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Encyclopedia of Applied and Computational Mathematics pp 1060–1066Cite as

Numerical Approaches for High-Dimensional PDEs for Quantum Chemistry

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The treatment of high-dimensional problems such as the Schrödinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics.

Introduction

Multiparticle Schrödinger-type equations are an important example of problems posed on high-dimensional tensor spaces. Numerical approximation of solutions of these problems suffers from the curse of dimensionality, i.e., the computational complexity scales exponentially with the dimension of the space. Circumventing this problem is a challenging topic in modern numerical analysis with a variety of applications, covering aside from the electronic and nuclear Schrödinger equation, e.g., the Fokker–Planck equation and the chemical master equation. Considerable progress in the treatment of such problems has been made...

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References

  1. Bardos, C., Catto, I., Mauser, N., Trabelsi, S.: Setting and analysis of the multi-configuration time-dependent Hartree-Fock equations. Arch. Ration. Mech. Anal. 198(1), 273–330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beylkin, G., Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26(6), 2133ff (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cancès, E., Ehrlacher, V., Lelivre, T.: Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Preprint available at http://arxiv.org/pdf/1004.0095 (2011). Accessed Feb 22, 2012

  4. Chan, G.K.-L., Head-Gordon, M.: Highly correlated calculations with a polynomial cost algorithm: a study of the density matrix renormalization group. J. Chem. Phys. 116, 4462 (2002)

    Article  Google Scholar 

  5. Falcó, A., Nuoy, A.: Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. (2011). doi: 10.1007/s00211-011-0437-5

    Google Scholar 

  6. Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM. J. Matrix Anal. Appl. 31, 2029 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. SSCM, vol. 42. Springer, Berlin/Heidelberg (2012)

    Book  MATH  Google Scholar 

  8. Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hillar, C.J., Lim, L.-H.: Most tensor problems are NP hard. Preprint, available at http://www.msri.org/people/members/chillar/files/tensorNPhardApril8.pdf. Accessed Feb 22, 2012

  10. Holtz, S., Rohwedder, T., Schneider, R.: On manifolds of tensors with fixed TT rank. Numer. Math. (2011). doi:10.1007/s00211-011-0419-7

    MathSciNet  MATH  Google Scholar 

  11. Holtz, S., Rohwedder, T., Schneider, R.: The Alternating Linear Scheme for Tensor Optimisation in the TT Format, SISC. Available at http://www.math.tu-berlin.de/fileadmin/i26/Bilder_Webseite/AG_ModNumDiff/FG_ModSimOpt/schneider/mals_110706.pdf. Accessed Feb 22, 2012

  12. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Landsberg, J.M., Qi, Y., Ye, K.: On the geometry of tensor network states. Preprint, available at http://arxiv.org/abs/1105.4449 (2011). Accessed Feb 22, 2012

  14. Legeza, Ö., Sólyom, J.: Optimizing the density-matrix renormalization group method using quantum information entropy. Phys. Rev. B 68(19), 195116 (2003)

    Article  Google Scholar 

  15. Lubich, C.: From Quantum to Classical Molecular Dynamics: Reduced methods and Numerical Analysis. Zürich Lectures in advanced mathematics. EMS, Zürich (2008)

    Book  MATH  Google Scholar 

  16. Marti, K.H., Bauer, B., Reiher, M., Troyer, M., Verstraete, F.: Complete-graph tensor network states: a new fermionic wave function ansatz for molecules. New J. Phys. 12, 103008 (2010)

    Article  Google Scholar 

  17. Moritz, G. Hess, B.A., Reiher, M.: Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings. J. Chem. Phys. 122, 024107 (2005)

    Article  Google Scholar 

  18. Murg, V., Verstraete, F., Legeza, Ö., Noack, R.M.: Simulating strongly correlated quantum systems with tree tensor networks. Phys. Rev. B 82, 205105 (2010)

    Article  Google Scholar 

  19. Oseledets, I.V.: Approximation of 2d × 2d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31, 2130–2145 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33, 2295–2317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schollwöck, U.: The density-matrix renormalization group in the age of matrix product states. Ann. Phys. (NY) 326, 96 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vidal, G.: Efficient classical simulation of slightly entagled quantum computation, Phy. Rev. Letters 14, 91 (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider & Thorsten Rohwedder

  2. Theoretical Solid State Physics, Hungarian Academy of Sciences, Budapest, Hungary

    Örs Legeza

Authors
  1. Reinhold Schneider
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  2. Thorsten Rohwedder
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  3. Örs Legeza
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Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Schneider, R., Rohwedder, T., Legeza, Ö. (2015). Numerical Approaches for High-Dimensional PDEs for Quantum Chemistry. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_245

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