmen, W., DeVore, R., Grasedyck, L. and Süli, E. (2016), Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations, Found. Comput. Math. 16, 813874.CrossRefGoogle Scholar
Dahmen, W., Kunoth, A. and Urban, K. (1999), Biorthogonal spline wavelets on the interval: Stability and moment conditions, Appl. Comput. Harmon. Anal. 6, 132196.CrossRefGoogle Scholar
Daubechies, I. (1988), Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math. 41, 909996.CrossRefGoogle Scholar
Daubechies, I. (1992), Ten Lectures on Wavelets, Vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.CrossRefGoogle Scholar
De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000), A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl. 21, 12531278.CrossRefGoogle Scholar
De Launey, W. and Seberry, J. (1994), The strong Kronecker product, J. Combin. Theory Ser. A 66, 192213.CrossRefGoogle Scholar
de Silva, V. and Lim, L.-H. (2008), Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl. 30, 10841127.CrossRefGoogle Scholar
Defant, A. and Floret, K. (1993), Tensor Norms and Operator Ideals, Vol. 176 of North-Holland Mathematics Studies, North-Holland.Google Scholar
Dektor, A., Rodgers, A. and Venturi, D. (2021), Rank-adaptive tensor methods for highdimensional nonlinear pdes, J. Sci. Comput. 88, 36.CrossRefGoogle Scholar
DeVore, R. A. (1998), Nonlinear approximation, Acta Numer. 7, 51150.CrossRefGoogle Scholar
Dijkema, T. J., Schwab, C. and Stevenson, R. (2009), An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constr. Approx. 30, 423455.CrossRefGoogle Scholar
Dirac, P. A. M. (1930), Note on exchange phenomena in the Thomas atom, Math. Proc. Camb. Phil. Soc. 26, 376385.CrossRefGoogle Scholar
Dolgov, S. V. and Khoromskij, B. N. (2013), Two-level QTT-Tucker format for optimized tensor calculus, SIAM J. Matrix Anal. Appl. 34, 593623.CrossRefGoogle Scholar
Dolgov, S. V. and Savostyanov, D. V. (2014), Alternating minimal energy methods for linear systems in higher dimensions, SIAM J. Sci. Comput. 36, A2248A2271.CrossRefGoogle Scholar
Dolgov, S. V. and Scheichl, R. (2019), A hybrid alternating least squares-TT-cross algorithm for parametric PDEs, SIAM/ASA J. Uncertain. Quantif. 7, 260291.CrossRefGoogle Scholar
Dolgov, S. V. and Vejchodský, T. (2021), Guaranteed a posteriori error bounds for low-rank tensor approximate solutions, IMA J. Numer. Anal. 41, 12401266.CrossRefGoogle Scholar
Dolgov, S. V., Khoromskij, B. N. and Oseledets, I. V. (2012), Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the Fokker–Planck equation, SIAM J. Sci. Comput. 34, A3016A3038.CrossRefGoogle Scholar
Dolgov, S. V., Khoromskij, B. N., Oseledets, I. V. and Savostyanov, D. V. (2014), Computation of extreme eigenvalues in higher dimensions using block tensor train format, Comput. Phys. Commun. 185, 12071216.CrossRefGoogle Scholar
Dolgov, S. V., Kressner, D. and Strössner, C. (2021), Functional Tucker approximation using Chebyshev interpolation, SIAM J. Sci. Comput. 43, A2190A2210.CrossRefGoogle Scholar
Dölz, J., Egger, H. and Schlottbom, M. (2021), A model reduction approach for inverse problems with operator valued data, Numer. Math 148, 889917.CrossRefGoogle Scholar
Donovan, G. C., Geronimo, J. S. and Hardin, D. P. (1996), Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets, SIAM J. Math. Anal. 27, 17911815.CrossRefGoogle Scholar
Donovan, G. C., Geronimo, J. S. and Hardin, D. P. (1999), Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets, SIAM J. Math. Anal. 30, 10291056.CrossRefGoogle Scholar
Dunnett, A. J. and Chin, A. W. (2021), Efficient bond-adaptive approach for finite-temperature open quantum dynamics using the one-site time-dependent variational principle for matrix product states, Phys. Rev. B 104, 214302.CrossRefGoogle Scholar
Eckart, C. and Young, G. (1936), The approximation of one matrix by another of lower rank, Psychometrika 1, 211218.CrossRefGoogle Scholar
Eigel, M., Marschall, M., Pfeffer, M. and Schneider, R. (2020), Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations, Numer. Math 145, 655692.CrossRefGoogle Scholar
Eigel, M., Pfeffer, M. and Schneider, R. (2017), Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Numer. Math 136, 765803.CrossRefGoogle Scholar
Einkemmer, L. and Lubich, C. (2018), A low-rank projector-splitting integrator for the Vlasov–Poisson equation, SIAM J. Sci. Comput. 40, B1330B1360.CrossRefGoogle Scholar
