Inverse distance functions

[ntessore]

How can we provide inverse distance functions (= redshift at which distance equals x) in the cosmology API?

CC: @nstarman

[nstarman]

Without thinking super deeply, there seem to be two ways:

1. Add a corresponding inverse to each method. Advantages: explicit, allows for method-specific speedups if there's an analytic solution. Drawbacks: doubles the API surface.
2. Add an inverse function, like Astropy's z_at_value. Advantages:small API surface addition. Drawback, method-specific speedups?

Inverting non-monotonic functions is hard. There's lots of kwargs — domain, minimizers, etc — that need to be handled carefully. It seems to me for this reason that option 2, where these details are handled once, is best. However, there should be a means to support method-specific speedups. I think that a dict-cache by function will suffice. So the Cosmology API should cover some aspects of this caching process: the storage, registration, (de-registration ?).

So a summary of the API additions are:

def redshift_at_value(func: Callable[[Redshift], ReturnArray], val: ReturnArray, ...) -> : ...

INVERSE_METHODS: Mapping[FunctionType, FunctionType]  # <- read only

def register_inverse_method(...):

What do you think?

[ntessore]

Right, I see the issue in the abstract, but here I guess I am counting that we have $\infty$ functions already, so $2\infty = \infty$.

If this is a major concern, we could explore other means, such as InvertibleXyzDistance which is XyzDistancethat accepts aninverse=True` keyword parameter or similar?

[ntessore]

My main concern about cosmo.z_at_value() is that there's no good way to provide this for anything which has the inverse distance functions directly, which mosts codes do IIRC.

[nstarman]

Why isn't there? Basically dispatch on the cosmo. The specific implementation is not part of the API, but in the docs we could suggest something like

FUNC_INVS = {}

def redshift_at_value(func, value, ...):
    key = (get_cosmology_class(func), func.__name__)
    if key in FUNC_INVS:
        return FUNC_INVS[key](*args, **kwargs)
     ...

[nstarman]

Right, I see the issue in the abstract, but here I guess I am counting that we have ∞ functions already, so 2 ∞ = ∞ .

Also true!

[nstarman]

we could explore other means, such as InvertibleXyzDistance which is XyzDistance that accepts an inverse=True keyword parameter or similar?

Also interesting. This might be hard to type annotate well. How do we deal with 2-arg distances, like comoving_distance(z1, z2) ?

[nstarman]

On a related note, we should also have an API for interpolated cosmologies. Within a monotonic region this is easily invertible to find the redshift.