Venera Khoromskaia
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Venera Khoromskaia has a total of 16 co-authors. Here are the top co-authors by number of shared publications:
- Boris N. Khoromskij
Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26 D-04103 Leipzig Germany
22 shared publications - Peter Benner
Max Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany
6 shared publications Boris N. Khoromskij
Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany
3 shared publications- Matthias Stein
Max Planck Institute for Dynamics of Complex Technical Systems
3 shared publications - Volker Schulz
RWTH Aachen University
3 shared publications - Alexander Litvinenko
RWTH Aachen
2 shared publications
Latest Publications
- Open Access
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.Open AccessFast 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. - Open Access
Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format
The Poisson-Boltzmann equation (PBE) is an implicit solvent continuum model for calculating the electrostatic potential and energies of ionic solvated biomolecules. However, its numerical solution remains a significant challenge due strong singularities and nonlinearity caused by the singular source terms and the exponential nonlinear terms, respectively. An efficient method for the treatment of singularities in the linear PBE was introduced in \cite{BeKKKS:18}, that is based on the RS tensor decomposition for both electrostatic potential and the discretized Dirac delta distribution. In this paper, we extend this regularization method to the nonlinear PBE. We apply the PBE only to the regular part of the solution corresponding to the modified right-hand side via extraction of the long-range part in the discretized Dirac delta distribution. The total electrostatic potential is obtained by adding the long-range solution to the directly precomputed short-range potential. The main computational benefit of the approach is the automatic maintaining of the continuity in the Cauchy data on the solute-solvent interface. The boundary conditions are also obtained from the long-range component of the precomputed canonical tensor representation of the Newton kernel. In the numerical experiments, we illustrate the accuracy of the nonlinear regularized PBE (NRPBE) over the classical variant.
Open AccessSolution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format
The Poisson-Boltzmann equation (PBE) is an implicit solvent continuum model for calculating the electrostatic potential and energies of ionic solvated biomolecules. However, its numerical solution remains a significant challenge due strong singularities and nonlinearity caused by the singular source terms and the exponential nonlinear terms, respectively. An efficient method for the treatment of singularities in the linear PBE was introduced in \cite{BeKKKS:18}, that is based on the RS tensor decomposition for both electrostatic potential and the discretized Dirac delta distribution. In this paper, we extend this regularization method to the nonlinear PBE. We apply the PBE only to the regular part of the solution corresponding to the modified right-hand side via extraction of the long-range part in the discretized Dirac delta distribution. The total electrostatic potential is obtained by adding the long-range solution to the directly precomputed short-range potential. The main computational benefit of the approach is the automatic maintaining of the continuity in the Cauchy data on the solute-solvent interface. The boundary conditions are also obtained from the long-range component of the precomputed canonical tensor representation of the Newton kernel. In the numerical experiments, we illustrate the accuracy of the nonlinear regularized PBE (NRPBE) over the classical variant.
- Open Access
- 19 April 2022
- Published by Frontiers Media SA in Frontiers in Applied Mathematics and Statistics
- Vol. 8
Ubiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing
Tensor numerical methods, based on the rank-structured tensor representation of d-variate functions and operators discretized on large n⊗d grids, are designed to provide O(dn) complexity of numerical calculations contrary to O(nd) 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 multi-dimensional operator and function calculus. The rank reduction scheme based on the Reduced Higher Order SVD (RHOSVD) introduced by the authors, 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 and demonstrate 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 multi-linear 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. We underline that RHOSVD proved to be the efficient rank reduction technique in numerous applications ranging from numerical treatment of multi-particle systems in material sciences up to a numerical solution of PDE constrained control problems in ℝd.
- 19 April 2022
- Published by Frontiers Media SA in Frontiers in Applied Mathematics and Statistics
- Vol. 8
Open AccessUbiquitous Nature of the Reduced Higher Order SVD in Tensor-Based Scientific Computing
Tensor numerical methods, based on the rank-structured tensor representation of d-variate functions and operators discretized on large n⊗d grids, are designed to provide O(dn) complexity of numerical calculations contrary to O(nd) 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 multi-dimensional operator and function calculus. The rank reduction scheme based on the Reduced Higher Order SVD (RHOSVD) introduced by the authors, 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 and demonstrate 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 multi-linear 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. We underline that RHOSVD proved to be the efficient rank reduction technique in numerous applications ranging from numerical treatment of multi-particle systems in material sciences up to a numerical solution of PDE constrained control problems in ℝd.
- 1 May 2021
- Published by Pleiades Publishing Ltd in Computational Mathematics and Mathematical Physics
- Vol. 61 (5) , 864-886
Prospects of Tensor-Based Numerical Modeling of the Collective Electrostatics in Many-Particle Systems
Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatics in many-particle systems and the respective interaction energy and forces. 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. Our approach, initially introduced for the rank-structured grid-based calculation of the interaction potentials on 3D lattices is generalized here 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 a result, the interaction potential is represented in a parametric low-rank canonical format in \(O(dLn)\) complexity. The total interaction energy can be then calculated in \(O(dL)\) operations. Electrostatics in large bio-molecular systems is discretized on a fine \({{n}^{{ \otimes 3}}}\) grid by using the novel range-separated (RS) tensor format, which maintains the long-range part of the 3D collective potential of a many-body system in a parametric low-rank form in \(O(n)\)-complexity. We show how the energy and 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 enables the construction of the efficient energy preserving (conservative) regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising in scientific computing. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics in bio-sciences, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.- 1 May 2021
- Published by Pleiades Publishing Ltd in Computational Mathematics and Mathematical Physics
- Vol. 61 (5) , 864-886
Prospects of Tensor-Based Numerical Modeling of the Collective Electrostatics in Many-Particle Systems
Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatics in many-particle systems and the respective interaction energy and forces. 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. Our approach, initially introduced for the rank-structured grid-based calculation of the interaction potentials on 3D lattices is generalized here 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 a result, the interaction potential is represented in a parametric low-rank canonical format in \(O(dLn)\) complexity. The total interaction energy can be then calculated in \(O(dL)\) operations. Electrostatics in large bio-molecular systems is discretized on a fine \({{n}^{{ \otimes 3}}}\) grid by using the novel range-separated (RS) tensor format, which maintains the long-range part of the 3D collective potential of a many-body system in a parametric low-rank form in \(O(n)\)-complexity. We show how the energy and 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 enables the construction of the efficient energy preserving (conservative) regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising in scientific computing. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics in bio-sciences, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.- Open Access
- 28 September 2020
- Published by Elsevier in Journal of Computational Physics
- Vol. 424, 109865
Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints
- 28 September 2020
- Published by Elsevier in Journal of Computational Physics
- Vol. 424, 109865
Open AccessTensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints
- Open Access
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.Open AccessTensor-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. - Open Access
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 $n$.Open AccessTensor 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 $n$. - Open Access
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 $n$.Open AccessTensor 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 $n$. - Open Access
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.Open AccessProspects 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. - Open Access
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.
Open AccessNumerical 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.