4.3 Smoothers and coarsest-level solvers

The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods, a hybrid version of the forward and backward Gauss-Seidel methods, and the additive Schwarz (AS) ones (see, e.g., [23, 24]).

The hybrid Gauss-Seidel version is considered because the original Gauss-Seidel method is inherently sequential. At each iteration of the hybrid version, each parallel process uses the most recent values of its own local variables and the values of the non-local variables computed at the previous iteration, obtained by exchanging data with other processes before the beginning of the current iteration.

In the AS methods, the index space Ω^k is divided into m _k subsets Ω_i^k of size n_k,i, possibly overlapping. For each i we consider the restriction operator R_i^k ∈ ℝ^n_k,i×n_k that maps a vector x^k to the vector x _i^k made of the components of x^k with indices in Ω _i^k, and the prolongation operator P _i^k = (R _i^k)^T . These operators are then used to build A_i^k = R _i^kA^kP _i^k, which is the restriction of A^k to the index space Ω _i^k. The classical AS preconditioner M _AS^k is defined as

where A_i^k is supposed to be nonsingular. We observe that an approximate inverse of A_i^k is usually considered instead of (A _i^k)^-1. The setup of M _AS^k during the multilevel build phase involves

• the definition of the index subspaces Ω_i^k and of the corresponding operators R_i^k (and P _i^k);
• the computation of the submatrices A_i^k;
• the computation of their inverses (usually approximated through some form of incomplete factorization).

The computation of z^k = M _AS^kw^k, with w^k ∈ ℝ^n_k, during the multilevel application phase, requires

• the restriction of w^k to the subspaces ℝ^n_k,i, i.e. w _i^k = R _i^kw^k;
• the computation of the vectors z_i^k = (A _i^k)^-1w _i^k;
• the prolongation and the sum of the previous vectors, i.e. z^k = ∑ _i=1^m_kP _i^kz _i^k.

Variants of the classical AS method, which use modifications of the restriction and prolongation operators, are also implemented in MLD2P4. Among them, the Restricted AS (RAS) preconditioner usually outperforms the classical AS preconditioner in terms of convergence rate and of computation and communication time on parallel distributed-memory computers, and is therefore the most widely used among the AS preconditioners [6].

Direct solvers based on sparse LU factorizations, implemented in the third-party libraries reported in Section 3.2, can be applied as coarsest-level solvers by MLD2P4. Native inexact solvers based on incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel, and block Jacobi preconditioners are also available. Direct solvers usually lead to more effective preconditioners in terms of algorithmic scalability; however, this does not guarantee parallel efficiency.