Quantum-mechanical calculations based on Kohn–Sham density functional theory (DFT) played a significant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn–Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean field that is governed by electron-density. Despite the reduced computational complexity of Kohn–Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally efficient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn–Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree–Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the finite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn–Sham DFT problem by using a separable approximation of the Kohn–Sham Hamiltonian. Further, the resulting Kohn–Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent field iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.
Quantum-mechanical calculations based on Kohn–Sham density functional theory (DFT) played a significant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn–Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean field that is governed by electron-density. Despite the reduced computational complexity of Kohn–Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally efficient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn–Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree–Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the finite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn–Sham DFT problem by using a separable approximation of the Kohn–Sham Hamiltonian. Further, the resulting Kohn–Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent field iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.
Abstract We present a tensor-structured algorithm for efficient large-scale density functional theory (DFT) calculations by constructing a Tucker tensor basis that is adapted to the Kohn–Sham Hamiltonian and localized in real-space. The proposed approach uses an additive separable approximation to the Kohn–Sham Hamiltonian and an L1 localization technique to generate the 1-D localized functions that constitute the Tucker tensor basis. Numerical results show that the resulting Tucker tensor basis exhibits exponential convergence in the ground-state energy with increasing Tucker rank. Further, the proposed tensor-structured algorithm demonstrated sub-quadratic scaling with system-size for both systems with and without a gap, and involving many thousands of atoms. This reduced-order scaling has also resulted in the proposed approach outperforming plane-wave DFT implementation for systems beyond 2000 electrons.
Electronic structure calculations have been one of the most successful scientific fields in the past 50 years. Density functional theory (DFT) with its nice balance between accuracy and efficiency has now become the standard technique for many materials research. On the other hand, the size of systems that wavefunction-based methods can handle has also been improved thanks to the recent developments in both computation power and more advanced methodologies. However, these methods are still suffering from the curse of dimensionality. The cubic scaling of a typical DFT calculation restricts the handleable system size of DFT to only a few thousand electrons. To this end, tensor-structured techniques provide a route to constructing a reduced-order algorithm and hence are promising to improve the computational efficiency. In this dissertation work, we aim at exploring the applications of the tensor-structured techniques on different aspects of electronic structure calculations. In the first part of this work, we present a tensor-structured algorithm for efficient large-scale DFT calculations by constructing a Tucker tensor basis that is adapted to the Kohn–Sham Hamiltonian and localized in real-space. The proposed approach uses an additive separable approximation to the Kohn–Sham Hamiltonian and an L1 localization technique to generate the 1-D localized functions that constitute the Tucker tensor basis. Numerical results show that the resulting Tucker tensor basis exhibits exponential convergence in the ground-state energy with increasing Tucker rank. Further, the proposed tensor-structured algorithm demonstrated sub-quadratic scaling with system-size for both systems with and without a gap, and involving many thousands of atoms. This reduced-order scaling has also resulted in the proposed approach outperforming plane-wave DFT implementation for systems beyond 2000 electrons. In the second part of this work, we present an acceleration scheme for the proposed tensor-structured algorithm. In the proposed algorithm, the discrete Kohn-Sham problem is solved using the Chebyshev filtering subspace iteration method that relies on matrix-matrix multiplications of a sparse symmetric Hamiltonian matrix and a dense wavefunction matrix, expressed in the localized Tucker tensor basis. These matrix-matrix multiplication operations, which constitute the most computationally intensive step of the solution procedure, are GPU accelerated providing ~8-fold GPU-CPU speedup for these operations on the largest systems studied. The computational performance of the GPU accelerated code is presented using benchmark studies on aluminum nano-particles and silicon quantum dots with system sizes ranging up to ~7,000 atoms. Finally, we present a computation kernel using tensor-structured techniques to evaluate the one-electron and two-electron integrals, which are the central quantities to construct the Hamiltonian matrix of a full configuration interaction (FCI) calculation, for any molecular orbitals projected on a tensor-structured finite-element mesh. The proposed computation kernel is used to compute the integrals using Hartree-Fock molecular orbitals, which is subsequently used to perform an FCI calculation. In our numerical study, the FCI energy using integrals from the proposed computation kernel is confirmed to be consistent with a standard FCI calculation using the same Hartree-Fock orbitals. The proposed computation kernel provides a useful tool to investigate the behavior of Kohn-Sham orbitals as a basis for FCI calculations in the future, which could be a route to construct a reduced-order basis for an FCI calculation to reduce the computational cost and enable FCI calculation for the previously computationally inaccessible systems.
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, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved 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 × n × 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, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. 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 excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, 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 × L × 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.
We present a brief survey on the modern tensor numerical methods for multidimensional stationary and time-dependent partial differential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable approximation of multivariate functions and operators represented on a grid. Recently, the traditional Tucker, canonical, and matrix product states (tensor train) tensor models have been applied to the grid-based electronic structure calculations, to parametric PDEs, and to dynamical equations arising in scientific computing. The essential progress is based on the quantics tensor approximation method proved to be capable to represent (approximate) function related d-dimensional data arrays of size Nd with log-volume complexity, O(dlog N). Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and differential equations using low-parametric rank-structured tensor formats. As the main example, we describe the grid-based tensor numerical approach for solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting point for the developments of tensor-structured numerical methods for large-scale computations in solving real-life multidimensional problems. We also discuss a new method for the fast 3D lattice summation of electrostatic potentials by assembled low-rank tensor approximation capable to treat the potential sum over millions of atoms in few seconds. We address new results on tensor approximation of the dynamical Fokker-Planck and master equations in many dimensions up to d = 20. Numerical tests demonstrate the benefits of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs. In particular, the use of grid-based tensor representations in the reduced basis of atomics orbitals yields an accurate solution of the Hartree-Fock equation on large N × N × N grids with a grid size of up to N = 105.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "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" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential 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 ab-initio 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 10 15 . Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle 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.
Abstract This article is a short introduction to the modern computational techniques used to tackle the many-body problem in materials. The aim is to present the basic ideas, using simple examples to illustrate strengths and weaknesses of each method. We will start from density-functional theory (DFT) and the Kohn–Sham construction—the standard computational tools for performing electronic structure calculations. Leaving the realm of rigorous density-functional theory, we will discuss the established practice of adopting the Kohn–Sham Hamiltonian as approximate model. After recalling the triumphs of the Kohn–Sham description, we will stress the fundamental reasons of its failure for strongly-correlated compounds, and discuss the strategies adopted to overcome the problem. The article will then focus on the most effective method so far, the DFT+DMFT technique and its extensions. Achievements, open issues and possible future developments will be reviewed. The key differences between dynamical (DFT+DMFT) and static (DFT+ U ) mean-field methods will be elucidated. In the conclusion, we will assess the apparent dichotomy between first-principles and model-based techniques, emphasizing the common ground that in fact they share.