Einkemmer, L. and Lubich, C. (2019), A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput. 41, B1061B1081.CrossRefGoogle Scholar
Eisert, J., Cramer, M. and Plenio, M. B. (2010), Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277306.CrossRefGoogle Scholar
Fackeldey, K., Oster, M., Sallandt, L. and Schneider, R. (2022), Approximative policy iteration for exit time feedback control problems driven by stochastic differential equations using tensor train format, Multiscale Model. Simul 20, 379403.CrossRefGoogle Scholar
Falcó, A. and Hackbusch, W. (2012), On minimal subspaces in tensor representations, Found. Comput. Math 12, 765803.CrossRefGoogle Scholar
Falcó, A. and Nouy, A. (2011), A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach, J. Math. Anal. Appl. 376, 469480.CrossRefGoogle Scholar
Falcó, A. and Nouy, A. (2012), Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numer. Math 121, 503530.CrossRefGoogle Scholar
Falcó, A., Hackbusch, W. and Nouy, A. (2019), On the Dirac–Frenkel variational principle on tensor Banach spaces, Found. Comput. Math. 19, 159204.CrossRefGoogle Scholar
Falcó, A., Hackbusch, W. and Nouy, A. (2021), Tree-based tensor formats, SeMA J. 78, 159173.CrossRefGoogle Scholar
Faulstich, F. M., Laestadius, A., Legeza, O., Schneider, R. and Kvaal, S. (2019), Analysis of the tailored coupled-cluster method in quantum chemistry, SIAM J. Numer. Anal. 57, 25792607.CrossRefGoogle Scholar
Feppon, F. and Lermusiaux, P. F. J. (2018), A geometric approach to dynamical model order reduction, SIAM J. Matrix Anal. Appl. 39, 510538.CrossRefGoogle Scholar
Feppon, F. and Lermusiaux, P. F. J. (2019), The extrinsic geometry of dynamical systems tracking nonlinear matrix projections, SIAM J. Matrix Anal. Appl. 40, 814844.CrossRefGoogle Scholar
Frenkel, J. (1934), Wave Mechanics: Advanced General Theory, Clarendon Press.Google Scholar
Friesecke, G. and Graswald, B. R. (2022), Two-electron wavefunctions are matrix product states with bond dimension three, J. Math. Phys. 63, 091901.CrossRefGoogle Scholar
Friesecke, G., Graswald, B. R. and Legeza, Ö. (2022), Exact matrix product state representation and convergence of a fully correlated electronic wavefunction in the infinite-basis limit, Phys. Rev. B 105, 165144.CrossRefGoogle Scholar
Gantumur, T., Harbrecht, H. and Stevenson, R. (2007), An optimal adaptive wavelet method without coarsening of the iterands, Math. Comp. 76, 615629.CrossRefGoogle Scholar
Gao, B. and Absil, P.-A. (2022), A Riemannian rank-adaptive method for low-rank matrix completion, Comput. Optim. Appl. 81, 6790.CrossRefGoogle Scholar
Gavrilyuk, I. and Khoromskij, B. N. (2019), Quasi-optimal rank-structured approximation to multidimensional parabolic problems by Cayley transform and Chebyshev interpolation, Comput. Methods Appl. Math 19, 5571.CrossRefGoogle Scholar
Gavrilyuk, I. P., Hackbusch, W. and Khoromskij, B. N. (2005), Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing 74, 131157.CrossRefGoogle Scholar
Golub, G. H. and Kahan, W. (1965), Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2, 205224.CrossRefGoogle Scholar
Golub, G. H. and Reinsch, C. (1970), Handbook Series Linear Algebra: Singular value decomposition and least squares solutions, Numer. Math 14, 403420.CrossRefGoogle Scholar
Gorodetsky, A., Karaman, S. and Marzouk, Y. (2019), A continuous analogue of the tensor-train decomposition, Comput. Methods Appl. Mech. Engrg 347, 5984.CrossRefGoogle Scholar
Grasedyck, L. (2004), Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing 72, 247265.CrossRefGoogle Scholar
Grasedyck, L. (2009/10), Hierarchical singular value decomposition of tensors, SIAM J. Matrix Anal. Appl. 31, 20292054.CrossRefGoogle Scholar
Grasedyck, L. (2010), Polynomial approximation in hierarchical Tucker format by vector-tensorization. DFG SPP 1324 Preprint 43.Google Scholar
Grasedyck, L. and Hackbusch, W. (2011), An introduction to hierarchical ( $\mathrm{\mathscr{H}}$ -) rank and TT-rank of tensors with examples, Comput. Methods Appl. Math. 11, 291304.Google Scholar
Grasedyck, L., Kressner, D. and Tobler, C. (2013), A literature survey of low-rank tensor approximation techniques, GAMM-Mitt. 36, 5378.CrossRefGoogle Scholar
