arXiv subjects

Boris N. Khoromskij

Publications and source records attributed to Boris N. Khoromskij.

At least 19 recordsLinked to original sources

arXiv2026

A hybrid Chebyshev-Tucker tensor format for approximation of multivariate functions

We introduce and analyze a mesh-free two-level hybrid Chebyshev-Tucker tensor representation for approximating multivariate functions, which combines tensor-product Chebyshev interpolation with the low-rank Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows to avoid the expensive rank-structured grid-based approximation of function-related tensors on large spatial grids, while benefiting from the Tucker decomposition of the moderate-sized core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy $\varepsilon >0$ without discretizing the function on a full fine grid in the domain, but only using its values at a small set of Chebyshev nodes computed either from the explicit analytic expression of the target function or from its data-sparse representation in a rank-structured tensor format with moderate rank parameter. Finally, we can represent the function in the algebraic Tucker format with optimal $\varepsilon$-rank on an arbitrarily large 3D tensor grid in the computational domain by discretizing the Chebyshev polynomials on that grid. The rank parameters of the nonlinear Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev linear interpolation in the function independent polynomial basis set. It is shown that our techniques can be gainfully applied to the long-range part of the singular electrostatic potential of multi-particle systems represented on a fine grid in the range-separated (RS) tensor format. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the collective electrostatic potential for large bio-molecular systems and lattice-type compounds.

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arXiv2025

Fast tensor-based electrostatic energy calculations in the perspective of protein-ligand docking problem

We propose and justify a new approach for fast calculation of the electrostatic interaction energy of clusters of charged particles in constrained energy minimization in the framework of rigid protein-ligand docking. Our ``blind search'' docking technique is based on the low-rank range-separated (RS) tensor-based representation of the free-space electrostatic potential of the biomolecule represented on large $n\times n\times n$ 3D grid. We show that both the collective electrostatic potential of a complex protein-ligand system and the respective electrostatic interaction energy can be calculated by tensor techniques in $O(n)$-complexity, such that the numerical cost for energy calculation only mildly (logarithmically) depends on the number of particles in the system. Moreover, tensor representation of the electrostatic potential enables usage of large 3D Cartesian grids (of the order of $n^3 \sim 10^{12}$), which could allow the accurate modeling of complexes with several large proteins. In our approach selection of the correct geometric pose predictions in the localized posing process is based on the control of van der Waals distance between the target molecular clusters. Here, we confine ourselves by constrained minimization of the energy functional by using only fast tensor-based free-space electrostatic energy recalculation for various rotations and translations of both clusters. Numerical tests of the electrostatic energy-based ``protein-ligand docking'' algorithm applied to synthetic and realistic input data present a proof of concept for rather complex particle configurations. The method may be used in the framework of the traditional stochastic or deterministic posing/docking techniques.

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arXiv2022

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing

Tensor numerical methods, based on the rank-structured tensor representation of $d$-variate functions and operators, are designed to provide $O(dn)$ complexity of numerical calculations on $n^{\otimes d }$ grids contrary to $O(n^d)$ scaling by conventional grid-based methods. However, multiple tensor operations may lead to enormous increase in the tensor ranks (curse of ranks) of the target data, making calculation intractable. Therefore one of the most important steps in tensor calculations is the robust and efficient rank reduction procedure which should be performed many times in the course of various tensor transforms in multidimensional operator and function calculus. The rank reduction scheme based on the Reduced Higher Order SVD (RHOSVD) introduced in [33] played a significant role in the development of tensor numerical methods. Here, we briefly survey the essentials of RHOSVD method and then focus on some new theoretical and computational aspects of the RHOSVD demonstrating that this rank reduction technique constitutes the basic ingredient in tensor computations for real-life problems. In particular, the stability analysis of RHOSVD is presented. We introduce the multilinear algebra of tensors represented in the range-separated (RS) tensor format. This allows to apply the RHOSVD rank-reduction techniques to non-regular functional data with many singularities, for example, to the rank-structured computation of the collective multi-particle interaction potentials in bio-molecular modeling, as well as to complicated composite radial functions. The new theoretical and numerical results on application of the RHOSVD in scattered data modeling are presented. RHOSVD proved to be the efficient rank reduction technique in numerous applications ranging from numerical treatment of multi-particle systems up to a numerical solution of PDE constrained control problems.

