Khoromskij, Boris N.
Overview
Works: 
25
works in
35
publications in
1
language and
257
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
223
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
)
1
edition published
in
2018
in
English
and held by
8
libraries
worldwide
Hierarchical Kronecker tensor product approximations by W Hackbusch (
Book
)
2
editions published
in
2003
in
Undetermined and English
and held by
3
libraries
worldwide
Blended kernel approximation in the Hmatrix techniques by W Hackbusch (
Book
)
2
editions published
in
2000
in
English
and held by
3
libraries
worldwide
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
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
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
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation by Boris N Khoromskij (
file
)
1
edition published
in
2011
in
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
Haar Wavelets by Boris N Khoromskij (
file
)
1
edition published
in
2013
in
English
and held by
1
library
worldwide
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
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
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
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
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
A sparse Hmatrix arithmetic by W Hackbusch (
Book
)
2
editions published
in
1999
in
Undetermined
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
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
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Related Identities

Wittum, Gabriel 1956

Khoromskaia, Venera Author

Hackbusch, Wolfgang Professor, Fachgebiet Informatik und praktische Mathematik Author

Dolgov, Sergey Author

Tyrtyšnikov, Evgenij E. 02.06.1955

Kazeev, Vladimir A. Author

Oseledets, Ivan V.

Khoromskaia, Verena Author

Schneider, Reinhold

Prössdorf, Siegfried

Alternative Names
Choromskij, B.N.
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