# Initial guesses for perturbed linear systems

Suppose you solve a linear system $Au = f$ by an iterative method, e.g. conjugate gradients or Richardson iteration. Then you try to solve a linear system that is slightly perturbed in the matrix and the right-hand side, say, $\tilde A \tilde u = \tilde f$.

Does it make sense to use the old solution as starting value $\tilde u_0 = u$ for the iterative method? "Makes sense" means that there is a relyable gain in the running time of the iterative method. I wonder whether this leads to an improvement in general, such that it can be regarded as advised practice.

An application that I have in mind comes from adaptive finite elements. If we have computed a solution $u$ on a coarse grid, and want to find a solution $\tilde u$ on a finer grid (which might have been generated based by an adaptive method), the starting value for any iterative algorithm can be the prolongation of $u$ onto the finer grid. Similarly, the Newton-method or Picard-iteration, which is involved in the solution of nonlinear problems, could be "boosted" that way, if it makes any sense at all.

• At least from a theoretical point of view, recycling a solution on a coarse grid as a starting value on a fine grid gives an initial error bound that grows smaller and smaller as the mesh is (uniformly) refined, while $0$ as a starting value leads to an ever increasing initial error. So very basic error estimates suggest it makes a tremendous difference. - So for me, I still wonder whether this is done in practice. In many communities, it is not standard. – shuhalo Mar 5 '12 at 23:32

• Good way to think about this! There is some value to thinking about what $(A-\tilde{A})(f-\tilde{f})$ looks like too I think. – meawoppl Mar 5 '12 at 21:33