PRECONDITIONING PARAMETRIZED LINEAR SYSTEMS

doi:10.1137/20M1331123

2021-06-17

Arielle Carr *

Eric de Sturler

Serkan Gugercin

DOI：10.1137/20M1331123

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摘要

En 中文

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. We consider recycling preconditioners for both the general case of sequences of linear systems A(p(k))x(k) = b(k) as well as the important special case of the type (s(k)E + A)x(k) = b(k). The right-hand sides may or may not change. We update preconditioners by defining a map from a new matrix to a previous matrix, for example, the first matrix in the sequence. We then combine the preconditioner for this previous matrix with the map to define the new preconditioner. This approach has several advantages. The update is 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 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.

Keyword:

preconditioning

recycling preconditioners

Krylov subspace methods

sparse approximate inverse

parametrized systems