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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 19, Issue 1

# Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation

Ivan Gavrilyuk
/ Boris N. Khoromskij
• Corresponding author
• Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
• Email
• Other articles by this author:
Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/cmam-2018-0021

## Abstract

In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ${\psi }^{\prime }\left(t\right)+A\psi =f\left(t\right)$ with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order $\mathcal{𝒪}\left({e}^{-N}\right)$ were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order $\mathcal{𝒪}\left({e}^{-N}\right)$ without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

MSC 2010: 65F30; 65F50; 65N35; 65F10

## References

• [1]

N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993. Google Scholar

• [2]

D. Z. Arov and I. P. Gavrilyuk, A method for solving initial value problems for linear differential equations in Hilbert space based on the Cayley transform, Numer. Funct. Anal. Optim. 14 (1993), no. 5–6, 459–473.

• [3]

D. Z. Arov, I. P. Gavrilyuk and V. L. Makarov, Representation and approximation of solutions of initial value problems for differential equations in Hilbert space based on the Cayley transform, Elliptic and Parabolic Problems (Pont-à-Mousson 1994), Pitman Res. Notes Math. Ser. 325, Longman Scientific & Technical, Harlow (1995), 40–50. Google Scholar

• [4]

H. Bateman and A. Erdelyi, Higher Transcendental Functions. Vol. 2, Mc Graw-Hill, New York, 1988. Google Scholar

• [5]

M. H. Beck, A. Jäckle, G. A. Worth and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324 (2000), 1–105.

• [6]

P. Benner, V. Khoromskaia and B. N. Khoromskij, Range-separated tensor format for many-particle modeling, SIAM J. Sci. Comput. 40 (2018), no. 2, A1034–A1062.

• [7]

S. Dolgov and B. Khoromskij, Simultaneous state-time approximation of the chemical master equation using tensor product formats, Numer. Linear Algebra Appl. 22 (2015), no. 2, 197–219.

• [8]

S. V. Dolgov, B. N. Khoromskij and I. V. Oseledets, 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 (2012), no. 6, A3016–A3038.

• [9]

I. P. Gavrilyuk, An algorithmic representation of fractional powers of positive operators, Numer. Funct. Anal. Optim. 17 (1996), no. 3–4, 293–305.

• [10]

I. P. Gavrilyuk, Strongly P-positive operators and explicit representations of the solutions of initial value problems for second-order differential equations in Banach space, J. Math. Anal. Appl. 236 (1999), no. 2, 327–349.

• [11]

I. P. Gavrilyuk, Super exponentially convergent approximation to the solution of the Schrödinger equation in abstract setting, Comput. Methods Appl. Math. 10 (2010), no. 4, 345–358. Google Scholar

• [12]

I. P. Gavrilyuk, Three recipes for constructing of exponentially convergent algorithms for operator equations, Proceedings of the Second International Conference “Supercomputer Technologies of Mathematical Modeling”, M. K. Ammosov North-Eastern Federal University, Yakutsk (2014), 182–192. Google Scholar

• [13]

I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij, Data-sparse approximation to a class of operator-valued functions, Math. Comp. 74 (2005), no. 250, 681–708. Google Scholar

• [14]

I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij, Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing 74 (2005), no. 2, 131–157.

• [15]

I. P. Gavrilyuk and B. Khoromskij, Quantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians, Comput. Methods Appl. Math. 11 (2011), no. 3, 273–290. Google Scholar

• [16]

I. P. Gavrilyuk and V. L. Makarov, Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach spaces, Z. Anal. Anwend. 15 (1996), no. 2, 495–527.

• [17]

I. P. Gavrilyuk and V. L. Makarov, Exact and approximate solutions of some operator equations based on the Cayley transform, Linear Algebra Appl. 282 (1998), no. 1–3, 97–121.

• [18]

M. Griebel and J. Hamaekers, Sparse grids for the Schrödinger equation, M2AN Math. Model. Numer. Anal. 41 (2007), no. 2, 215–247.

• [19]

M. Griebel, D. Oeltz and P. Vassilevski, Space-time approximation with sparse grids, SIAM J. Sci. Comput. 28 (2006), no. 2, 701–727.

• [20]

W. Hackbusch and B. N. Khoromskij, Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions, Computing 76 (2006), no. 3–4, 177–202.

