CFD Online Discussion Forums - Discrete Adjoint derivative computation

I am looking for some guidance on the implementation of the Discrete Adjoint method, specifically the computation of the partial derivatives via Finite Differencing.

source(s):
DAFoam: An Open-Source Adjoint Framework for Multidisciplinary Design Optimization with OpenFOAM
https://arc.aiaa.org/doi/10.2514/1.J058853

The first major step in solving the Discrete Adjoint is computing the partial derivatives of the vector of flow residuals ( $\textbf{R}$ ) with respect to the vector state variables ( $\textbf{w}$ ). This is possible via Finite Difference:

Set $\textbf{w}=[u,v,w,p,\nu,\phi]$ and $\textbf{R}=[R_u,R_v,R_w,R_p,R_\nu,R_\phi]$ ,

$\left(\frac{\partial\textbf{R}}{\partial\textbf{w}}\right)_{ij}=\frac{R_i(\textbf{w}+\epsilon\textbf{e}_{j})-R_i(\textbf{w})}{\epsilon}$

where the subscripts

and

are the row and column indices, respectively, $\epsilon$ is the step size, and $\textbf{e}_j$ is a unit vector with unity in row

. As I understand it, this involves sequentially perturbing each element of the state variable vector in a reference cell and recomputing the residual vector. This should produce a $n_{\textbf{w}}\times n_{\textbf{w}}$ matrix, with $n_{\textbf{w}}$ calls to residual function (equal to the number of columns). Is this correct?

In the DAFoam implementation, a graph-colouring method is used to accelerate the computation of the partial derivatives via Finite Difference. This works by exploiting the sparsity of a Jacobian matrix to simultaneously perturb sets of columns that influence independent rows. An example is given for a $5\times 5$ diagonal Jacobian matrix but I am not sure what step this relates to in the context of computing the partial derivates for $\frac{\partial\textbf{R}}{\partial\textbf{w}}$ , particularly for a 3D problem.

Assumptions:
1. For an unperturbed and converged flow solution, each cell contains a $1\times n_{\textbf{w}}$ vector of flow variables and a $1\times n_{\textbf{w}}$ vector of residuals.

Questions:
1. when the source(s) discusses graph-colouring, what matrix are they referring to? Is the $n_{\textbf{w}}\times n_{\textbf{w}}$ Jacobian of $\frac{\partial\textbf{R}}{\partial\textbf{w}}$ in each cell?
2. when perturbing a flow variable e.g. the u-component of velocity, is $\epsilon$ added to the reference value in a specific cell only, or does this perturbation affect the reference values in neighbouring cells?
3. how does perturbing a flow variable in a cell influence other rows in the matrix?

Appreciate any input,

Chris