Khoromskij, Boris N.
Overview
Works: 
26
works in
42
publications in
2
languages and
356
library holdings

Roles: 
Author

Most widely held works by
Boris N Khoromskij
Numerical solution of elliptic differential equations by reduction to the interface by Boris N Khoromskij (
Book
)
8
editions published
in
2004
in
English
and held by
226
libraries
worldwide
This is the first book that deals systematically with the numerical solution of elliptic partial differential equations by their reduction to the interface via the Schur complement. Inheriting the beneficial features of finite element, boundary element and domain decomposition methods, our approach permits solving iteratively the Schur complement equation with linearlogarithmic cost in the number of the interface degrees of freedom. The book presents the detailed analysis of the efficient datasparse approximation techniques to the nonlocal PoincaréSteklov interface operators associated with the Laplace, biharmonic, Stokes and Lamé equations. Another attractive topic are the robust preconditioning methods for elliptic equations with highly jumping, anisotropic coefficients. A special feature of the book is a unified presentation of the traditional iterative substructuring and multilevel methods combined with modern matrix compression techniques applied to the Schur complement on the interface
Tensor numerical methods in quantum chemistry by Venera Khoromskaia (
file
)
4
editions published
in
2018
in
English
and held by
55
libraries
worldwide
The conventional numerical methods when applied to multidimensional problems suffer from the socalled "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rankstructured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integraldifferential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in abinitio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensorbased HartreeFock solver, incorporating the gridbased lowrank factorization of the twoelectron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient gridbased numerical treatment of the longrange electrostatic potentials on large 3D finite lattices with defects.The novel rangeseparated tensor format applies to interaction potentials of multiparticle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing
Tensor numerical methods in scientific computing by Boris N Khoromskij (
file
)
5
editions published
between
2016
and
2018
in
English
and held by
55
libraries
worldwide
The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rankstructured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rankstructured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides functionoperator calculus in higher dimensions in logarithmic complexity rendering superfast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sincbased separable approximation of the analytic radial basis functions including Green's and Helmholtz kernels. The efficient tensorbased techniques for computational problems in electronic structure calculations and for the gridbased evaluation of longrange interaction potentials in multiparticle systems are considered. We also discuss the QTT numerical approach in manyparticle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highlyoscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations
Haar Wavelets by Boris N Khoromskij (
file
)
1
edition published
in
2013
in
English
and held by
1
library
worldwide
The tensorstructured solution of onedimensional elliptic differential equations with highdimensional parameters by Sergey Dolgov (
file
)
1
edition published
in
2012
in
English
and held by
1
library
worldwide
We consider a onedimensional secondorder elliptic equation with a highdimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the KarhunenLoève expansion of a stochastic PDE posed in a onedimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensorproduct grid. The paper is focused on the tensorstructured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the recently introduced Tensor Train and Quantized Tensor Train formats. We also discuss the efficient tensorstructured preconditioning of the entire multiparametric family of onedimensional elliptic problems, which leads us to a direct solution formula. We compare this method to a tensorstructured preconditioned GMRES solver in a series of numerical experiments
Multilevel Toeplitz matrices generated by QTT tensorstructured vectors and convolution with logarithmic complexity by Vladimir A Kazeev (
file
)
1
edition published
in
2011
in
English
and held by
1
library
worldwide
We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that lowrank QTT structure of the input is preserved in the output and propose efficient algorithms for these operations in the QTT format
Tensorproduct approach to global timespaceparametric discretization of chemical master equation by Sergey Dolgov (
file
)
1
edition published
in
2012
in
English
and held by
1
library
worldwide
