More draft comments on the operators overview

[jlperla]

NOTE: Made a few additions to the list. I have made many of the changes in #2 from @ChrisRackauckas

A few more to take care of. I have modified equation numbers from the #2 post to refer to the latest version:

• [x] Note that I put in convenience matrices for the zeros and identity at the bottom of section 1. Look for places to use these to simplify notation.
• [x] Note that the notation has moved to use a $\tilde{y}(x)$, etc. whenever talking about the continuous fucntions, so that $y$ can represent the discretized vector on the interior, and $\bar{y} on the closure. Look for any other use of continuous functions to use this notation.
• [x] In general, try to remove text and definitions as you go through that is redundant. The tighter the better for this.
• [x] In particular, go through to remove repetitions of the sizes of matrices/vectors unless you are repeating it to illustrate a point. Note that I put the sizes of everything near the top of Section 2, so feel free to put anything I missed there.
• [x] Never use \nonumber, since inevitably you want to refer to the equation in a comment and you don't have a number for it!
• [x] From #2: (8) is missing a uhat. If you move the uhats by taking the interior then you see that it essentially goes away. ^ud = LQu What you actually want to solve has uhat on both sides, which is why there's Q (since then you know the derivative of all of the quantities you're solving for!). Ping him if this is unclear
• [x] From #2: For (32) and (38), just use R to go from the full to the interior.
• [x] Putting the $\Delta$ on top with a negative exponent is odd. There are a few places like this, all of which should be removed. I think they will be cleaner after you simplify the definitions.
• [x] In general, lets remove unnecessary expansion of algebra. If we have defined a matrix/vector, then no need to substitute.
  • In particular, I do not see the benefit of expanding out the P in equation (28) and (29). Just define it as $P = r \mathbf{I} - A$, or whatever it is, using the identity matrix. No need to even denote the dimensions of the identity matrix since it is unambiguous.
  • Similarly (47), (50), and (53)
  • Also, better to have the composition of the A matrices from the different underlying operators than to combine them ourselves. In particular, the A in (43), (49), (52) should be composed of the sub-A rather than the combination ourselves. This will also be how the software implementation works.
• [x] Just to repeat this in the case of (49) and (50), We want to represent $A$ as the composition of equations (12) to (14), potentially with multiplication by vectors of constants/etc.
• [x] Where possible, we should define reusable matrices for the examples. I added some obvious ones in (12) to (19), and we should go through and use them in all the examples. I suspect the examples sections come become far shorter after we do this (and also when we compose the A matrices with these composable matrices).
• [x] Not sure why there is the $\underline{\tilde{x}}$, etc. in section 3.3. Should these just be changes to use $x^{\max}$ and $x^{\min}$? (to use the new notation). Recall that I had to change things because of the use of $\bar{x}$ to represent the closure of the vector $x$, and that $\tilde{}$ is reserved for continuous functions.
• [x] Equation (5) doesn't seem to conform, unless I am missing something. Go through those carefully.
• [x] Below (5) Also the ...in an iterative solver converge to... doesn't make sense to me. I believe this came from something @ChrisRackauckas wrote, so you can ask him to clarify given the iterated notation.
• [x] "Section 2.1, Affine relations" needs some work to be more clear on the notation/etc.
  • In particular, I would go through it carefully to make sure it uses the new notation and rewrite things as you go through to make sure things are clear
  • It says "with some abuse of notation, that such expression means both the discretzed and non-discretized forms". I think this is why it is confusing. I would go through this being clear on the separation between $u$, $\bar{u}$, and $\tilde{u}(x)$.
• [x] Not sure that Section 2.2 shoudl survive, unless I am missing something. I would instead get rid of this and just point out the connection to Dirichlet0, Dirchlet, Neumann0, and Neumann in the examples of Section 3 and probably in the equations (17) to (19) and beyond. With this, it might be worth putting in a $b$ to go with equations (17) to (19) to distinguish Dirichlet0 from Dirichlet, etc. But I leave that to you.
• [x] In general, I am confused with the algebra of the examples since so much involves $Q_A$ and $Q_b$ instead of directly using $Q$. In fact, I don't even see the $Q$ defined directly, which makes it hard to connection to the equations at the beginning of Section 3.
• [ ] Look for additional TODO in the document to see some other tasks and questions. There are some things where matrices don't conform, and I suspect a mistake is there.

[jlperla]

Also, worth looking through the examples using the prototype interface in https://nbviewer.jupyter.org/github/MSeeker1340/PDERoadmap/blob/operator_interface_prototype/Discretizing%20Linear%20Operators.ipynb to look for inconsistencies.

[jlperla]

Note that these tasks do not implement the #7 notation, and the equation numbers may be off. However, I think that made a few of the changes above when implementing the notation fix.

[jlperla]

@stevenzhangdx @FernandoKuwer I think this issue is almost complete. I am closing out the affine relations in the issue list for now. There is a good chance that those parts of the algebra would change if #13 ends up working... Which may be a precursor for jump diffusion processes.

I would go through the rest of the issue and close it if you think you have done everything else. Luckily, our notation and most of Section 2 seems to survive if we do the #13 approach, but that is too be proven.