Chronos: A General Purpose Classical AMG Solver for High Performance Computing

Abstract

The numerical simulation of physical systems has become in recent years a fundamental tool to perform analyses and predictions in several application fields, spanning from industry to the academy. As far as large-scale simulations are concerned, one of the most computationally expensive tasks is the solution of linear systems of equations arising from the discretization of the partial differential equations governing physical processes. This work presents Chronos, a collection of linear algebra functions specifically designed for the solution of large, sparse linear systems on massively parallel computers. Its emphasis is on modern, effective, and scalable Algebraic Multigrid (AMG) preconditioners for high performance computing (HPC). This work describes the numerical algorithms and the main structures of this software suite, especially from an implementation standpoint. Several numerical results arising from practical mechanics and fluid dynamics applications with hundreds of millions of unknowns are addressed and compared with other state-of-the-art linear solvers, proving Chronos's efficiency and robustness.

A Reverse Augmented Constraint preconditioner for Lagrange multiplier methods in contact mechanics

“…Most of the operations that need to be parallelized in RACP, such as approximating the (1,1) block or performing sparse matrix by matrix products, have been already studied by several authors and can be borrowed from existing libraries. In this work, we build RACP relying on the parallel Chronos software package [59]. Chronos is a library designed to solve linear algebra problems on high performance computers, providing the parallel preconditioners and sparse linear algebra kernels required to form and approximate the Schur complement.…”

“…In fact, the Schur complement is simply formed by calling the distributed sparse matrix-by-sparse matrix product, which manages all the necessary communications on its own. As a preconditioner for S u , we choose AMG with the special set-up designed for structural problems [59], which includes a Factorized Sparse Approximate Inverse (FSAI) as smoother and adaptive interpolation. The source files needed to store the saddle-point matrix, compute the RACP preconditioner and solve the linear system can be found at https://github.com/matteofrigo5/HPC_ ReverseAugmentedConstrained and they can be easily modified to test other approaches or use another preconditioner for S u .…”

“…The iterations are stopped when the 2-norm of the initial residual is reduced by 8 orders of magnitude. The RACP(1) approach is used as a preconditioner, with the application of the inverse of the primal Schur complement, S −1 u , carried out approximately by the AMG method proposed in [59,62]. For the sake of the comparison, we solve the system with the leading block A alone by accelerating GMRES(100) with the same multigrid method used to approximate S −1 u .…”

“…• the inverse of the leading block A is approximately applied by using either the same AMG approach used with RACP(1) [59,62], or an adaptive FSAI (aFSAI) [63,49];…”

“…Depending on the choice for the inner preconditioner of the leading block, we distinguish between MCP+AMG and MCP+FSAI in Table 4. For these approaches, the set of user-specified parameters providing the empirically optimal performance, as suggested in the relevant literature [63,62,64,59], is used. It can be noticed that RACP is able to solve all the test cases with acceptable computational costs.…”