Preconditioner conflicts with starting guess

In many applications, for many reasons, we have a starting guess $\bold m_0$ of the solution. It is usually a wise idea to use the starting guess rather than begin anew from $\bold m = \bold 0$.This presents a problem. To do preconditioning, we begin by defining (implicitly) the new underlying model variables $\bold p$by the substitution $\bold m = \bold S \bold p$.When we are given an initial $\bold m_0$,then the corresponding $\bold p_0$ should be $\bold p_0= \bold S^{-1}\bold m_0$but we might not know $\bold S^{-1}$.So a noninvertible preconditioner is in conflict with a nonzero starting model. You cannot have both. (If you want to explore a null space by trying various starting solutions, it might suffice to try various starting $\bold p_0$;you need not try various starting $\bold m_0$.)