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_37: We present a fully grid-based approach for solving Hartree-Fock and all-electron Kohn-Sham equations based on low-rank approximation of three-dimensional electron orbitals. Due to the low-rank structure the total complexity of the algorithm depends linearly with respect to the one-dimensional grid size. Linear complexity allows for the usage of fine grids, e.g. 81923 and, thus, cheap extrapolation procedure.We test the proposed approach on closed-shell atoms up to the argon, several molecules and clusters of hydrogen atoms. All tests show systematical convergence with the required accuracy.
@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_44: When an algorithm in dimension one is extended to dimension d, in nearly every case its computational cost is taken to the power d. This fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality. For numerical analysis in dimension d, we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy. Basic linear algebra operations can be performed in this representation using one-dimensional operations, thus bypassing the exponential scaling with respect to the dimension. Although not all operators and algorithms may be compatible with this representation, we believe that many of the most important ones are. We prove that the multiparticle Schrodinger operator, as well as the inverse Laplacian, can be represented very efficiently in this form. We give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground-state eigenfunction for a simplified Schrodinger operator with 30 particles. We conjecture and provide numerical evidence that functions of operators inherit this property, in which case numerical operator calculus in higher dimensions becomes feasible.
@cite_25: We propose a new cross-conv algorithm for approximate computation of convolution in different low-rank tensor formats (tensor train, Tucker, hierarchical Tucker). It has better complexity with respect to the tensor rank than previous approaches. The new algorithm has a high potential impact in different applications. The key idea is based on applying cross approximation in the “frequency domain,” where convolution becomes a simple elementwise product. We illustrate efficiency of our algorithm by computing the three-dimensional Newton potential and by presenting preliminary results for solution of the Hartree--Fock equation on tensor-product grids.
@cite_20: We present an algorithm for learning (or estimating) a function of many variables from scattered data. The function is approximated by a sum of separable functions, following the paradigm of separated representations. The central fitting algorithm is linear in both the number of data points and the number of variables and, thus, is suitable for large data sets in high dimensions. We present numerical evidence for the utility of these representations. In particular, we show that our method outperforms other methods on several benchmark data sets.
We would also like to discuss tensor algorithms for solving eigenvalue problems developed in mathematical community. There are two natural directions of solving eigenvalue problems in tensor formats. One direction is straightforward generalization of iterative methods to tensor arithmetics with rank reduction on each iteration. For the canonical decomposition power method with shifts was generalized in @cite_44 @cite_20 and used in the RRBPM method. The preconditioned inverse iteration (PINVIT) for tensor formats was considered in @cite_42 @cite_37 @cite_25 . The inverse iteration used in this paper differs from the PINVIT, which is basically preconditioned steepest descent. Tensor version of the inverse iteration based on iterative solution of arising linear systems was considered in @cite_30 .