Seminar in Numerical Analysis: Reinhold Schneider (TU Berlin)

The QDMRG algorithm and recent advances in tensor approximation

The DMRG algorithm (density matrix renormalization group algorithm) introduced by S. White provides a powerful  tool for the numerical treatment of spin systems. The DMRG version for the electronic Schrödinger equation, the  QDMRG (quantum chemistry density matrix renormalization group) algorithm is less known.  Although it provides an approximaiton of the full CI solution within polynomial complexity. Concepts known from spin systems, e.g. matrix product states, tree  tensor networks have been rediscovered recently in tensor product approximation, under a different perspective as  hierarchical Tucker representation introduced by Hackbusch and coworkers. and on  TT-tensors  (tensor trains) by Oseledets & Tyrtishnikov, offering a promising approach for the numerical treatment of high dimensional differential equation.  We have shown that under a full rank condition TT tensors form a manifold and characterize its tangent space, e.g.  to apply the Dirac-Frenkel variartional  principle. We propose an alternating linear scheme  (ALS alternating linear scheme) approach for optimization in the TT format. A modified alternating linear scheme (MALS) applied to the electronic Schrödinger equation resembles exactly the density matrix renormalization group algorithm (QDMRG). Identifying the discretized Fock space with the tensor product space ⊗R² (⊗C²),  the formalism of second quantization is directly implemented in the tensor treatment for numerical computation.

Joint work with Th. Rohwedder and S. Holtz

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