Graswald, B. R. and Friesecke, G. (2021), Electronic wavefunction with maximally entangled mps representation, Europ. Phys. J. D 75, 14.CrossRefGoogle Scholar
Griebel, M. and Harbrecht, H. (2014), Approximation of bi-variate functions: Singular value decomposition versus sparse grids, IMA J. Numer. Anal. 34, 2854.CrossRefGoogle Scholar
Griebel, M. and Harbrecht, H. (2019), Singular value decomposition versus sparse grids: Refined complexity estimates, IMA J. Numer. Anal. 39, 16521671.CrossRefGoogle Scholar
Griebel, M. and Harbrecht, H. (2023), Analysis of tensor approximation schemes for continuous functions, Found. Comput. Math 23, 219240.CrossRefGoogle Scholar
Griebel, M., Harbrecht, H. and Schneider, R. (2022), Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness. Available at arXiv:2203.04100.Google Scholar
Gu, M. and Eisenstat, S. C. (1995), A divide-and-conquer algorithm for the bidiagonal SVD, SIAM J. Matrix Anal. Appl. 16, 7992.CrossRefGoogle Scholar
Hackbusch, W. (2005), Entwicklungen nach Exponentialsummen. Technical Report 4, MPI MIS Leipzig.Google Scholar
Hackbusch, W. (2011), Tensorisation of vectors and their efficient convolution, Numer. Math. 119, 465488.CrossRefGoogle Scholar
Hackbusch, W. (2014), Numerical tensor calculus, Acta Numer. 23, 651742.CrossRefGoogle Scholar
Hackbusch, W. (2015a), Hierarchical Matrices: Algorithms and Analysis, Vol. 49 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Hackbusch, W. (2015b), Solution of linear systems in high spatial dimensions, Comput. Vis. Sci. 17, 111118.CrossRefGoogle Scholar
Hackbusch, W. (2018), On the representation of symmetric and antisymmetric tensors, in Contemporary Computational Mathematics: A Celebration of the 80th Birthday of Ian Sloan (Dick, J. et al., eds), Springer, pp. 483515.Google Scholar
Hackbusch, W. (2019), Tensor Spaces and Numerical Tensor Calculus, Vol. 56 of Springer Series in Computational Mathematics, second edition, Springer.CrossRefGoogle Scholar
Hackbusch, W. and Khoromskij, B. N. (2006a), Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators, I: Separable approximation of multi-variate functions, Computing 76, 177202.CrossRefGoogle Scholar
Hackbusch, W. and Khoromskij, B. N. (2006b), Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators, II: HKT representation of certain operators, Computing 76, 203225.CrossRefGoogle Scholar
Hackbusch, W. and Kühn, S. (2009), A new scheme for the tensor representation, J. Fourier Anal. Appl. 15, 706722.CrossRefGoogle Scholar
Hackbusch, W., Khoromskij, B. N. and Tyrtyshnikov, E. E. (2008), Approximate iterations for structured matrices, Numer. Math 109, 365383.CrossRefGoogle Scholar
Hackbusch, W., Khoromskij, B. N., Sauter, S. and Tyrtyshnikov, E. E. (2012), Use of tensor formats in elliptic eigenvalue problems, Numer. Linear Algebra Appl. 19, 133151.CrossRefGoogle Scholar
Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. and Verstraete, F. (2016), Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116.CrossRefGoogle Scholar
Haegeman, J., Osborne, T. J. and Verstraete, F. (2013), Post-matrix product state methods: To tangent space and beyond, Phys. Rev. B 88, 075133.CrossRefGoogle Scholar
Helgaker, T., Jørgensen, P. and Olsen, J. (2000), Molecular Electronic-Structure Theory, Wiley.CrossRefGoogle Scholar
Helmke, U. and Shayman, M. A. (1995), Critical points of matrix least squares distance functions, Linear Algebra Appl. 215, 119.CrossRefGoogle Scholar
Hesthaven, J. S., Rozza, G. and Stamm, B. (2016), Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer; BCAM Basque Center for Applied Mathematics, Bilbao.CrossRefGoogle Scholar
Hillar, C. J. and Lim, L.-H. (2013), Most tensor problems are NP-hard, J. Assoc. Comput. Mach. 60, Art. 45, 39.CrossRefGoogle Scholar
Hitchcock, F. L. (1927), The expression of a tensor or a polyadic as a sum of products, J. Math. Phys. 6, 164189.CrossRefGoogle Scholar
Hitchcock, F. L. (1928), Multiple invariants and generalized rank of a p-way matrix or tensor, J. Math. Phys. 7, 3979.Google Scholar
Holtz, S., Rohwedder, T. and Schneider, R. (2012a), The alternating linear scheme for tensor optimization in the tensor train format, SIAM J. Sci. Comput. 34, A683A713.CrossRefGoogle Scholar
Holtz, S., Rohwedder, T. and Schneider, R. (2012b), On manifolds of tensors of fixed TT-rank, Numer. Math 120, 701731.CrossRefGoogle Scholar
