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Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics and many others. The numerical treatment of such kind of problems is difficult due to the curse of dimensionality: the number of unknowns and computational complexity grows exponentially with the dimension of the problem. To break the curse of dimensionality,… (More)

We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high–dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors… (More)

We propose algorithms for the solution of high-dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right-hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To… (More)

We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one– and multi-dimensional vectors (m–tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm.… (More)

In this paper we accomplish the development of the fast rank–adaptive solver for tensor–structured symmetric positive definite linear systems in higher dimensions. In [9] this problem is approached by alternating minimization of the energy function, which we combine with steps of the basis expansion in accordance with the steepest descent algorithm. In this… (More)

We study the application of the novel tensor formats (TT, QTT, QTT-Tucker) to the solution of d-dimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage-λ). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The time is… (More)

Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to the Fokker-Planck equation Abstract. In this paper we propose two schemes of using the so-called QTT-approximation for the solution of multidimensional parabolic problems. First, we present a simple one-step implicit time integration scheme using an… (More)

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… (More)