We study the application of the novel tensor formats (TT, QTT, QTTTucker) to the solution of ddimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage?). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The time is treated as an additional variable, with the Quantized tensor representations (QTT, QTTTucker) employed, leading to the logcomplexity in the system size. This global spacetime (d + 1)dimensional system, approximated in the QTT or QTTTucker formats, is solved in the blockdiagonal form by the ALStype iterations. Another issue considered is the quantification of uncertainty, which means that some model parameters are not known exactly, but only their ranges can be estimated. It occurs frequently in reallife systems. In this case, we introduce the unknown parameters as auxiliary variables discretized on the corresponding grids, and solve the global spaceparametric system at once in the tensor formats
Fast solution of multidimensional parabolic problems in the TT/QTTformat with initial application to the FokkerPlanck equation by Sergey Dolgov (
file
)
1
edition published
in
2011
in
English
and held by
1
library
worldwide
In this paper we propose two schemes of using the QTT tensor approximations for solution of multidimensional parabolic problems. First, we present a simple onestep implicit time integration scheme and modify it using the matrix multiplication and a linear ALStype solver in the TT format. As the second approach, we propose the global spacetime formulation, resulting in a large block linear system, encapsulating all time steps, and solve it at once in the QTT format. We prove the QTTrank estimate for certain classes of multivariate potentials and respective solutions in (x, t) variables. We observe the loglinear complexity of storage and the solution algorithm in both spatial and time grid sizes, and at most cubic scaling in the QTT ranks of the discretized operator matrix and solution. The method is applied to the FokkerPlanck equation arising from the beadssprings models of polymeric liquids. For the dumbbell model numerical experiments are shown to demonstrate logarithmic behavior of computational time versus number of grid points in space and time, as well as accuracy. However, in numerical tests for the case of multispring Hookean potential we observe, that the rank properties of more general models might make the straightforward application of the tensor product approximations inefficient, requiring modifications in model descriptions and tensor discretizations
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations by Venera Khoromskaia (
file
)
1
edition published
in
2011
in
English
and held by
1
library
worldwide
In this paper, the tensorstructured numerical evaluation of the Coulomb and exchange operators in the HartreeFock equation is supplemented by the usage of recent quanticsTT (QTT) formats. It leads to O(log n) complexity at computationally extensive stages in the rankstructured calculation of the respective 3D and 6D integral operators including the Newton convolving kernel, and discretized on the n x n x n Cartesian grid. The numerical examples for some volumetric organic molecules show that the QTT ranks of the Coulomb and exchange operators are nearly independent on the onedimension grid size n. Thus, paradoxically, the complexity of the gridbased evaluation of the 3D integral operators becomes almost independent on the grid size, being regulated only by the structure of a molecular system. Hence, the gridbased approximation of the HartreeFock equation allows to gain a guaranteed accuracy. In numerical illustrations we present the QTT approximation of the Hartree and exchange operators for some moderate size molecules
Gridbased lattice summation of electrostatic potentials by lowrank tensor approximation by Verena Khoromskaia (
file
)
1
edition published
in
2013
in
English
and held by
1
library
worldwide
Twolevel TuckerTTQTT format for optimized tensor calculus by Sergey Dolgov (
file
)
1
edition published
in
2012
in
English
and held by
1
library
worldwide
We propose a combined tensor format, which encapsulates the benefits of Tucker, Tensor Train (TT) and Quantized TT (QTT) formats. The structure is composed of subtensors in TT representations, so the approximation problem is proven to be stable. We describe all important algebraic and optimization operations, which are recast to the TT routines. Several examples on explicit function and operator representations are provided. The asymptotic storage complexity is at most cubic in the rank parameter, that is larger than for the QTT format, but the numerical examples manifest, that the ranks in the twolevel format increase usually slower with the approximation accuracy than the QTT ones. In particular, we observe, that high rank peaks, which usually occur in the QTT representation, are significantly relaxed. Thus the reduced costs can be achieved
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation by Boris N Khoromskij (
file
)
2
editions published
in
2011
in
German and English
and held by
1
library
worldwide
We consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains. Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper we use TT and QTT tensor approximations to increase the efficiency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integrals
On explicit QTT representation of Laplace operator and its inverse by Vladimir A Kazeev (
file
)
1
edition published
in
2010
in
English
and held by
1
library
worldwide
Ranks and explicit structure of some matrices in the Quantics Tensor Train format, which allows representation with logarithmic complexity in many cases, are investigated. The matrices under consideration are Laplace operator with various boundary conditions in D dimensions and inverse Laplace operator with Dirichlet and DirichletNeumann boundary conditions in one dimension. The minimalrank explicit QTT representations of these matrices presented are suitable for any high mode sizes and, in the multidimensional case, for any high dimensions
MøllerPlesset (MP2) Energy Correction Using Tensor Factorizations of the Gridbased Twoelectron Integrals by Venera and Boris N Khoromskaia (
file
)
1
edition published
in
2013
in
English
and held by
1
library
worldwide
We present a tensorstructured method to calculate the MøllerPlesset (MP2) correction to the HartreeFock energy with reduced computational consumptions. The approach originates from the 3D gridbased lowrank factorization of the twoelectron integrals performed by the purely algebraic optimization. The computational scheme benefits from fast multilinear algebra implemented on the separable representations of the molecular orbital transformed twoelectron integrals, the doubles amplitude tensors and other fours order dataarrays involved. The separation rank estimates are discussed. The socalled quantized approximation of the long skeleton vectors comprising the tensor factorizations of the main entities allows to reduce the storage costs. The detailed description of tensor algorithms for evaluation of the MP2 energy correction is presented. The efficiency of these algorithms is illustrated in the framework of HartreeFock calculations for compact molecules, including alanine and glycine amino acids
QuantizedTTCayley transform to compute dynamics and spectrum of highdimensional Hamiltonians by Boris N Khoromskij (
file
)
1
edition published
in
2011
in
English
and held by
1
library
worldwide
In the present paper we propose and analyse a class of tensor methods for the efficient numerical computation of dynamics and spectrum of highdimensional Hamiltonians. We mainly focus on the complextime evolution problems. We apply the recent quanticsTT (QTT) matrix product states tensor approximation that allows to represent Nd tensors generated by ddimensional functions and operators with logvolume complexity, O(dlog N), where N is the univariate discretization parameter in space. We apply the truncated Cayley transform method that allows to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and spatial parts of solution to the highdimensional parabolic equation. We show the exponential convergence of the mterm time separation scheme and describe the efficient tensorstructured preconditioners for the class of multidimensional Hamiltonians
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator by Boris N Khoromskij (
file
)
1
edition published
in
2011
in
English
and held by
1
library
worldwide
In this paper we discuss how to combine two approaches: a Quantized Tensor Train (QTT) model and an advanced optimization method the Density Matrix Renormalization Group (DMRG) to obtain efficient numerical algorithms for highdimensional eigenvalue problems arising in quantum molecular dynamics. The QTTformat is used to approximate a multidimensional Hamiltonian, including the potential energy surface (PES), and the DMRG is applied for the solution of the arising eigenvalue problem in high dimension. The numerical experiments are presented for the approximation of a 6dimensional PES of HONO molecule, as well as for the computation of the ground state of a HenonHeiles potential with a large number of degrees of freedom up to 256
TensorStructured Galerkin Approximation of parametric and stochastic Elliptic PDEs by Boris N Khoromskij (
file
)
2
editions published
in
2010
in
English
and held by
1
library
worldwide
We investigate the convergence rate of approximations by finite sums of rank1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based for example, on the Mterm truncated expansion. Our approach could be regarded as either a class of compressed approximations of these solution or as a new class of iterative elliptic problem solvers for high dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension M of the parameter space. It is based on rankreduced, tensorformatted separable approximations of the highdimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent lowrank tensorstructured preconditioners to the parametric matrices resulting from a Finite Element discretization of the highdimensional parametric, deterministic problems
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Alternative Names
Choromskij, B.N.
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