Add generic diagnostics capability

[briochemc]

Let's try to add some diagnostic functionality to AIBECS that generalizes the work done in Pasquier and Holzer, 2018.

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Preliminary notes

Let's start from the generic equation

(∂ₜ + T) x = G(x).

x could represent a multitude of tracers and processes. Within x, there may be separate groups of tracers with a group per compound or element. E.g., one could be tracking many elements including, e.g., phosphorus, whose group could be composed of three pools, like PO₄, DOP, and POP. For the diagnostics that I am interested in, we will assume for simplicity and without any loss of generality, that there is only one element (one group) here.

We can then express the group's system without loss of generality as a bunch of

• source terms s_p(x) that inject x into the system,
• of "transfers" terms that exchange the element between tracers of our group J_i→j(x),
• and of "death" processes d_q(x), which ultimately remove x from the system.

Mathematically, that means we can write

G(x) = ∑_p s_p(x) + ∑_ij J_i→j(x) + ∑_q d_q(x).

We construct the LEM by first creating linear-equivalent terms for J_i→j(x) and d_q(x), evaluated at the steady-state solution x given by

T x = G(x)

In other words, the LEM is built such that

G(x) = ∑_p s_p + ∑_ij L_i→j x + ∑_q L_q x

when x is the steady-state, and where

• L_i→j is a block matrix where only 2 blocks are non-zero, (i,j) and (j,i), which have diagonals -(J_i→j(x))_i / x_i, and +(J_i→j(x))_j / x_i, respectively. (Note we use x_i to "linearize" J_i→j so that the rate of transfer is specific to the removed tracer.)
• and L_q is a diagonal matrix with diagonal d_q(x) / x.

We then construct the LEM simply as

(∂ₜ + H) x = ∑_p s_p

where

H = T - ∑_ij L_i→j - ∑_q L_q

and we can then exploit this for powerful diagnostics as in Pasquier and Holzer, 2018. Fractions that came from source s_p, or fraction that will be removed via process d_q, are available from a single backslash with H. One can also further partition according to each i→j passage by removing J_i→j from H into a F operator and iteratively reapplying the source term. Direct computations leveraging the classical identities ∑_n xⁿ and ∑_n n xⁿ also allow for direct computations of, e.g., the number of i→j passages.

[briochemc]

Maybe classify diagnostics in different groups (passages, age, and so on) and lay down a plan to implenent them. E.g., automate the extraction of the equivalent linear models and use dispatch to solve diagnostic equations of Solved and Unsolved problems (e.g., to solve the main problem and the diagnostic in one go).