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I have the energy function that I would like to minimize as: $$\sum_{\{i, j\}}((h_i - h_j) - q_{ij})^2$$ This is applied over a 2D grid, where $q_{ij}$ is the relative height between two cells $i$ $j$, and $h_i$ is the absolute height of the cell that I am trying to find. Note that $q_{ij}$ is only non-zero for the 4-neighborhood of each cell i.e. directly adjacent ones. I was given a hint that this can be formulated as a Poisson problem but I've not been able to get there.

This is my current attempt:

We can expand this given energy to get: $$\sum_{\{i, j\}} (h_i^2 -2h_i h_j -2h_i q_{ij} + h_j^2 +2h_j q_{ij} + q_{ij}^2)$$

Which is then equivalent to the individual sums: $$\sum_{\{i, j\}}(h_i^2) -2\sum_{\{i, j\}}(h_i h_j) -2\sum_{\{i, j\}}(h_i q_{ij}) + \sum_{\{i, j\}}(h_j^2) +2\sum_{\{i, j\}}(h_j q_{ij}) + \sum_{\{i, j\}}(q_{ij}^2)$$

I then define the column vector $H$ as the vector with all values of $h_i$ stacked on top of each other. I also define $M$ as the adjacency matrix between cells, which in this case would be a grid yielding the 4-neighborhood. Finally I define $Q$ as the matrix with the same sparsity pattern as $M$, but with values representing the relative height between $i$ and $j$.

With $H$, $M$ and $Q$ I can write these sums in matrix form:

$$4H^TH -2H^T(MH) -2(QH)^TO + (MH)^T(MH) + 2(QMH)^TO + O^TQ^2O$$ Where $O$ is a vector of ones. My reasoning for the $1^{st}$ coefficient is that each $h_i$ will appear on the left of the sum 4 times.

From this point I can group similar terms:

$$H^T(4I -2M + M^2)H + H^T(2QM -2Q)O + O^TQ^2O$$

Which I write as: $$H^TAH + H^T(2QM -2Q)O + O^TQ^2O$$

I then use the reasoning from this question (which I admit to not fully understanding), to get my final system:

$$AH = (QM - Q)O,\quad A = 4I -2M + M^2$$

W̶e̶ ̶c̶a̶n̶ ̶w̶o̶r̶k̶o̶u̶t̶ ̶t̶h̶a̶t̶ ̶$̶A̶$̶ ̶i̶s̶ ̶a̶c̶t̶u̶a̶l̶l̶y̶ ̶$̶4̶I̶ ̶-̶ ̶L̶$̶ ̶w̶h̶e̶r̶e̶ ̶$̶L̶$̶ ̶i̶s̶ ̶t̶h̶e̶ ̶l̶a̶p̶l̶a̶c̶i̶a̶n̶ ̶m̶a̶t̶r̶i̶x̶. This to me doesn't look like a poisson problem, so I would like to know where I went wrong in the above workings? My understanding of what a poisson problem looks like is from the wiki page, so perhaps I've misunderstood?

Any hints or advice would be great, thank you.

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My mistake was in the conversion to vectors and matrices. $(MH)^T(MH)$ should instead be: $4H^TH$

The reason being, that each cell shall be present as a neighbor to 4 other cells. This gives us: $$8H^TH -2H^T(MH) -2(QH)^TO + 2(QMH)^TO + O^TQ^2O$$ And when we work through with this equation, a standard Poisson problem does emerge. I also had a mistake on the final line, $QM - Q$ should instead be $Q - QM$. $$AH = \frac{1}{2}(Q - QM)O, \quad A = 4I - M$$

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