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MLD2P4: A Package of Parallel Algebraic Multilevel Domain Decomposition Preconditioners in Fortran 95

Domain decomposition ideas have long been an essential tool for the solution of PDEs on parallel computers. In recent years many research efforts have been focused on recursively employing domain decomposition methods to obtain multilevel preconditioners to be used with Krylov solvers. In this context, we developed MLD2P4 (MultiLevel Domain Decomposition Parallel Preconditioners Package based on PSBLAS), a package of parallel multilevel preconditioners that combines additive Schwarz domain decomposition methods with a smoothed aggregation technique to build a hierarchy of coarse-level corrections in an algebraic way. The design of MLD2P4 was guided by objectives such as extensibility, flexibility, performance, portability, and ease of use. They were achieved by following an object-based approach while using the Fortran 95 language, as well as by employing the PSBLAS library as a basic framework. In this article, we present MLD2P4 focusing on its design principles, software architecture, and use.

2LEV-D2P4: a package of high-performance preconditioners for scientific and engineering applications

We present a package of parallel preconditioners which implements one-level and two-level Domain Decomposition algorithms on the top of the PSBLAS library for sparse matrix computations. The package, named 2LEV-D2P4 (Two-LEVel Domain Decomposition Parallel Preconditioners Package based on PSBLAS), currently includes various versions of additive Schwarz preconditioners that are combined with a coarse-level correction to obtain two-level preconditioners. A pure algebraic formulation of the preconditioners is considered. 2LEV-D2P4 has been written in Fortran~95, exploiting features such as abstract data type creation, functional overloading and dynamic memory management, while providing a smooth path towards the integration in legacy application codes. The package, used with Krylov solvers implemented in PSBLAS, has been tested on large-scale linear systems arising from model problems and real applications, showing its effectiveness.

Scalable algebraic multilevel preconditioners with application to CFD

The solution of large and sparse linear systems is one of the main computational kernels in CFD applications and is often a very time-consuming task, thus requiring the use of effective algorithms on high-performance computers. Preconditioned Krylov solvers are the methods of choice for these systems, but the availability of “good” preconditioners is crucial to achieve efficiency and robustness. In this paper we discuss some issues concerning the design and the implementation of scalable algebraic multilevel preconditioners, that have shown to be able to enhance the performance of Krylov solvers in parallel settings. In this context, we outline the main objectives and the related design choices of MLD2P4, a package of multilevel preconditioners based on Schwarz methods and on the smoothed aggregation technique, that has been developed to provide scalable and easy-to-use preconditioners in the Parallel Sparse BLAS computing framework. Results concerning the application of various MLD2P4 preconditioners within a large eddy simulation of a turbulent channel flow are discussed.

The impact of different stiff ODE solvers in parallel simulation of diesel combustion

In this paper we analyze the behaviour of two stiff ODE solvers in the solution of chemical kinetics systems arising from detailed models of Diesel combustion. We consider general-purpose solvers, based on Backward Differentiation Formulas or Runge-Kutta methods and compare their impact, in terms of reliability and efficiency, on the solution of two different chemical kinetics systems, modeling combustion in the context of realistic simulations of Common Rail Diesel Engines. Numerical experiments have been carried out by using an improved version of KIVA3V, interfacing a parallel combustion solver. The parallel combustion solver is based on the CHEMKIN package for evaluating chemical reaction rates and on the general-purpose stiff ODE solvers for solving chemical kynetic equations.

PARALLEL NUMERICAL SIMULATION OF AIR POLLUTION IN SOUTHERN ITALY

In this paper we present the Parallel Naples Airshed Model (PNAM),a parallel software package for the numerical simulation of air pollution episodes on urban scale domains, using MIMD distributed-memory machines. This is a first result of a research activity aimed at developing a system software to simulate air pollution episodes in the Campania Region, in Southern Italy. PNAM is based on an Eulerian model of the transport and photochemical transformations of air pollutants and uses a time-splitting approach, which separates the advection from the (coupled) diffusion and chemistry phenomena. The parallel implementation is based on grid partitioning and the use of dynamic load balancing techniques is currently under experiment. It is written in Fortran 90 and is based on the parallel Runtime System Library (RSL) to implement domain decomposition, data communication and dynamic load balancing. Numerical experiments have been carried out on a realistic test case, using an IBM SP, to evaluate the parallel performance of PNAM. Execution times, speedup and efficiency have been measured, obtaining a speedup of more than 7 on 12 processors. Preliminary results obtained with a dynamic load balancing strategy have been also analyzed, gaining suggestions for future work.

αAMG based on Weighted Matching for Systems of Elliptic PDEs Arising From Displacement and Mixed Methods

Adaptive Algebraic Multigrid (or Multilevel) Methods (αAMG) are introduced to improve robustness and efficiency of classical algebraic multigrid methods in dealing with problems where no a priori knowledge or assumptions on the near-null kernel of the underlined matrix are available. Recently we proposed an adaptive (bootstrap) AMG method, αAMG, aimed to obtain a composite solver with a desired convergence rate. Each new multigrid component relies on a current (general) smooth vector and exploits pairwise aggregation based on weighted matching in a matrix graph to define a new automatic, general-purpose coarsening process, which we refer to as “the compatible weighted matching”. In this work, we present results that broaden the applicability of our method to different finite element discretizations of elliptic PDEs. In particular, we consider systems arising from displacement methods in linear elasticity problems and saddle-point systems that appear in the application of the mixed method to Darcy problems.

BootCMatch: A Software Package for Bootstrap AMG Based on Graph Weighted Matching

This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process. Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.

Parallel Aggregation Based on Compatible Weighted Matching for AMG

We focus on the extension of the MLD2P4 package of parallel Algebraic MultiGrid (AMG) preconditioners, with the objective of improving its robustness and efficiency when dealing with sparse linear systems arising from anisotropic PDE problems on general meshes. We present a parallel implementation of a new coarsening algorithm for symmetric positive definite matrices, which is based on a weighted matching approach. We discuss preliminary results obtained by combining this coarsening strategy with the AMG components available in MLD2P4, on linear systems arising from applications considered in the Horizon 2020 Project “Energy oriented Centre of Excellence for computing applications” (EoCoE).