In the problem I am dealing with, I require to repeatedly solve $Ax=b$ where $A$ is a weighted Laplacian matrix of a sparse graph. The right-hand side remains constant. However each time I solve the system, only one weight of the graph changes, effectively changing 4 coefficients in the Laplacian matrix (2 diagonal and 2 off-diagonal entries).
I am currently using the GMRES solver in Petsc, and using the previous solution as the initial guess. However the coefficient changes by 1 or 2 orders of magnitude, and it is not as fast as if the coefficient were changing slightly.
I was wondering if there is anyway to solve this problem any faster than what I have been currently doing. Perhaps something which takes advantage of the linear nature of the problem and involves a direct method.