A PETSc parallel-in-time solver based on MGRIT algorithm

We address the development of a modular implementation of the MGRIT (MultiGrid-In-Time) algorithm to solve linear and nonlinear systems that arise from the discretization of evolutionary models with a parallel-in-time approach in the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the opportunity of predicting the performance gain achievable when using the MGRIT approach instead of the Time Stepping integrator (TS). To this end, we analyze the performance parameters of the algorithm that provide a-priori the best number of processing elements and grid levels to use to address the scaling of MGRIT, regarded as a parallel iterative algorithm proceeding along the time dimensio

Algebraic Temporal Blocking for Sparse Iterative Solvers on Multi-Core CPUs

Sparse linear iterative solvers are essential for many large-scale simulations. Much of the runtime of these solvers is often spent in the implicit evaluation of matrix polynomials via a sequence of sparse matrix-vector products. A variety of approaches has been proposed to make these polynomial evaluations explicit (i.e., fix the coefficients), e.g., polynomial preconditioners or s-step Krylov methods. Furthermore, it is nowadays a popular practice to approximate triangular solves by a matrix polynomial to increase parallelism. Such algorithms allow to evaluate the polynomial using a so-called matrix power kernel (MPK), which computes the product between a power of a sparse matrix A and a dense vector x, or a related operation. Recently we have shown that using the level-based formulation of sparse matrix-vector multiplications in the Recursive Algebraic Coloring Engine (RACE) framework we can perform temporal cache blocking of MPK to increase its performance. In this work, we demonstrate the application of this cache-blocking optimization in sparse iterative solvers. By integrating the RACE library into the Trilinos framework, we demonstrate the speedups achieved in preconditioned) s-step GMRES, polynomial preconditioners, and algebraic multigrid (AMG). For MPK-dominated algorithms we achieve speedups of up to 3x on modern multi-core compute nodes. For algorithms with moderate contributions from subspace orthogonalization, the gain reduces significantly, which is often caused by the insufficient quality of the orthogonalization routines. Finally, we showcase the application of RACE-accelerated solvers in a real-world wind turbine simulation (Nalu-Wind) and highlight the new opportunities and perspectives opened up by RACE as a cache-blocking technique for MPK-enabled sparse solvers.Comment: 25 pages, 11 figures, 3 table