Deflation and the low modes of the Dirac operator

The condition number of the Dirac operator D is governed by the ratio of its extreme eigenvalues, \kappa(D) = \lambda_{\max} / \lambda_{\min}. Because the smallest eigenvalues sit near zero, the condition number is large and Krylov solvers converge slowly.

Deflation removes the k smallest modes from the operator that the solver sees:

D_{\text{defl}} = D \left( I - \sum_{i=1}^{k} v_i v_i^{\dagger} \right)

The effective condition number becomes \lambda_{\max} / \lambda_{k+1}, which is dramatically smaller.