# General Overview
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General Overview

The MULTILEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE BASED ON PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain Decomposition preconditioners (see, e.g., [3,23,21]), to be used in the iterative solution of linear systems,

| (1)

where is a square, real or complex, sparse matrix. The name of the package comes from its original implementation, containing multilevel additive and hybrid Schwarz preconditioners, as well as one-level additive Schwarz preconditioners. The current version extends the original plan by including multilevel cycles and smoothers widely used in multigrid methods.

The multilevel preconditioners implemented in MLD2P4 are obtained by combining AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and K-cycles [3,19] are available, which allow to define almost all the preconditioners in the package, including the multilevel hybrid Schwarz ones; a specific cycle is implemented to obtain multilevel additive Schwarz preconditioners. The Jacobi, hybridforward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods are available as smoothers. An algebraic approach is used to generate a hierarchy of coarse-level matrices and operators, without explicitly using any information on the geometry of the original problem, e.g., the discretization of a PDE. To this end, the smoothed aggregation technique [2,25] is applied. Either exact or approximate solvers can be used on the coarsest-level system. Specifically, different sparse LU factorizations from external packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, and block-Jacobi solvers are available. All smoothers can be also exploited as one-level preconditioners.

MLD2P4 is written in Fortran 2003, following an object-oriented design through the exploitation of features such as abstract data type creation, type extension, functional overloading, and dynamic memory management. The parallel implementation is based on a Single Program Multiple Data (SPMD) paradigm. Single and double precision implementations of MLD2P4 are available for both the real and the complex case, which can be used through a single interface.

MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) computational framework [15,14]. PSBLAS provides basic linear algebra operators and data management facilities for distributed sparse matrices, as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been mainly motivated by the need of having a portable and efficient software infrastructure implementing ``de facto'' standard parallel sparse linear algebra kernels, to pursue goals such as performance, portability, modularity ed extensibility in the development of the preconditioner package. On the other hand, the implementation of MLD2P4 has led to some revisions and extentions of the original PSBLAS kernels. The inter-process comunication required by MLD2P4 is encapsulated in the PSBLAS routines;therefore, MLD2P4 can be run on any parallel machine where PSBLAS implementations are available.

MLD2P4 has a layered and modular software architecture where three main layers can be identified. The lower layer consists of the PSBLAS kernels, the middle one implements the construction and application phases of the preconditioners, and the upper one provides a uniform interface to all the preconditioners. This architecture allows for different levels of use of the package: few black-box routines at the upper layer allow all users to easily build and apply any preconditioner available in MLD2P4; facilities are also available allowing expert users to extend the set of smoothers and solvers for building new versions of the preconditioners (see Section 7).

We note that the user interface of MLD2P4 2.1 has been extended with respect to the previous versions in order to separate the construction of the multilevel hierarchy from the construction of the smoothers and solvers, and to allow for more flexibility at each level. The software architecture described in [8] has significantly evolved too, in order to fully exploit the Fortran 2003 features implemented in PSBLAS 3. However, compatibility with previous versions has been preserved.

This guide is organized as follows. General information on the distribution of the source code is reported in Section 2, while details on the configuration and installation of the package are given in Section 3. A short description of the preconditioners implemented in MLD2P4 is provided in Section 4, to help the users in choosing among them. The basics for building and applying the preconditioners with the Krylov solvers implemented in PSBLAS are reported in Section 5, where the Fortran codes of a few sample programs are also shown. A reference guide for the user interface routines is provided in Section 6. Information on the extension of the package through the addition of new smoothers and solvers is reported in Section 7. The error handling mechanism used by the package is briefly described in Section 8. The copyright terms concerning the distribution and modification of MLD2P4 are reported in Appendix A.

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