Janzamin, M., Ge, R., Kossaifi, J., Anandkumar, A. et al. (2019), Spectral learning on matrices and tensors, Found. Trends Mach. Learn. 12, 393536.CrossRefGoogle Scholar
Kahlbacher, M. and Volkwein, S. (2007), Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems, Discuss. Math. Differ. Incl. Control Optim 27, 95117.CrossRefGoogle Scholar
Kazashi, Y. and Nobile, F. (2021), Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval, Stoch. Partial Differ. Equ. Anal. Comput 9, 603629.Google ScholarPubMed
Kazashi, Y., Nobile, F. and Vidličková, E. (2021), Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations, Numer. Math 149, 9731024.CrossRefGoogle Scholar
Kazeev, V. A. (2015), Quantized tensor structured finite elements for second-order elliptic PDEs in two dimensions. PhD thesis, ETH Zürich.Google Scholar
Kazeev, V. A. and Khoromskij, B. N. (2012), Low-rank explicit QTT representation of the Laplace operator and its inverse, SIAM J. Matrix Anal. Appl. 33, 742758.CrossRefGoogle Scholar
Kazeev, V. A. and Schwab, C. (2018), Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions, Numer. Math. 138, 133190.CrossRefGoogle Scholar
Kazeev, V. A., Khoromskij, B. N. and Tyrtyshnikov, E. E. (2013a), Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity, SIAM J. Sci. Comput. 35, A1511A1536.CrossRefGoogle Scholar
Kazeev, V. A., Oseledets, I. V., Rakhuba, M. and Schwab, C. (2017), QTT-finite-element approximation for multiscale problems, I: Model problems in one dimension, Adv. Comput. Math. 43, 411442.CrossRefGoogle Scholar
Kazeev, V. A., Oseledets, I. V., Rakhuba, M. V. and Schwab, C. (2022), Quantized tensor FEM for multiscale problems: Diffusion problems in two and three dimensions, Multiscale Model. Simul 20, 893935.CrossRefGoogle Scholar
Kazeev, V. A., Reichmann, O. and Schwab, C. (2013b), Low-rank tensor structure of linear diffusion operators in the TT and QTT formats, Linear Algebra Appl. 438, 42044221.CrossRefGoogle Scholar
Keller, S., Dolfi, M., Troyer, M. and Reiher, M. (2015), An efficient matrix product operator representation of the quantum chemical Hamiltonian, J. Chem. Phys. 143, 244118.CrossRefGoogle ScholarPubMed
Khoromskaia, V. and Khoromskij, B. N. (2015), Tensor numerical methods in quantum chemistry: From Hartree–Fock to excitation energies, Phys. Chem. Chem. Phys. 17, 3149131509.CrossRefGoogle ScholarPubMed
Khoromskaia, V. and Khoromskij, B. N. (2016), Fast tensor method for summation of long-range potentials on 3D lattices with defects, Numer. Linear Algebra Appl 23, 249271.CrossRefGoogle Scholar
Khoromskaia, V. and Khoromskij, B. N. (2018), Tensor Numerical Methods in Quantum Chemistry, De Gruyter.CrossRefGoogle Scholar
Khoromskaia, V., Khoromskij, B. N. and Schneider, R. (2013), Tensor-structured factorized calculation of two-electron integrals in a general basis, SIAM J. Sci. Comput. 35, A987A1010.CrossRefGoogle Scholar
Khoromskij, B. N. (2009), Tensor-structured preconditioners and approximate inverse of elliptic operators in ${R}^d$ , Constr. Approx 30, 599620.CrossRefGoogle Scholar
Khoromskij, B. N. (2011), $O\left(d\log N\right)$ -quantics approximation of N-d tensors in high-dimensional numerical modeling, Constr. Approx. 34, 257280.CrossRefGoogle Scholar
Khoromskij, B. N. (2015), Tensor numerical methods for multidimensional PDEs: Theoretical analysis and initial applications, ESAIM Proc. Surveys 48, 128.CrossRefGoogle Scholar
Khoromskij, B. N. (2018), Tensor Numerical Methods in Scientific Computing, Vol. 19 of Radon Series on Computational and Applied Mathematics, De Gruyter.CrossRefGoogle Scholar
Khoromskij, B. N. and Miao, S. (2014), Superfast wavelet transform using quantics-TT approximation, I: Application to Haar wavelets, Comput. Methods Appl. Math. 14, 537553.CrossRefGoogle Scholar
Khoromskij, B. N. and Oseledets, I. V. (2010), Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs, Comput. Methods Appl. Math 10, 376394.CrossRefGoogle Scholar
Khoromskij, B. N. and Oseledets, I. V. (2011), QTT approximation of elliptic solution operators in higher dimensions, Russian J. Numer. Anal. Math. Modelling 26, 303322.CrossRefGoogle Scholar
Khoromskij, B. N. and Schwab, C. (2011), Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs, SIAM J. Sci. Comput. 33, 364385.CrossRefGoogle Scholar
