Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems

Abstract

We introduce a family of numerical algorithms for the solution of linear system in higher dimensions with the matrix and right hand side given and the solution sought in the tensor train format. The proposed methods are rank--adaptive and follow the alternating directions framework, but in contrast to ALS methods, in each iteration a tensor subspace is enlarged by a set of vectors chosen similarly to the steepest descent algorithm. The convergence is analyzed in the presence of approximation errors and the geometrical convergence rate is estimated and related to the one of the steepest descent. The complexity of the presented algorithms is linear in the mode size and dimension and the convergence demonstrated in the numerical experiments is comparable to the one of the DMRG--type algorithm.

Submitted to SIAM J Num Anal

References in corpus (7)

• The density-matrix renormalization group in the age of matrix product states
• Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems
• Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems
• Computation of extreme eigenvalues in higher dimensions using block tensor train format
• Computations in Quantum Tensor Networks
• Exact NMR simulation of protein-size spin systems using tensor train formalism
• On the Convergence of Alternating Least Squares Optimisation in Tensor Format Representations

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