Preconditioning

Preconditioner with a starting guess

In many applications, for many reasons, we have a starting guess $\bold m_0$ of the solution. You might worry that you could not find the starting preconditioned variable $\bold p_0= \bold S^{-1}\bold m_0$ because you did not know the inverse of $\bold S$ . The way to avoid this problem is to reformulate the problem in terms of a new variable $\tilde {\bold m}$ where $\bold m = \tilde {\bold m} + \bold m_0$ . Then $\bold 0\approx \bold F \bold m - \bold d$ becomes $\bold 0\approx \bold F \tilde {\bold m} - (\bold d - \bold F \bold m_0)$ or $\bold 0\approx \bold F \tilde {\bold m} - \tilde {\bold d}.$ Thus we have accomplished the goal of taking a problem with a nonzero starting model and converting it a problem of the same type with a zero starting model. Thus we do not need the inverse of $\bold S$ because the iteration starts from $\tilde {\bold m}=\bold 0$ so $\bold p_0 = \bold 0$ .