Computing $L_i$ instead of $L$

[orkolorko]

A first modification we could do, in the long refactor of Dual+iterate could be the following.

If $T$ satisfies the following assumption there exists a partition ${P_i}$ of $M$ of full measure such that

1. $T_i: = T|_{P_i}$ is injective
2. $|DT_i|>0$

then we can write

$$ L f(x) =\sum_{i=1}^n L_i f $$

where

$$ L_i f(x) = \frac{f(T_i^{-1}(x))}{|DT(T_i^{-1}(x))|}\chi_{T_i(P_i)}(x) $$

to each branch is associated an operator, and the discretized operator is the sum of these operators.

[fph]

Maybe I am missing some motivation; what would be the advantage of doing this?

Here $n$ is the number of branches, right? And that expression is for the hat assembler only, not for Ulam?

Anyway I think we already do something very similar: the preimages are computed branch-by-branch, so we compute those terms separately, store them in the I, J, nzvals vectors and add them up when we create the sparse matrix in assemble.

In other words, the elements of our "dual" are already the functions $L_i f(x_k)$ for all $i$ and $k$.

[fph]

Oh OK maybe I get it now; your plan is having this part when we split the assembly over branches as a "common sub-factor" of assemble for all basis?

For the future my suggestion would be instead having completely separated assemble methods for each basis, dispatched over the base type: especially looking at future Chebyshev / FFT strategies, they seem different enough that introducing a common interface is just a complication. Indeed all that iterate business with the duals makes the current code really hard to read.

And in a sense preimages is already a common part of all of them that we have identified and abstracted out. Note that calling preimages on a dynamic already concatenates the preimages over each branch in a single vector, so it already plays that role, more or less.

[orkolorko]

I am thinking about the fact that each one of the branches can be computed in parallel, mostly, but I need to think more about it...

[fph]

Note that at least for ComposedDynamics, one should not split into branches; see #153 (comment) for an explanation.

Apart from that, I agree that preimages along different branches could be computed in parallel, and then afterwards also $L_i$ as you suggest; that is something to study.