A New Projected Variant of the Deflated Block Conjugate Gradient Method

The deflated block conjugate gradient (D-BCG) method is an attractive approach for the solution of symmetric positive definite linear systems with multiple right-hand sides. However, the orthogonality between the block residual vectors and the deflation subspace is gradually lost along with the process of the underlying algorithm implementation, which usually causes the algorithm to be unstable or possibly have delayed convergence. In order to maintain such orthogonality to keep certain level, full reorthogonalization could be employed as a remedy, but the expense required is quite costly. In this paper, we present a new projected variant of the deflated block conjugate gradient (PD-BCG) method to mitigate the loss of this orthogonality, which is helpful to deal with the delay of convergence and thus further achieve the theoretically faster convergence rate of D-BCG. Meanwhile, the proposed PD-BCG method is shown to scarcely have any extra computational cost, while having the same theoretical properties as D-BCG in exact arithmetic. Additionally, an automated reorthogonalization strategy is introduced as an alternative choice for the PD-BCG method. Numerical experiments demonstrate that PD-BCG is more efficient than its counterparts especially when solving ill-conditioned linear systems or linear systems suffering from rank deficiency.