Section: New Results

Algebraic preconditioners

Our work focused on the design of robust algebraic preconditioners and domain decomposition methods to accelerate the convergence of iterative methods.

In [8] we introduce the block filtering decomposition, a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We present experimental results that demonstrate the efficiency of the proposed preconditioner on a set of matrices arising from the discretization of partial differential equations on two-dimensional and three-dimensional grids. We also show that the numerical efficiency of the preconditioner does not suffer from the reordering of the unknowns for the matrices in our test set, which can have highly heterogeneous and anisotropic coefficients.

In [9] we discuss the usage of overlapping techniques for improving the convergence of preconditioners based on incomplete factorizations. To enable parallelism, these preconditioners are usually applied after the input matrix is permuted into nested bordered block diagonal form. We use k-way partitioning with vertex separator (KPVS) to recursively partitions the corresponding graph of the input matrix into $k$ subgraphs using a subset of its vertices called separators. In the case where $k=2$, it is called nested dissection. The overlapping technique is then based on algebraically extending the associated subdomains of these subgraphs and their corresponding separators obtained from KPVS by their direct neighbours. This approach is known to accelerate the convergence of domain decomposition methods, where the input matrix is partitioned into a number of independent subdomains using k-way graph partitioning, a different graph decomposition technique. We discuss the effect of the overlapping technique on the convergence of two classes of preconditioners, based on nested factorization and block incomplete LDU factorization.

In [22] we introduce LORASC, a robust algebraic preconditioner for solving sparse linear systems of equations involving symmetric and positive definite matrices. The graph of the input matrix is partitioned by using k-way partitioning with vertex separators into $N$ disjoint domains and a separator formed by the vertices connecting the $N$ domains. The obtained permuted matrix has a block arrow structure. The preconditioner relies on the Cholesky factorization of the first $N$ diagonal blocks and on approximating the Schur complement corresponding to the separator block. The approximation of the Schur complement involves the factorization of the last diagonal block and a low rank correction obtained by solving a generalized eigenvalue problem or a randomized algorithm. The preconditioner can be build and applied in parallel. Numerical results on a set of matrices arising from the discretization by the finite element method of linear elasticity models illustrate the robusteness and the efficiency of our preconditioner.

The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating.

In [5] , we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. In [12] , we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

Multiphase, compositional porous media flow models lead to the solution of highly heterogeneous systems of Partial Differential Equations (PDEs). In [7] , we focus on overlapping Schwarz type methods on parallel computers and on multiscale methods. We recall a coarse space that is robust even when there are such heterogeneities. The two-level domain decomposition approach is compared to multiscale methods.

In [16] , we investigate two-level preconditioners on the extended linear system arising from the domain decomposition method. The additive Schwarz method is used as a smoother, and the coarse grid space is constructed by using the Ritz vectors obtained in the Arnoldi process. The coarse grid space can be improved adaptively as the Ritz vectors become a better approximation of the eigenvectors. Numerical tests on the model problem demonstrate the efficiency.