Optimization Problems in Contracted Tensor Networks
Mike Espig, Wolfgang Hackbusch, Stefan Handschuh,and Reinhold Schneider
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Submission date: 17. Oct. 2011 (revised version: April 2012)
published in: Computing and visualization in science, 14 (2012) 6, p. 271-285
DOI number (of the published article): 10.1007/s00791-012-0183-y
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We discuss the calculus of variations in tensor representations with a special focus on tensor networks and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the representation rank compression in tensor networks. For the numerical treatment, we use the nonlinear block Gauss-Seidel method. We demonstrate the rate of convergence in numerical tests.