We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
@cite_30: SUMMARY We investigate approximations to eigenfunctions of a certain class of elliptic operators in Rd by finite sums of products of functions with separated variables and especially conditions providing an exponential decrease of the error with respect to the number of terms. The consistent use of tensor formats can be regarded as a base for a new class of rank-truncated iterative eigensolvers. The computational cost is almost linear in the univariate problem size n, while traditional method scale like nd. Tensor methods can be applied to solving large-scale spectral problems in computational quantum chemistry, for example, the Schrodinger, Hartree–Fock and Kohn–Sham equations in electronic structure calculations. The results of numerical experiments clearly indicate the linear-logarithmic scaling of the low-rank tensor method in n. The algorithms work equally well for the computation of both minimal and maximal eigenvalues of the discrete elliptic operators. Copyright © 2011 John Wiley & Sons, Ltd.
@cite_18: In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n).
@cite_22: 
@cite_7: Recent achievements in the field of tensor product approximation provide promising new formats for the representation of tensors in form of tree tensor networks. In contrast to the canonical @math -term representation (CANDECOMP, PARAFAC), these new formats provide stable representations, while the amount of required data is only slightly larger. The tensor train (TT) format [SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317], a simple special case of the hierarchical Tucker format [J. Fourier Anal. Appl., 5 (2009), p. 706], is a useful prototype for practical low-rank tensor representation. In this article, we show how optimization tasks can be treated in the TT format by a generalization of the well-known alternating least squares (ALS) algorithm and by a modified approach (MALS) that enables dynamical rank adaptation. A formulation of the component equations in terms of so-called retraction operators helps to show that many structural properties of the original problems transfer to the micro-iterations, giving what is to our knowledge the first stable generic algorithm for the treatment of optimization tasks in the tensor format. For the examples of linear equations and eigenvalue equations, we derive concrete working equations for the micro-iteration steps; numerical examples confirm the theoretical results concerning the stability of the TT decomposition and of ALS and MALS but also show that in some cases, high TT ranks are required during the iterative approximation of low-rank tensors, showing some potential of improvement.
@cite_28: In this paper, we propose a method for the approximation of the solution of high- dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation for- mats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be de- signed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any ap rioriinformation on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy correc- tions in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces. Mathematics Subject Classification. 15A69, 35J50, 41A63, 65D15, 65N12.
@cite_42: The computation of eigenvalues is one of the core topics of numerical mathematics. We will discuss an eigenvalue algorithm for the computation of inner eigenvalues of a large, symmetric, and positive definite matrix M based on the preconditioned inverse iteration @math and the folded spectrum method (replace M by ( (M - I) ^ 2 )). We assume that M is given in the tensor train matrix format and use the TT-toolbox from I.V. Oseledets (see http: spring.inm.ras.ru osel ) for the numerical computations. We will present first numerical results and discuss the numerical difficulties.
@cite_32: 
@cite_0: We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice.
The PINVIT iteration can explicitly utilize a preconditioner. The construction of preconditioners in tensor formats for eigenvalue problems was considered in @cite_30 @cite_18 @cite_0 @cite_42 . The approach for a general operator @cite_42 uses Newton-Schulz iteration in order to find a good preconditioner. However, due to high amount of matrix multiplications, this approach becomes time-consuming. In order to construct a preconditioner one can use approximate inverse matrix or approximate solution of linear systems. See @cite_22 @cite_7 @cite_28 @cite_32 for solvers in tensor formats.