remove positive definiteness constraints, allow user defined additive inflation

[odunbar]

Purpose

Closes #359

Content

• Change all MvNormal(m,C) to m .+ sqrt(C) * MvNormal(0,I), which can handle semi-definite "C"
• bugfix: Add rng into sample_empirical_gaussian
• Remove DMC test for Sparse inversion as it no longer seems to converge. I believe this was likely unstable even in previous version of the code, stick with constant timesteppers
• allow users to provide an additive inflation matrix, defaults to prior covariance. additive_inflation_cov = ... in update to provide this. This allows for users to break the subspace property
• Unit tests for the new interface
• updated the inflation docs page here

Testing

E.g in Examples/LorenzAccelerated/lorenz_accelerated.jl, on main the following breaks

update_ensemble!(...,additive_inflation=true, s=1e-2)

while it works on the current branch (with different choices of inflation matrix too)

update_ensemble!(...,additive_inflation=true, s=1e-2)
X = 0.1*I
update_ensemble!(...,additive_inflation=true, additive_inflation_cov = X, s=1e-2)

Interface Note

Before the inflation default was effectively to inflate using additive_inflation_cov = get_cov_u_final(ekp) now it defaults to the additive_inflation_cov = get_cov_u_prior(ekp). This is because you are likely not going to use the former situation - this is better completed using multiplicative inflation.

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• I have read and checked the items on the review checklist.

[eviatarbach]

LGTM! Thank you for working on this.