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Although some preconditioners are available for solving dense linear systems,there are still many matrices for which preconditioners are lacking, inparticular in cases where the size of the matrix $N$ becomes very large. Thereremains hence a great need to develop general purpose preconditioners whosecost scales well with the matrix size $N$. In this paper, we propose apreconditioner with broad applicability and with cost $\mathcal{O}(N)$ fordense matrices, when the matrix is given by a smooth kernel. Extending themethod using the same framework to general $\mathcal{H}^2$-matrices isrelatively straightforward. These preconditioners have a controlled accuracy(machine accuracy can be achieved if needed) and scale linearly with $N$. Theyare based on an approximate direct solve of the system. The linear scaling ofthe algorithm is achieved by means of two key ideas. First, the$\mathcal{H}^2$-structure of the dense matrix is exploited to obtain anextended sparse system of equations. Second, fill-ins arising when performingthe elimination are compressed as low-rank matrices if they correspond towell-separated interactions. This ensures that the sparsity pattern of theextended sparse matrix is preserved throughout the elimination, hence resultingin a very efficient algorithm with $\mathcal{O}(N \log(1/\varepsilon)^2 )$computational cost and $\mathcal{O}(N \log 1/\varepsilon )$ memory requirement,for an error tolerance $0 < \varepsilon < 1$. The solver is inexact, althoughthe error can be controlled and made as small as needed. These solvers arerelated to ILU in the sense that the fill-in is controlled. However, in ILU,most of the fill-in is simply discarded whereas here it is approximated usinglow-rank blocks, with a prescribed tolerance. Numerical examples are discussedto demonstrate the linear scaling of the method and to illustrate itseffectiveness as a preconditioner.

The inverse fast multipole method: using a fast approximate direct solver as a preconditioner for dense linear systems