Preliminary list of abstracts

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Optimization in novel tensor formats and the DMRG and related algorithms
Keywords: HT and TT Tensors; direct optimization; Alternating least squares and DMRG
Mon, 14:00--14:25
  • Schneider, Reinhold (Mathematical Institute, TU Berlin, Germany)
  • Holtz, Sebastian
  • Rohwedder, Thorsten

We will discuss recent progress in tensor product approximation concerning hierarchical Tucker representation introduced by Hackbusch. We will focus mainly on TT-tensors (Oseledets & Tyrtishnikov), which can be written by a matrix product representation (matrix product states (MPS) in quantum information theory). We consider numerical methods solving an optimization problem within a prescribed format, focusing on i) $L_2$-approximation, ii) linear equations and iii) eigenvalue problems. We consider the alternating linear scheme which is a generalzation of an alternating least square (ALS) approach for optimization in TT format. A modification (MALS) applied to N-body Fermn ionic systems resembles the density matrix renormalization group algorithm (DMRG). Furthermore, we propose an iterative linearization scheme projection onto the tangent space of the manifold of TT tensors. Finally convergence behaviour will investigated, and local linear convergence and even sometimes quadratical convergence could be shown und certain assumptions.