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arXiv2021

Tensor numerical method for optimal control problems constrained by an elliptic operator with general rank-structured coefficients

We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a preconditioned iterative method with an adaptive rank truncation for solving the equation for the control function, governed by a sum of the elliptic operator and its inverse $M=A + A^{-1}$, both discretized over large $n^{\otimes d}$, $d=2,3$, spatial grids. Two basic solution schemes are proposed and analyzed. In the first approach, one solves iteratively the initial linear system of equations with the matrix $M$ such that the matrix vector multiplication with the elliptic operator inverse, $y=A^{-1} u,$ is performed as an embedded iteration by using a rank-structured solver for the equation of the form $A y=u$. The second numerical scheme avoids the embedded iteration by reducing the initial equation to an equivalent one with the polynomial system matrix of the form $A^2 +I$. For both schemes, a low Kronecker rank spectrally equivalent preconditioner is constructed by using the corresponding matrix valued function of the anisotropic Laplacian diagonalized in the Fourier basis. Numerical tests for control problems in 2D setting confirm the linear-quadratic complexity scaling of the proposed method in the univariate grid size $n$. Further, we numerically demonstrate that for our low rank solution method, a cascadic multigrid approach reduces the number of PCG iterations considerably, however the total CPU time remains merely the same as for the unigrid iteration.

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arXiv2020

Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in $\mathbb{R}^d$, $d=2,3$. We consider the stochastic realizations using checkerboard configuration of the equation coefficients built on a large $L \times L \times L$ lattice, where $L$ is the size of representative volume elements. The Kronecker tensor product scheme is introduced for fast generation of the stiffness matrix for FDM discretization on a tensor grid. We describe tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in periodic setting to be used in the framework of PCG iteration. In our construction the diagonal matrix of the discrete Laplacian inverse represented in the Fourier basis is reshaped into a 3D tensor, which is then approximated by a low-rank canonical tensor, calculated by the multigrid Tucker-to-canonical tensor transform. The FDM discretization scheme on a tensor grid is described in detail, and the computational characteristics in terms of $L$ for the 3D Matlab implementation of the PCG iteration are illustrated. The present work continues the developments in [22], where the numerical primer to study the asymptotic convergence rate vs. $L$ for the homogenized matrix for 2D elliptic PDEs with random coefficients was investigated numerically. The presented elliptic problem solver can be applied for calculation of long sequences of stochastic realizations in numerical analysis of 3D stochastic homogenization problems, for solving 3D quasi-periodic geometric homogenization problems, as well as in the numerical simulation of dynamical many body interaction processes and multi-particle electrostatics.

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arXiv2020

Tensor Method for Optimal Control Problems Constrained by Fractional 3D Elliptic Operator with Variable Coefficients

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional 2D and 3D elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D $n\times n \times n$ spacial grids. Using the diagonalization of the arising matrix valued functions in the eigenbasis of the 1D Sturm-Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an $\varepsilon$-threshold. This method reduces the numerical cost for solving the control problem to $O(n \log n)$ (plus the quadratic term $O(n^2)$ with a small weight), which is superior to the approaches based on the traditional linear algebra tools that yield at least $O(n^3 \log n)$ complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by $O(n \log n)$. This essentially outperforms the traditional methods operating with fully populated $n^3 \times n^3$ matrices and vectors in $\mathbb{R}^{n^3}$. Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated PCG iteration in the univariate grid size $n$.

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arXiv2020

Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the fractional $d$-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor grid are diagonalized by using the fast Fourier transform (FFT). Then the low rank approximation of the $d$-dimensional tensors obtained by folding of the corresponding large diagonal matrices of eigenvalues are computed, which allows to solve the governing equation for the control function in a tensor-structured format. The existence of low rank canonical approximation to the class of matrix valued functions involved is justified by using the sinc quadrature approximation method applied to the Laplace transform of the generating function. The linear system of equations for the control function is solved by the PCG iterative method with the rank truncation at each iteration step, where the low Kronecker rank preconditioner is precomputed. The right-hand side, the solution vector, and the governing system matrix are maintained in the rank-structured tensor format which beneficially reduces the numerical cost to $O(n\log n)$, outperforming the standard FFT based methods of complexity $O(n^3\log n)$ for 3D case. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size $n$.

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arXiv2020

Prospects of tensor-based numerical modeling of the collective electrostatic potential in many-particle systems

Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatic potentials in many-particle systems and the respective interaction energy and forces [39,40,2]. In this paper, we outline the prospects for tensor-based numerical modeling of the collective electrostatic potential on lattices and in many-particle systems of general type. We generalize the approach initially introduced for the rank-structured grid-based calculation of the collective potentials on 3D lattices [39] to the case of many-particle systems with variable charges placed on $L^{\otimes d}$ lattices and discretized on fine $n^{\otimes d}$ Cartesian grids for arbitrary dimension $d$. As result, the interaction potential is represented in a parametric low-rank canonical format in $O(d L n)$ complexity. The energy is then calculated in $O(d L)$ operations. Electrostatics in large biomolecules is modeled by using the novel range-separated (RS) tensor format [2], which maintains the long-range part of the 3D collective potential of the many-body system represented on $n\times n \times n$ grid in a parametric low-rank form in $O(n)$-complexity. We show that the force field can be easily recovered by using the already precomputed electric field in the low-rank RS format. The RS tensor representation of the discretized Dirac delta [45] enables the efficient energy preserving regularization scheme for solving the 3D elliptic PDEs with strongly singular right-hand side arising in bio-sciences. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics, protein docking and classification problems and for low-parametric interpolation of scattered data in data science.

