arXiv subjects

Maxim Rakhuba

Publications and source records attributed to Maxim Rakhuba.

41 records · Page 3Linked to original sources

arXiv2018

Low-rank Riemannian eigensolver for high-dimensional Hamiltonians

Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as high-dimensional eigenvalue problems, i.e., when the eigenvector can be naturally represented as a multidimensional tensor. Tensor methods have proven to be an efficient tool for the approximation of solutions of high-dimensional eigenvalue problems, however, their performance deteriorates quickly when the number of eigenstates to be computed increases. We address this issue by designing a new algorithm motivated by the ideas of Riemannian optimization (optimization on smooth manifolds) for the approximation of multiple eigenstates in the tensor-train format, which is also known as matrix product state representation. The proposed algorithm is implemented in TensorFlow, which allows for both CPU and GPU parallelization.

math.NA↗

arXiv2017

Vico-Greengard-Ferrando quadratures in the tensor solver for integral equations

Convolution with Green's function of a differential operator appears in a lot of applications e.g. Lippmann-Schwinger integral equation. Algorithms for computing such are usually non-trivial and require non-uniform mesh. However, recently Vico, Greengard and Ferrando developed method for computing convolution with smooth functions with compact support with spectral accuracy, requiring nothing more than Fast Fourier Transform (FFT). Their approach is very suitable for the low-rank tensor implementation which we develop using Quantized Tensor Train (QTT) decomposition.

math.NA↗

arXiv2017

Jacobi-Davidson method on low-rank matrix manifolds

In this work we generalize the Jacobi-Davidson method to the case when eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits advantages of the original Jacobi-Davidson method, has lower complexity and requires less storage. We also introduce low-rank version of the Rayleigh quotient iteration which naturally arises in the Jacobi-Davidson method.

math.NA↗

arXiv2016

Calculating vibrational spectra of molecules using tensor train decomposition

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 (LOBPCG, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH$_3$CN on a laptop in one hour using only $100$ MB of memory to represent all computed eigenfunctions.

math.NA↗

arXiv2015

Grid-based electronic structure calculations: the tensor decomposition approach

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. $8192^3$ 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.

math.NA↗