Efficient Preconditioner Updates for Shifted Linear Systems
Stefania Bellavia, Valentina De Simone, Daniela di Serafino, Benedetta Morini · SIAM Journal on Scientific Computing · 2011
We present a technique for building effective and low cost preconditioners for sequences of shifted linear systems $(A + \alpha I) x_\alpha = b$, where A is symmetric positive definite and $\alpha > 0$. This technique updates a preconditioner for A, available in the form of an $LDL^T$ factorization, by modifying only the nonzero entries of the L factor in such a way that the resulting preconditioner mimics the diagonal of the shifted matrix and reproduces its overall behavior. This approach is supported by a theoretical analysis as well as by numerical experiments, showing that it works efficiently for a broad range of values of $\alpha$.