Preconditioning Parametrized Linear Systems

TL;DR

This paper presents a simple, effective method for recycling and updating preconditioners in sequences of parametrized linear systems, reducing computational costs and maintaining convergence efficiency.

Contribution

The authors introduce a novel preconditioner update technique based on mapping previous matrices to new ones, applicable to any preconditioner and suitable for slowly changing matrix sequences.

Findings

Effective preconditioner updates for sequences of linear systems

Reduced computational cost for preconditioning in iterative methods

Successful application with algebraic multigrid preconditioners

Abstract

Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems, it is advantageous to recycle preconditioners, that is, update a previous preconditioner and reuse the updated version. In this paper, we introduce a simple and effective method for doing this. Although our approach can be used for matrices changing slowly in any way, we focus on the important case of sequences of the type $(s_k\textbf{E}(\textbf{p}) + \textbf{A}(\textbf{p}))\textbf{x}_k = \textbf{b}_k$, where the right hand side may or may not change. More general changes in matrices will be discussed in a future paper. We update preconditioners by defining a map from a new matrix to a previous matrix, for example the first matrix in the sequence, and combine the preconditioner for this previous matrix with the map to define the new preconditioner. This approach has several advantages. The update is entirely independent from the original preconditioner, so it can be applied to any preconditioner. The possibly high cost of an initial preconditioner can be amortized over many linear solves. The cost of updating the preconditioner is more or less constant and independent of the original preconditioner. There is flexibility in balancing the quality of the map with the computational cost. In the numerical experiments section we demonstrate good results for several applications, in particular when using an algebraic multigrid preconditioner.