ERA Chemistry - Generalized tensor methods in quantum chemistry

• Antragsteller: Professor Dr. Reinhold Schneider
• Fachliche Zuordnung: Theoretische Chemie: Elektronenstruktur, Dynamik, Simulation ; Mathematik ; Physikalische Chemie von Molekülen, Flüssigkeiten und Grenzflächen, Biophysikalische Chemie
• Förderung: Förderung von 2013 bis 2017
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 234056486

• Erstellungsjahr: 2018

Zusammenfassung der Projektergebnisse

The present group of the ERA project has contributed significantly to the understanding of DMRG and Tensor Network Approximations in quantum chemistry. It has embedded the Tree Tensor Network state approximation in a general low rank tensor approximation with Hierarchical Tensors (HT) and linked the development of tensor product approximation to other problems in high-dimensional PDE’s. It supplements the development of numerical methods in quantum chemistry, in particular QC-DMRG, methods by providing rigorous analysis. After the leave of V. Murg from the project, the development of the special tree code was interrupted for a certain time. As a consequence, we mainly avoided the implementational difficulties when dealing with the topological more difficult structure of arbitrary trees. Therefore, an implementation of the general tree is still in its infancy. In particular it has not been under the responsibility of the Berlin group in the ERA project. Although the TT is a good model for HT, i.e. for Tree Tensor Networks, there are various situations where other trees are superior. However the development for linear trees, i.e. Tensor Trains (or MPS) had still been in its infancy before. Supplied with further strategies for adapting appropriate ranks, optimizing the orbital basis functions, e.g., by appropriate reordering etc. the traditional (one side and/or two side) DMRG (ALS/MALS), is still a very powerful linear optimization technique for TT (MPS) and even for general HT or tree networks. To our experience, further Riemannian optimization techniques have not shown enough advantages. The QC-DMRG developed partly by this project is cutting edge for computations. The project has tremendous impact to transfer the knowledge from computational quantum physics to other areas where low rank tensor product approximation provide substantial progress, particularly for high-dimensional PDE’s. In our part of the project, possible alternatives to avoid existing drawbacks of the low rank tensor product approximation in a Hierarchical- or Tree Tensor Network format have been shown. The tensor network approximation and particularly the Tree Tensor Networks equivalent to HT, are at least as good as traditional adaptive approximations which rest on sparsity. In many cases (or most cases), it is superior due to the underlying hierarchical structure. It may be only by-passed by modern deep neuronal networks, which are the recent standard in machine learning. However the relation to deep convolutionary networks has been observed quite recently. This may be an interesting subject for future research.

Projektbezogene Publikationen (Auswahl)

• Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry, Numer. Math. 128, no. 1, 137165 (2014)
  M. Bachmayr, H. Chen and R. Schneider
  
• Tensor spaces and hierarchical tensor representations. In: Extraction of quantifiable information from complex systems in Stephan Dahlke ... (eds.) New York : Springer, 2014. - P. 237-261 ( Lecture notes in computational science and engineering ; 102)
  W. Hackbusch and R. Schneider
  
• Error estimates of some numerical atomic orbitals in molecular simulations. Commun. Comput. Phys. 18, no. 1, 125146, (2015)
  H. Chen and R. Schneider
  
• Numerical analysis of augmented plane wave methods for fullpotential electronic structure calculations. ESAIM Math. Model. Numer. Anal. 49 no. 3, 755785 (2015)
  H. Chen and R. Schneider
  
• Tensor product methods and entanglement optimization for ab initio quantum chemistry, Int. J. Quant. Chem. 115, 1342 -1391 (2015)
  S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, Ors Legeza
  
• Tree tensor network state with variable tensor order: An efficient multi-reference method for strongly correlated systems, Journal of chemical theory and computation 11 (3), 1027-1036 (2015)
  V. Murg, F. Verstraete, R. Schneider, P.R. Nagy, O. Legeza
  
• Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Numerische Mathematik, pp. 1-39 (2016)
  M. Eigel, M. Pfeffer and R. Schneider
  
• Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality, SIAM journal on optimization, 25 1, p. 622-646 (2016)
  R. Schneider and A. Uschmajew
  
• Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations, Foundations of Computational Mathematics 16 (6), 1423-1472, (2016)
  M. Bachmayr, R. Schneider and A. Uschmajew
  
• Variational tensor approach for approximating the rare-event kinetics of macromolecular systems, J. Chem. Phys. 144, 054105 (2016)
  F. Nüske, R. Schneider, F. Vitalini, F. and F. Noé
  
• Iterative methods based on soft thresholding of hierarchical tensors, Foundations of Computational Mathematics 17 (4), 1037-1083 (2017)
  M. Bachmayr and R. Schneider
  
• Tensor Methods for the Numerical Solution of High-Dimensional Parametric Partial Differential Equations. Dissertation, (2018). 95 S.
  M. Pfeffer