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arXiv2019

Numerical study in stochastic homogenization for elliptic PDEs: convergence rate in the size of representative volume elements

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned CG iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver we numerically investigate the convergence of the Representative Volume Element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in [6]. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

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arXiv2018

Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse

In this paper, we introduce the operator dependent range-separated tensor approximation of the discretized Dirac delta in $\mathbb{R}^d$. It is constructed by application of the discrete elliptic operator to the range-separated decomposition of the associated Green kernel discretized on the Cartesian grid in $\mathbb{R}^d$. The presented operator dependent local-global splitting of the Dirac delta can be applied for solving the potential equations in non-homogeneous media when the density in the right-hand side is given by the large sum of pointwise singular charges. We show how the idea of the operator dependent RS splitting of the Dirac delta can be extended to the closely related problem on the range separated tensor representation of the elliptic resolvent. The numerical tests confirm the expected localization properties of the obtained operator dependent approximation of the Dirac delta represented on a tensor grid. As an example of application, we consider the regularization scheme for solving the Poisson-Boltzmann equation for modeling the electrostatics in bio-molecules.

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arXiv2018

Tucker Tensor analysis of Matern functions in spatial statistics

In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential $\mathcal{O}(n^d)$ to a linear scaling $\mathcal{O}(drn)$, where $d$ is the spatial dimension, $n$ is the number of mesh points in one direction, and $r$ is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, $\Vert x-y \Vert$.

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arXiv2018

Computing the density of states for optical spectra by low-rank and QTT tensor approximation

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter equation (BSE). The presented approach for approximating the DOS is based on two main techniques. First, we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block-diagonal plus low-rank matrix structure described in [6, 3] for the BSE/TDA problem. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size $N$ can be accurately represented by a low rank quantized tensor train (QTT) tensor that can be determined through a least squares fitting procedure. The latter provides good approximation properties for strictly oscillating DOS functions with multiple gaps, and requires asymptotically much fewer ($O(\log N)$) functional calls compared with the full grid size $N$. This approach allows us to overcome the computational difficulties of the traditional schemes by avoiding both the need of stochastic sampling and interpolation by problem independent functions like polynomials etc. Numerical tests indicate that the QTT approach yields accurate recovery of DOS associated with problems that contain relatively large spectral gaps. The QTT tensor rank only weakly depends on the size of a molecular system which paves the way for treating large-scale spectral problems.

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arXiv2017

Quantized-CP Approximation and Sparse Tensor Interpolation of Function Generated Data

In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the QTT method [16] developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach the target vector of length $2^{L}$ is reshaped to a $L^{th}$ order tensor with two entries in each mode (Quantized representation) and then approximated by the QTT tenor including $2r^2 L$ parameters, where $r$ is the maximal TT rank. In what follows, we consider the Alternating Least-Squares (ALS) iterative scheme to compute the rank-$r$ CP approximation of the quantized vectors, which requires only $2 r L\ll 2^L$ parameters for storage. In the earlier papers [17] such a representation was called Q$_{Can}$ format, while in this paper we abbreviate it as the QCP representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank-$r$ QCP format using the reduced number the functional samples, calculated only at $O(2rL)$ grid points, is presented. The special version of ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP-interpolation method that allows to recover all $2r L$ representation parameters of the rank-$r$ QCP tensor. Numerical examples show the efficiency of the QCP-ALS type iteration and indicate the exponential convergence rate in $r$.

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arXiv2016

Range-separated tensor formats for numerical modeling of many-particle interaction potentials

We introduce and analyze the new range-separated (RS) canonical/Tucker tensor format which aims for numerical modeling of the 3D long-range interaction potentials in multi-particle systems. The main idea of the RS tensor format is the independent grid-based low-rank representation of the localized and global parts in the target tensor which allows the efficient numerical approximation of $N$-particle interaction potentials. The single-particle reference potential like $1/\|x\|$ is split into a sum of localized and long-range low-rank canonical tensors represented on a fine 3D $n\times n\times n$ Cartesian grid. The smoothed long-range contribution to the total potential sum is represented on the 3D grid in $O(n)$ storage via the low-rank canonical/Tucker tensor. We prove that the Tucker rank parameters depend only logarithmically on the number of particles $N$ and the grid-size $n$. Agglomeration of the short range part in the sum is reduced to an independent treatment of $N$ localized terms with almost disjoint effective supports, calculated in $O(N)$ operations. Thus, the cumulated sum of short range clusters is parametrized by a single low-rank canonical reference tensor with a local support, accomplished by a list of particle coordinates and their charges. The RS canonical/Tucker tensor representations reduce the cost of multi-linear algebraic operations on the 3D potential sums arising in modeling of multi-dimensional data by radial basis functions, say, in computation of the electrostatic potential of a protein, in 3D integration and convolution transforms, computation of gradients, forces and the interaction energy of a many-particle systems, and in low parametric fitting of multi-dimensional scattered data by reducing all of them to 1D calculations.

