Sparse SVD

[dallan-keylogic]

Summary/Motivation:

Presently, the SVDToolbox isn't widely utilized, in part because of the limited options for calculating the SVD. Scipy's dense SVD is time-consuming on problems larger than 1000x1000, while its sparse SVD does not work well to find the singular values of the smallest magnitude.

Therefore, I've implemented a custom version of the SVD that uses simultaneous inverse iteration. Due to limited options for sparse matrix factorization available in Scipy, we must explicitly form a normal matrix A.T @ A or A @ A.T in order to invert it, which reduces numerical precision of the results from ~1e-16 to ~1e-8. Therefore a second refinement step is used, in order to project A into its near-null space computed through the inverse iteration and then computes a dense SVD of that matrix, reduces an n by n matrix calculation to an n by n_vec matrix computation, which can be computed rapidly.

In order to try out the new method, check out this branch, optionally merge it into your working branch, then set the svd callback:

import pyomo.environ as pyo
from idaes.core.util.linalg import svd_explicit_normal
from idaes.core.util.model_diagnostics import SVDToolbox

...

m_scaled = pyo.TransformationFactory("core.scale_model").create_using(m, rename=False)
svd_tbx = SVDToolbox(m_scaled)
svd_tbx.config.svd_callback = svd_explicit_normal
svd_tbx.display_underdetermined_variables_and_constraints()
# If you want to view the singular values in addition to the variables and constraints
svd_tbx.s

I'd appreciate testers for this method before we merge it.

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