Sixth deal.II Users and Developers Workshop

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

Jul 26, 2018, 9:00 AM

45m

Meeting Room -- VII Floor (SISSA Main Building)

Keynote presentations Developers' track Keynotes

Speaker

Denis Davydov (Friedrich-Alexander University Erlangen-Nuremberg)

Description

Traditional solution approaches for problems in quantum mechanics scale as $\mathcal{O}(N^3)$, where $N$ is the number of electrons. Various methods have been proposed to address this issue and obtain linear scaling $\mathcal{O}(N)$. 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. This is often called the near-sightedness principle of matter. In this talk we present numerically efficient implementation of sparse parallel vectors within the deal.II open-source finite element library suitable for matrix-free operator evaluation. Based on the a-priori chosen support for each vector, we develop algorithms and data structures to perform (i) matrix-free sparse matrix multi-vector products (SpMM) (ii) projection of an operator onto a sparse sub-space (inner products) (iii) post-multiplication with a matrix. Strong and weak scaling results are reported for a typical benchmark problem using quadratic and quartic finite element bases.