I'm trying to solve a linear system of equations $$Ax = b,$$$A\in \mathbb R^{n\times n}, x,b\in\mathbb R^n$. The linear system is an iteration in a nonlinear solution procedure. The nonlinear system is very sensitive to the initial guess.
I can solve it on a very small problem, let's call this the solution to the problem $A_1u_1 = b_1,\ u_1\in \mathbb R^m, \ m<n$. I want to use a projection of $u_1$ as an initial guess for the underlying original system. I introduce the matrix $P\ \in\mathbb R^{n\times m}$ such that it minimizes the residual $$ ||APu_1-b||_2.$$ I was wondering if there exists a known procedure in order to find $P$? I have tried to do some kind of adapted minimal residual method, but know minimizing in $P$ in stead of $u$, but this lead nowhere.
Any help is appreciated, a reference to useful literature as much as a real solution to my problem,