This question is a follow-up to this one.
Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to the largest eigenvalues, i.e. $\{( \lambda_i, \vec{q}_i) \}_{i=1}^m \subset \mathbb{R}^+\times \mathbb{R}^n$.
I need to solve $A \vec{x}=\vec{b}$ for $x$.
The idea is that we expand $\vec{x}$ as $\vec{x}=\sum_{i=1}^m \chi_i \vec{q}_i + \sum_{i=m+1}^n \chi_i \vec{q}_i$. Now since the eigenvectors are orthogonal, we can take the dot product of both sides of the equation with $\vec{q}_j$, say, and immediately recover that $\chi_j = \vec{q}_j\cdot \vec{b}/ \lambda_j$. So we define
$$\vec{x}_\mathrm{guess} = \sum_{i=1}^m \frac{\vec{q}_i\cdot \vec{b}}{\lambda_i} \vec{q}_i,$$ and use it as the first guess to any iterative solver.
I would be shocked if this isn't a standard idea, but I don't know what it is called. What is it's name? I would especially like to know what papers to cite.