Algebraic Domain Decomposition Methods for Highly Heterogeneous Problems

Abstract

We consider the solving of linear systems arising from porous media flow simulations with high heterogeneities. Using a Newton algorithm to handle the non-linearity leads to the solving of a sequence of linear systems with different but similar matrices and right hand sides. The parallel solver is a Schwarz domain decomposition method. The unknowns are partitioned with a criterion based on the entries of the input matrix. This leads to substantial gains compared to a partition based only on the adjacency graph of the matrix. From the information generated during the solving of the first linear system, it is possible to build a coarse space for a two-level domain decomposition algorithm that leads to an acceleration of the convergence of the subsequent linear systems. We compare two coarse spaces: a classical approach and a new one adapted to parallel implementation.

Accelerating the cosmic microwave background map-making procedure through preconditioning

“…Approximations of eigenvectors/eigenvalues of a matrix in the context of deflation techniques are discussed in e.g., Morgan (1995) and Chapman & Saad (1996). Havé et al (2013) gives a recent example of applications of all these techniques in the context of preconditioners of linear systems. There are several different classes of iterative methods for solving the linear system of equations,…”

A Spectral Analysis of Subspace Enhanced Preconditioners

“…The authors of [9] used these two types of preconditioners in a multiplicative fashion PMAx = PMb, in which the fine space preconditioner M removes the large eigenvalues of A, and the spectral preconditioner P removes the small eigenvalues of MA. In the combined preconditioner, the coarse space used in P should be close enough to the space spanned by the eigenvectors associated with the small eigenvalues of MA.…”

“…by the two-level multiplicative Schwarz method [9,23]. Here, κ is the diffusion function of x and y.…”

“…It is well-known that the convergence of Krylov subspace methods for solving linear systems depends on the eigenvalue distribution of A. Recently, several studies [4,5,[7][8][9][13][14][15][16][17][18][19][20][21][25][26][27] have shown that by removing the subspace spanned by the eigenvectors corresponding to several small eigenvalues from the Krylov search space makes the spectrum more clustered and consequently the convergence is improved. Preconditioners based on deflation and coarse space correc-B Tao Zhao zhaot.cn@gmail.com tion techniques are two main types of spectral preconditioners that deflate or shift small eigenvalues.…”

“…The subspace spanned by the columns of Z is referred to as a coarse space [9,27]. In an ideal situation, the coarse space contains the vectors corresponding to the lower part of the spectrum that is responsible for the stagnation, or the slow convergence, of Krylov subspace methods.…”