Khoromskij, B. N., Khoromskaia, V. and Flad, H.-J. (2011), Numerical solution of the Hartree–Fock equation in multilevel tensor-structured format, SIAM J. Sci. Comput. 33, 4565.CrossRefGoogle Scholar
Khoromskij, B. N., Khoromskaia, V., Chinnamsetty, S. R. and Flad, H.-J. (2009), Tensor decomposition in electronic structure calculations on 3D Cartesian grids, J. Comput. Phys. 228, 57495762.CrossRefGoogle Scholar
Kieri, E. and Vandereycken, B. (2019), Projection methods for dynamical low-rank approximation of high-dimensional problems, Comput. Methods Appl. Math 19, 7392.CrossRefGoogle Scholar
Kieri, E., Lubich, C. and Walach, H. (2016), Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal. 54, 10201038.CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2007a), Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl. 29, 434454.CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2007b), Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics, M2AN Math. Model. Numer. Anal. 41, 315331.CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2010), Dynamical tensor approximation, SIAM J. Matrix Anal. Appl. 31, 23602375.CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2011), Variational-splitting time integration of the multi-configuration time-dependent Hartree–Fock equations in electron dynamics, IMA J. Numer. Anal. 31, 379395.CrossRefGoogle Scholar
Kolda, T. G. and Bader, B. W. (2009), Tensor decompositions and applications, SIAM Rev. 51, 455500.CrossRefGoogle Scholar
Kressner, D. and Tobler, C. (2011a), Low-rank tensor Krylov subspace methods for parametrized linear systems, SIAM J. Matrix Anal. Appl. 32, 12881316.CrossRefGoogle Scholar
Kressner, D. and Tobler, C. (2011b), Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems, Comput. Methods Appl. Math 11, 363381.CrossRefGoogle Scholar
Kressner, D. and Uschmajew, A. (2016), On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems, Linear Algebra Appl. 493, 556572.CrossRefGoogle Scholar
Kressner, D., Steinlechner, M. and Uschmajew, A. (2014a), Low-rank tensor methods with subspace correction for symmetric eigenvalue problems, SIAM J. Sci. Comput. 36, A2346A2368.CrossRefGoogle Scholar
Kressner, D., Steinlechner, M. and Vandereycken, B. (2014b), Low-rank tensor completion by Riemannian optimization, BIT Numer. Math. 54, 447468.CrossRefGoogle Scholar
Kressner, D., Steinlechner, M. and Vandereycken, B. (2016), Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure, SIAM J. Sci. Comput. 38, A2018A2044.CrossRefGoogle Scholar
Krumnow, C., Pfeffer, M. and Uschmajew, A. (2021), Computing eigenspaces with low rank constraints, SIAM J. Sci. Comput. 43, A586A608.CrossRefGoogle Scholar
Krumnow, C., Veis, L., Legeza, O. and Eisert, J. (2016), Fermionic orbital optimization in tensor network states, Phys. Rev. Lett. 117, 210402.CrossRefGoogle ScholarPubMed
Kühn, S. (2012), Hierarchische Tensordarstellung. PhD thesis, Universität Leipzig.Google Scholar
Landsberg, J. M. (2012), Tensors: Geometry and Applications, American Mathematical Society.Google Scholar
Landsberg, J. M., Qi, Y. and Ye, K. (2012), On the geometry of tensor network states, Quantum Inf. Comput 12, 346354.Google Scholar
Lee, K. and Elman, H. C. (2017), A preconditioned low-rank projection method with a rank-reduction scheme for stochastic partial differential equations, SIAM J. Sci. Comput. 39, S828S850.CrossRefGoogle Scholar
Light, W. A. and Cheney, E. W. (1985), Approximation Theory in Tensor Product Spaces, Vol. 1169 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Lim, L.-H. (2021), Tensors in computations, Acta Numer. 30, 555764.CrossRefGoogle Scholar
Lubich, C. (2005), On variational approximations in quantum molecular dynamics, Math. Comp. 74, 765779.CrossRefGoogle Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS).CrossRefGoogle Scholar
Lubich, C. (2015), Time integration in the multiconfiguration time-dependent Hartree method of molecular quantum dynamics, Appl. Math. Res. Express. AMRX 2015, 311328.CrossRefGoogle Scholar
Lubich, C. and Oseledets, I. V. (2014), A projector-splitting integrator for dynamical low-rank approximation, BIT Numer. Math. 54, 171188.CrossRefGoogle Scholar
Lubich, C., Oseledets, I. V. and Vandereycken, B. (2015), Time integration of tensor trains, SIAM J. Numer. Anal. 53, 917941.CrossRefGoogle Scholar
