2019/07/01 by Denis Davydov, Davydov, Denis, Martin Kronbichler +1 · 1 citation
Computer Science · Engineering · #Advanced Numerical Methods in Computational Mathematics #Computational Physics (physics.comp-ph) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Mathematical Software (cs.MS) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Parallel Computing and Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1907.01005
openalex publication_date 2019/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Traditional solution approaches for problems in quantum mechanics scale as\n mathcal O(M3), where M is the number of electrons. Various methods have\nbeen proposed to address this issue and obtain linear scaling mathcal O(M).\nOne promising formulation is the direct minimization of energy. Such methods\ntake advantage of physical localization of the solution, namely that the\nsolution can be sought in terms of non-orthogonal orbitals with local support.\nIn this work a numerically efficient implementation of sparse parallel vectors\nwithin the open-source finite element library deal.II is proposed. The main\nalgorithmic ingredient is the matrix-free evaluation of the Hamiltonian\noperator by cell-wise quadrature. Based on an a-priori chosen support for each\nvector we develop algorithms and data structures to perform (i) matrix-free\nsparse matrix multivector products (SpMM), (ii) the projection of an operator\nonto a sparse sub-space (inner products), and (iii) post-multiplication of a\nsparse multivector with a square matrix. The node-level performance is analyzed\nusing a roofline model. Our matrix-free implementation of finite element\noperators with sparse multivectors achieves the performance of 157 GFlop/s on\nIntel Cascade Lake architecture. Strong and weak scaling results are reported\nfor a typical benchmark problem using quadratic and quartic finite element\nbases.\n