Algorithms and data structures for matrix-free finite element operators with MPI-parallel sparse multi-vectors

Traditional solution approaches for problems in quantum mechanics scale as $\mathcal O(M^3)$, where $M$ is the number of electrons. Various methods have been proposed to address this issue and obtain linear scaling $\mathcal O(M)$. One promising formulation is the direct minimization of energy. Such methods take advantage of physical localization of the solution, namely that the solution can be sought in terms of non-orthogonal orbitals with local support. In this work a numerically efficient implementation of sparse parallel vectors within the open-source finite element library deal.II is proposed. The main algorithmic ingredient is the matrix-free evaluation of the Hamiltonian operator by cell-wise quadrature. Based on an a-priori chosen support for each vector we develop algorithms and data structures to perform (i) matrix-free sparse matrix multivector products (SpMM), (ii) the projection of an operator onto a sparse sub-space (inner products), and (iii) post-multiplication of a sparse multivector with a square matrix. The node-level performance is analyzed using a roofline model. Our matrix-free implementation of finite element operators with sparse multivectors achieves the performance of 157 GFlop/s on Intel Cascade Lake architecture. Strong and weak scaling results are reported for a typical benchmark problem using quadratic and quartic finite element bases.Comment: 29 pages, 12 figure

Compact Sparse Coulomb Integrals using a Range-Separated Potential

The efficient calculation of so-called two-electron integrals is an important component for electronic structure calculations on large molecules and periodic systems at both mean field and post-HF, correlated, levels. In this thesis, a new and fairly complicated representation of the Coulomb interaction is presented. The Coulomb potential is partitioned into short and long-range parts. The short-range interactions are treated analytically using conventional density fitting methods. The long-range interactions are treated numerically through either a Fourier transform in spherical coordinates or through a Cartesian multipole expansion. The Fourier transform is used for intermediate distances, while multipole expansions (up to octupole) are used for longer range, with a switching algorithm to decide between the two. In this range-separated representation, the corresponding two-electron Coulomb integrals can be calculated efficiently and the amount of data scales linearly with respect to system size. Hartree-Fock theory is used as an extensive test of the range-separated method, but the same building blocks can be used in correlated calculations like Second-order Moller-Plesset perturbation theory and (Cluster in Molecule) type Coupled Cluster calculations