Low-rank updates of balanced incomplete factorization preconditioners

Abstract

Let Ax = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + P Q T where P , Q ∈ R n×k are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system Bx = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.

Updating preconditioners for modified least squares problems

“…Step 2 in Algorithm 1 represents the extra cost in the application of the preconditioner with respect to the case of non-updating an existing one. If R 12 and R S are kept sparse and the number of added or removed equations is small compared with the problem size, this overhead is small and can be amortized even for moderate reductions on the number of iterations, see [7].…”

“…It is important to note that in our algorithm, to compute an update with moderate fill-in, element dropping was applied in three different steps. First, a sparsification of the new block of rows (equations) added to the matrix was done before computing the block column R 12 in equation (7). Then, the computation of the block R 12 itself was done incompletely by dropping small entries.…”

“…Then, the computation of the block R 12 itself was done incompletely by dropping small entries. Finally, an incomplete factorization was computed for the Schur-complement block S, see equation (7). The respective tolerances were 1.0, 1.0 and 0.1 for all the matrices.…”

“…The cases in which we are interested correspond to LS modified problems whose normal equations have a coefficient matrix that does not change in size but their entries do. These problems can be formulated as a low-rank update of the original normal equations and we propose updating the preconditioner following the ideas presented in [7]. The goal is computing the update with smaller cost than obtaining a new preconditioner from scratch, but with comparable performance.…”

On Updating Preconditioners for the Iterative Solution of Linear Systems

“…The general framework of the technique is discused in Chapter 2. It is based on the work presented in [24]. Basically, it is a low-rank update of an incomplete LU factorization of the symmetric part of the system matrix H by a bordering method.…”