Lubich, C., Rohwedder, T., Schneider, R. and Vandereycken, B. (2013), Dynamical approximation by hierarchical Tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl. 34, 470494.CrossRefGoogle Scholar
Marcati, C., Rakhuba, M. and Schwab, C. (2022a), Tensor rank bounds for point singularities in ${R}^3$ , Adv. Comput. Math. 48, 18.CrossRefGoogle Scholar
Marcati, C., Rakhuba, M. and Ulander, J. E. M. (2022b), Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension, Calcolo 59, 2.CrossRefGoogle Scholar
Markeeva, L., Tsybulin, I. and Oseledets, I. V. (2021), QTT-isogeometric solver in two dimensions, J. Comput. Phys. 424, 109835.CrossRefGoogle Scholar
Matthies, H. G. and Zander, E. (2012), Solving stochastic systems with low-rank tensor compression, Linear Algebra Appl. 436, 38193838.CrossRefGoogle Scholar
Meyer, H.-D., Gatti, F. and Worth, G. A. (2009), Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Wiley.CrossRefGoogle Scholar
Meyer, H.-D., Manthe, U. and Cederbaum, L. S. (1990), The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett. 165, 7378.CrossRefGoogle Scholar
Michel, B. and Nouy, A. (2022), Learning with tree tensor networks: Complexity estimates and model selection, Bernoulli 28, 910936.CrossRefGoogle Scholar
Mohlenkamp, M. J. (2010), A center-of-mass principle for the multiparticle Schrödinger equation, J. Math. Phys. 51, 022112.CrossRefGoogle Scholar
Moreau, J.-J. (1965), Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93, 273299.CrossRefGoogle Scholar
Murg, V., Verstraete, F., Legeza, O. and Noack, R. M. (2010), Simulating strongly correlated quantum systems with tree tensor networks, Phys. Rev. B 82, 205105.CrossRefGoogle Scholar
Musharbash, E. and Nobile, F. (2018), Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, J. Comput. Phys. 354, 135162.CrossRefGoogle Scholar
Musharbash, E., Nobile, F. and Vidličková, E. (2020), Symplectic dynamical low rank approximation of wave equations with random parameters, BIT Numer. Math. 60, 11531201.CrossRefGoogle Scholar
Musharbash, E., Nobile, F. and Zhou, T. (2015), Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comput. 37, A776A810.CrossRefGoogle Scholar
Novikov, A., Trofimov, M. and Oseledets, I. (2018), Exponential machines, Bull. Pol. Acad. Sci. Math 66, 789797.Google Scholar
Orús, R. (2014), A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117158.CrossRefGoogle Scholar
Oseledets, I. V. (2009a), Approximation of matrices with logarithmic number of parameters, Dokl. Math. 80, 653654.CrossRefGoogle Scholar
Oseledets, I. V. (2009b), On a new tensor decomposition, Dokl. Akad. Nauk 427, 168169. In Russian; English translation in Dokl. Math. 80, 495–496 (2009).Google Scholar
Oseledets, I. V. (2011a), DMRG approach to fast linear algebra in the TT-format, Comput. Methods Appl. Math 11, 382393.CrossRefGoogle Scholar
Oseledets, I. V. (2011b), Tensor-train decomposition, SIAM J. Sci. Comput. 33, 22952317.CrossRefGoogle Scholar
Oseledets, I. V. (2013), Constructive representation of functions in low-rank tensor formats, Constr. Approx. 37, 118.CrossRefGoogle Scholar
Oseledets, I. V. and Dolgov, S. V. (2012), Solution of linear systems and matrix inversion in the TT-format, SIAM J. Sci. Comput. 34, A2718A2739.CrossRefGoogle Scholar
Oseledets, I. V. and Tyrtyshnikov, E. E. (2009a), Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput. 31, 37443759.CrossRefGoogle Scholar
Oseledets, I. V. and Tyrtyshnikov, E. E. (2009b), Recursive decomposition of multidimensional tensors, Dokl. Akad. Nauk 427, 1416. In Russian; English translation in Dokl. Math. 80, 460–462 (2009).Google Scholar
Oseledets, I. V. and Tyrtyshnikov, E. E. (2010), TT-cross approximation for multidimensional arrays, Linear Algebra Appl. 432, 7088.CrossRefGoogle Scholar
Oseledets, I. V., Rakhuba, M. V. and Uschmajew, A. (2018), Alternating least squares as moving subspace correction, SIAM J. Numer. Anal. 56, 34593479.CrossRefGoogle Scholar
Oster, M., Sallandt, L. and Schneider, R. (2022), Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats, SIAM J. Sci. Comput. 44, B746B770.CrossRefGoogle Scholar
Ostermann, A., Piazzola, C. and Walach, H. (2019), Convergence of a low-rank Lie–Trotter splitting for stiff matrix differential equations, SIAM J. Numer. Anal. 57, 19471966.CrossRefGoogle Scholar
Parlett, B. N. (1998), The Symmetric Eigenvalue Problem, Vol. 20 of Classics in Applied Mathematics, SIAM. Corrected reprint of the 1980 original.Google Scholar
