Briggs, "A Multigrid Tutorial" (pg. 35) has the following expressed as 2-D interpolation: \begin{align*} v^h_{2i,2j} &= v_{i,j}^{2h}\\ v^h_{2i+1,2j} &= 0.5\cdot(v_{i,j}^{2h} + v_{i+1,j}^{2h})\\ v^h_{2i,2j+1} &= 0.5\cdot(v_{i,j}^{2h} + v_{i,j+1}^{2h})\\ v^h_{2i+1,2j+1} &= 0.25\cdot(v_{i,j}^{2h} + v_{i+1,j}^{2h} + v_{i,j+1}^{2h} + v_{i+1,j+1}^{2h}), \quad 0 \leq i,j \leq \frac{n}{2} -1 \end{align*}

I believe it is number of discretization steps as otherwise $n/2$ wouldn't be an integer if it were number of points in on axis direction. This means that $i,j$ correspond to coarse gird points. But if that's the case, then the interpolation isn't for the entire grid -- the bound on $i,j$, it only goes up to half the grid.

Can someone explain to me why this is or perhaps what I'm misreading? I'd really appreciate a visual display to make sure I know what's going on (if possible).

  • 1
    $\begingroup$ I think a link to the paper or book you are referring to, along with a page number, would be appropriate here. Some in the community will know exactly what you're talking about, but not everyone will. $\endgroup$
    – Bill Barth
    Mar 25 '13 at 22:16
  • $\begingroup$ I agree with @BillBarth... It would be very helpful to have the link to the document that you're referring to. $\endgroup$
    – Paul
    Mar 28 '13 at 4:52
  • $\begingroup$ @Paul I have the physical book, so I'm not sure how to provide a link except for something like amazon? $\endgroup$ Mar 28 '13 at 6:11

From the formula, $i$ and $j$ are coarse grid indices. The output of interpolation is the fine grid $0 \le i',j' \le n$.

  • $\begingroup$ But on say a $2^3$ by $2^3$ grid, that means $i,j$ stop at 3, which doesn't make sense. $\endgroup$ Mar 25 '13 at 22:27
  • $\begingroup$ Briggs sometimes uses 1-based indexing and sometimes does not include boundary points, but this is just bilinear interpolation, which is also the (row-normalized) transpose of the "full weighted" restriction. $\endgroup$
    – Jed Brown
    Mar 26 '13 at 2:56

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