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arXiv2016

Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

In this paper, we study and implement the structural iterative eigensolvers for the large-scale eigenvalue problem in the Bethe-Salpeter equation (BSE) based on the reduced basis approach via low-rank factorizations in generating matrices, introduced in the previous paper. The approach reduces numerical costs down to $\mathcal{O}(N_b^2)$ in the size of atomic orbitals basis set, $N_b$, instead of practically intractable $\mathcal{O}(N_b^6)$ complexity scaling for the direct diagonalization of the BSE matrix. As an alternative to rank approximation of the static screen interaction part of the BSE matrix, we propose to restrict it to a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate that the enhanced reduced-block approximation exhibits higher precision within the controlled numerical cost, providing as well a distinct two-sided error estimate for the BSE eigenvalues. It is shown that further reduction of the asymptotic computational cost is possible due to ALS-type iteration in block tensor train (TT) format applied to the quantized-TT (QTT) tensor representation of both long eigenvectors and rank-structured matrix blocks. The QTT-rank of these entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, $N_o$, hence the overall asymptotic complexity for solving the BSE problem can be estimated by $\mathcal{O}(\log(N_o) N_o^{2})$. We confirm numerically a considerable decrease in computational time for the presented iterative approach applied to various compact and chain-type molecules, while supporting sufficient accuracy.

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arXiv2015

A fast iteration method for solving elliptic problems with quasiperiodic coefficients

The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients $A_ε$ specified by the small parameter $ε>0$. We use an iteration method generated by an elliptic operator, associated with a certain simplified (e.g., homogenized) problem. On each step of this procedure it is required to solve an auxiliary elliptic boundary value problem with non--oscillating coefficients $A_0$. All the information related to complicated coefficients of the original differential problem is encompasses in the linear functional, which forms the right hand side of the auxiliary problem. Therefore, explicit inversion of the original operator associated with oscillating coefficients is avoided. The only operation used instead is multiplication of the operator by a vector (vector function), which can be efficiently performed due to the low-rank QTT tensor operations with the rank parameter controlled by the given precision $δ>0$ independent on the parameter $ε$. We deduce two--sided a posteriori error estimates that do not use $A^{-1}_ε$ and provide guaranteed two sided bounds of the distance to the exact solution of the original problem for any step of the iteration process. The second part is concerned with realisations of the iteration method. For a wide class of oscillating coefficients, we obtain sharp QTT rank estimates for the stiffness matrix in tensor representation. In practice, this leads to the logarithmic complexity scaling of the approximation and solution process in both the FEM grid-size, and $O(\vert\logε\vert)$ cost in terms of $ε$. Numerical tests in 1D confirm the logarithmic complexity scaling of our method applied to a class of complicated quasiperiodic coefficients.

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arXiv2015

A reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations

The Bethe-Salpeter equation (BSE) is a reliable model for estimating the absorption spectra in molecules and solids on the basis of accurate calculation of the excited states from first principles. This challenging task includes calculation of the BSE operator in terms of two-electron integrals tensor represented in molecular orbital basis, and introduces a complicated algebraic task of solving the arising large matrix eigenvalue problem. The direct diagonalization of the BSE matrix is practically intractable due to $O(N^6)$ complexity scaling in the size of the atomic orbitals basis set, $N$. In this paper, we present a new approach to the computation of Bethe-Salpeter excitation energies which can lead to relaxation of the numerical costs up to $O(N^3)$. The idea is twofold: first, the diagonal plus low-rank tensor approximations to the fully populated blocks in the BSE matrix is constructed, enabling easier partial eigenvalue solver for a large auxiliary system relying only on matrix-vector multiplications with rank-structured matrices. And second, a small subset of eigenfunctions from the auxiliary eigenvalue problem is selected to build the Galerkin projection of the exact BSE system onto the reduced basis set. We present numerical tests on BSE calculations for a number of molecules confirming the $\varepsilon$-rank bounds for the blocks of BSE matrix. The numerics indicates that the reduced BSE eigenvalue problem with small matrices enables calculation of the lowest part of the excitation spectrum with sufficient accuracy.

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arXiv2015

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n $ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches.

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