Pfeffer, M. (2018), Tensor methods for the numerical solution of high-dimensional parametric partial differential equations. PhD thesis, Technische Universität Berlin.Google Scholar
Primbs, M. (2010), New stable biorthogonal spline-wavelets on the interval, Results Math. 57, 121162.CrossRefGoogle Scholar
Prud’homme, C. and Patera, A. T. (2004), Reduced-basis output bounds for approximately parameterized elliptic coercive partial differential equations, Comput. Vis. Sci 6, 147162.CrossRefGoogle Scholar
Rakhuba, M. (2021), Robust alternating direction implicit solver in quantized tensor formats for a three-dimensional elliptic PDE, SIAM J. Sci. Comput. 43, A800A827.CrossRefGoogle Scholar
Rakhuba, M., Novikov, A. and Oseledets, I. V. (2019), Low-rank Riemannian eigensolver for high-dimensional Hamiltonians, J. Comput. Phys. 396, 718737.CrossRefGoogle Scholar
Rakhuba, M. V. and Oseledets, I. V. (2016), Grid-based electronic structure calculations: The tensor decomposition approach, J. Comput. Phys. 312, 1930.CrossRefGoogle Scholar
Rakhuba, M. V. and Oseledets, I. V. (2018), Jacobi–Davidson method on low-rank matrix manifolds, SIAM J. Sci. Comput. 40, A1149A1170.CrossRefGoogle Scholar
Richter, L., Sallandt, L. and Nüsken, N. (2021), Solving high-dimensional parabolic PDEs using the tensor train format, in 38th International Conference on Machine Learning, Vol. 139 of Proceedings of Machine Learning Research, PMLR, pp. 89989009.Google Scholar
Rohwedder, T. and Uschmajew, A. (2013), On local convergence of alternating schemes for optimization of convex problems in the tensor train format, SIAM J. Numer. Anal. 51, 11341162.CrossRefGoogle Scholar
Rozza, G., Huynh, D. B. P. and Patera, A. T. (2008), Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics, Arch. Comput. Methods Engrg 15, 229275.CrossRefGoogle Scholar
Ryan, R. A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer.CrossRefGoogle Scholar
Sapsis, T. P. and Lermusiaux, P. F. (2009), Dynamically orthogonal field equations for continuous stochastic dynamical systems, Phys. D 238, 23472360.CrossRefGoogle Scholar
Schmidt, E. (1907), Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann. 63, 433476.CrossRefGoogle Scholar
R. Schneider and Uschmajew, A. (2014), Approximation rates for the hierarchical tensor format in periodic Sobolev spaces, J. Complexity 30, 5671.Google Scholar
R. Schneider and Uschmajew, A. (2015), Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality, SIAM J. Optim. 25, 622646.Google Scholar
Schollwöck, U. (2011), The density-matrix renormalization group in the age of matrix product states, Ann. Phys 326, 96192.CrossRefGoogle Scholar
Scholz, S. and Yserentant, H. (2017), On the approximation of electronic wavefunctions by anisotropic Gauss and Gauss–Hermite functions, Numer. Math 136, 841874.CrossRefGoogle Scholar
Schwab, C. and Gittelson, C. J. (2011), Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numer. 20, 291467.CrossRefGoogle Scholar
Schwab, C. and Stevenson, R. (2009), Space–time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78, 12931318.CrossRefGoogle Scholar
Shi, Y.-Y., Duan, L.-M. and Vidal, G. (2006), Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74, 022320.CrossRefGoogle Scholar
Sidiropoulos, N. D., De Lathauwer, L., Fu, X., Huang, K., Papalexakis, E. E. and Faloutsos, C. (2017), Tensor decomposition for signal processing and machine learning, IEEE Trans. Signal Process. 65, 35513582.CrossRefGoogle Scholar
Signoretto, M., Dinh, Q. Tran, De Lathauwer, L. and Suykens, J. A. K. (2014), Learning with tensors: A framework based on convex optimization and spectral regularization, Mach. Learn 94, 303351.CrossRefGoogle Scholar
Singh, S., Pfeifer, R. N. C. and Vidal, G. (2011), Tensor network states and algorithms in the presence of a global U(1) symmetry, Phys. Rev. B 83, 115125.Google Scholar
Steinlechner, M. (2016), Riemannian optimization for high-dimensional tensor completion, SIAM J. Sci. Comput. 38, S461S484.CrossRefGoogle Scholar
Stenger, F. (1993), Numerical Methods Based on Sinc and Analytic Functions, Vol. 20 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Stevenson, R. (2009), Adaptive wavelet methods for solving operator equations: An overview, in Multiscale, Nonlinear and Adaptive Approximation (DeVore, R. and Kunoth, A., eds), Springer, pp. 543597.CrossRefGoogle Scholar
