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.