gp: efficient low-rank inverse update

[asher-m]

In Gaussian process kriging, it is necessary to compute the inverse of the quantity $k_{XX} + \sigma I$.

As of 4ec65a0, that quantity is computed as a dense inverse every time the conditioning data are updated and stored in the UncertainSCI/gp subpackage for scalar- and vector-valued Gaussian processes respectively as

self.inv = np.linalg.inv(self.k(x) + sigma * np.eye(len(x)))

(here) and

self.inv = np.linalg.inv(self.k(x) + sigma.flatten() * np.eye(len(x) * self.cdim))

(here).

This could be improved by performing a low-rank update of the inverse when new data are added.

[asher-m]

Here are some preliminary thoughts:

• It is likely not possible to avoid a dense inverse of all conditioning data when the process is initialized.
• Instead, when the process is iteratively updated, the new data point(s) should be appended to the existing data and a low-rank update of the existing inverse can be computed (e.g., via Schur complement).