Stoudenmire, E. and Schwab, D. J. (2016), Supervised learning with tensor networks, in Advances in Neural Information Processing Systems 29 (Lee, D. et al., eds), Curran Associates.Google Scholar
Sun, Y. and Kumar, M. (2014), Numerical solution of high dimensional stationary Fokker–Planck equations via tensor decomposition and Chebyshev spectral differentiation, Comput. Math. Appl. 67, 19601977.CrossRefGoogle Scholar
Szalay, S., Pfeffer, M., Murg, V., Barcza, G., Verstraete, F., Schneider, R. and Legeza, Ö. (2015), Tensor product methods and entanglement optimization for ab initio quantum chemistry, Internat. J. Quantum Chem. 115, 13421391.CrossRefGoogle Scholar
Tobler, C. (2012), Low-rank tensor methods for linear systems and eigenvalue problems. PhD thesis, ETH Zürich.Google Scholar
Tucker, L. R. (1964), The extension of factor analysis to three-dimensional matrices, in Contributions to Mathematical Psychology (Gulliksen, H. and Frederiksen, N., eds), Holt, Rinehart & Winston, pp. 109127.Google Scholar
Tucker, L. R. (1966), Some mathematical notes on three-mode factor analysis, Psychometrika 31, 279311.CrossRefGoogle ScholarPubMed
Tyrtyshnikov, E. E. (2003), Tensor approximations of matrices generated by asymptotically smooth functions, Mat. Sb. 194, 147160.Google Scholar
Uschmajew, A. (2010), Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations, Numer. Math 115, 309331.CrossRefGoogle Scholar
Uschmajew, A. (2012), Local convergence of the alternating least squares algorithm for canonical tensor approximation, SIAM J. Matrix Anal. Appl. 33, 639652.CrossRefGoogle Scholar
Uschmajew, A. (2013), Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen. PhD thesis, Technische Universität Berlin.Google Scholar
Uschmajew, A. and Vandereycken, B. (2013), The geometry of algorithms using hierarchical tensors, Linear Algebra Appl. 439, 133166.CrossRefGoogle Scholar
Uschmajew, A. and Vandereycken, B. (2015), Greedy rank updates combined with Riemannian descent methods for low-rank optimization, in 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 420424.Google Scholar
Uschmajew, A. and Vandereycken, B. (2020), Geometric methods on low-rank matrix and tensor manifolds, in Handbook of Variational Methods for Nonlinear Geometric Data (Grohs, P. et al., eds), Springer, pp. 261313.Google Scholar
Vidal, G. (2003), Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902.CrossRefGoogle ScholarPubMed
Wang, H. and Thoss, M. (2003), Multilayer formulation of the multiconfiguration time-dependent Hartree theory, J. Chem Phys. 119, 12891299.CrossRefGoogle Scholar
Wang, L. and Chu, M. T. (2014), On the global convergence of the alternating least squares method for rank-one approximation to generic tensors, SIAM J. Matrix Anal. Appl. 35, 10581072.CrossRefGoogle Scholar
Weidmann, J. (1980), Linear Operators in Hilbert Spaces, Vol. 68 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar
White, S. R. (1992), Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 28632866.CrossRefGoogle ScholarPubMed
White, S. R. (2005), Density matrix renormalization group algorithms with a single center site, Phys. Rev. B 72, 180403.CrossRefGoogle Scholar
White, S. R. and Martin, R. L. (1999), Ab initio quantum chemistry using the density matrix renormalization group, J. Chem. Phys. 110, 41274130.CrossRefGoogle Scholar
Wilkinson, J. H. (1968), Global convergence of tridiagonal QR algorithm with origin shifts, Linear Algebra Appl. 1, 409420.CrossRefGoogle Scholar
Wouters, S. and Van Neck, D. (2014), The density matrix renormalization group for ab initio quantum chemistry, Europ. Phys. J. D 68, 120.Google Scholar
Yang, M. and White, S. R. (2020), Time-dependent variational principle with ancillary Krylov subspace, Phys. Rev. B 102, 094315.CrossRefGoogle Scholar
Yserentant, H. (2010), Regularity and Approximability of Electronic Wave Functions, Springer.CrossRefGoogle Scholar
Zeiser, A. (2010), Direkte Diskretisierung der Schrödinger-Gleichung auf dünnen Gittern. PhD thesis, TU Berlin.Google Scholar
Zhou, G., Huang, W., Gallivan, K. A., Van Dooren, P. and Absil, P.-A. (2016), A Riemannian rank-adaptive method for low-rank optimization, Neurocomput. 192, 7280.CrossRefGoogle Scholar