• [21]

W. Hackbusch, B. N. Khoromskij, S. Sauter and E. E. Tyrtyshnikov, Use of tensor formats in elliptic eigenvalue problems, Numer. Linear Algebra Appl. 19 (2012), no. 1, 133–151.

• [22]

W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov, Approximate iterations for structured matrices, Numer. Math. 109 (2008), no. 3, 365–383.

• [23]

V. A. Kazeev and B. N. Khoromskij, Low-rank explicit QTT representation of the Laplace operator and its inverse, SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 742–758.

• [24]

B. N. Khoromskij, Structured rank-$\left({R}_{1},\mathrm{\dots },{R}_{D}\right)$ decomposition of function-related tensors in ${ℝ}^{D}$, Comput. Methods Appl. Math. 6 (2006), no. 2, 194–220. Google Scholar

• [25]

B. N. Khoromskij, Tensor-structured preconditioners and approximate inverse of elliptic operators in ${ℝ}^{d}$, Constr. Approx. 30 (2009), no. 3, 599–620. Google Scholar

• [26]

B. N. Khoromskij, $O\left(d\mathrm{log}N\right)$-quantics approximation of N-d tensors in high-dimensional numerical modeling, Constr. Approx. 34 (2011), no. 2, 257–280. Google Scholar

• [27]

B. N. Khoromskij, Tensors-structured numerical methods in scientific computing: Survey on recent advances, Chem. Intell. Lab. Syst. 110 (2012), 1–19.

• [28]

B. N. Khoromskij, Tensor Numerical Methods in Scientific Computing, Radon Ser. Comput. Appl. Math. 19, De Gruyter, Berlin, 2018. Google Scholar

• [29]

B. N. Khoromskij and I. Oseledets, DMRG$+$QTT approach to high-dimensional quantum molecular dynamics, Preprint 68/2010, Max Planck Institute for Mathematics in the Sciences, Leipzig, 2010. Google Scholar

• [30]

B. N. Khoromskij and I. V. Oseledets, QTT approximation of elliptic solution operators in higher dimensions, Russian J. Numer. Anal. Math. Modelling 26 (2011), no. 3, 303–322.

• [31]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), no. 3, 455–500.

• [32]

U. Langer, S. E. Moore and M. Neumüller, Space-time isogeometric analysis of parabolic evolution problems, Comput. Methods Appl. Mech. Engrg. 306 (2016), 342–363.

• [33]

C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2008. Google Scholar

• [34]

C. Lubich, I. V. Oseledets and B. Vandereycken, Time integration of tensor trains, SIAM J. Numer. Anal. 53 (2015), no. 2, 917–941.

• [35]

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), no. 2, 470–494.

• [36]

H.-D. Meyer, F. Gatti and G. A. Worth, Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Willey-VCH, Wienheim, 2009. Google Scholar

• [37]

I. V. Oseledets, Approximation of ${2}^{d}×{2}^{d}$ matrices using tensor decomposition, SIAM J. Matrix Anal. Appl. 31 (2009/10), no. 4, 2130–2145. Google Scholar

• [38]

I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33 (2011), no. 5, 2295–2317.

• [39]

I. V. Oseledets and E. E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput. 31 (2009), no. 5, 3744–3759.

• [40]

D. Perez-Garcia, F. Verstraete, M. M. Wolf and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7 (2007), no. 5–6, 401–430. Google Scholar

• [41]

F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer Ser. Comput. Math. 20, Springer, New York, 1993. Google Scholar

• [42]

P. K. Suetin, Classical Orthogonal Polynomials (in Russian), “Nauka”, Moscow, 1979. Google Scholar

• [43]

G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, American Mathematical Society, Providence, 1959. Google Scholar

• [44]

E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), no. 1, 1–10.

• [45]

G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91 (2003), no. 14, Article ID 147902. Google Scholar

• [46]

T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions, M2AN Math. Model. Numer. Anal. 38 (2004), no. 1, 93–127.

• [47]

H. Wang and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent Hartree theory, J. Chem. Phys. 119 (2003), 1289–1299.

• [48]

S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48 (1993), no. 14, 10345–10356.

Revised: 2018-02-21

Accepted: 2018-05-02

Published Online: 2018-07-12

Published in Print: 2019-01-01

Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 55–71, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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