Update in MFCK/ SMFCK

[mcastanoUQ]

Summary

Extends the MFCK and SMFCK multi-fidelity surrogate models with gradient-based
hyperparameter optimization, heteroscedastic noise prediction, and a sequential
optimization scheme. Also includes an internal refactoring that makes the shared
co-Kriging machinery reusable and inspectable.

Modularization

Shared logic previously duplicated across the MFCK and SMFCK implementations has
been extracted into helper methods:

• Cholesky decomposition of the covariance matrix
• Cross-covariance assembly between fidelity levels
• Diagonal prior covariance construction

Beyond removing duplication, this exposes the intermediate quantities so that
internal behaviour (conditioning, cross-level correlations) can be inspected
directly rather than only through the final prediction.

Gradients for MFCK

Analytical gradients of the log marginal likelihood with respect to the kernel
hyperparameters are now implemented for the RBF kernel. This allows MFCK to be
trained with gradient-based optimizers such as TNC.

Heteroscedastic noise prediction

Both MFCK and SMFCK can now predict input-dependent noise

Sequential optimization
An option that integrates hyperparameter optimization level by level rather
than jointly. This option reduces the training complexity of MFCK to make it comparable to that of MFK.

Notebooks

Updated to use the sequential optimization option in all tutorials associated:

• tutorial/MultiFi/SMT_MFCK_tutorial.ipynb
• tutorial/MultiFi/SMT_SMFCK_tutorial.ipynb
• tutorial/MultiFi/SMT_SMFK_tutorial.ipynb
• tutorial/MultiFi/Paper_SMFCK_Notebook_Rosenbrock.ipynb