Analysis of the Deflated Conjugate Gradient Method Based on Symmetric Multigrid Theory

Deflation techniques for Krylov subspace methods and in particular the conjugate gradient method have seen a lot of attention in recent years. They provide means to improve the convergence speed of the methods in a rather straight forward way by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors. However, there are many situations where a more general deflation subspace is advisable. We derive an estimate for the speed of convergence of the deflated conjugate gradient method using theory originally developed for algebraic multigrid methods. Our result holds for general deflation subspaces and is based on the weak approximation property---known from multigrid methods---and a measure of the A invariance of the subspace by the strengthened Cauchy-Schwarz inequality. In addition the result suggests that the techniques developed to construct efficient interpolation operators in algebraic multigrid methods can also be applied to improve deflation subspaces.

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