Algopaul/greedy_quadratic_manifolds

Greedy quadratic manifolds

Code for the computation of quadratic manifolds using a greedy column selection as proposed in

@Misc{SchwerdtnerP2024Greedy,
    title	= {Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction},
    author	= {Paul Schwerdtner and Benjamin Peherstorfer},
    doi		= {10.48550/arXiv.2403.06732},
    year	= {2024},
    archiveprefix={arXiv},
    primaryclass= {math.NA}
}

Installation

Requires python3.11. Create new virtual environment and install requirements:

python3.11 -m venv .venv
source .venv/bin/activate
pip install -e .

Usage

The greedy column selection is demonstrated in the notebook greedy_quadratic_manifolds.ipynb; alternatively, the code can be run as follows from the command line:

python greedyqm/quadmani.py

This computes a quadratic manifold of dimension 20 for the linear advection dataset. Parameters are passed as flags, e.g.

python greedyqm/quadmani.py --reduced_dimension=10 --n_vectors_to_check=100 --reg_magnitude=1e-8

This computes a quadratic manifold of dimension 10 for the linear advection dataset with a regularization magnitude of 1e-8 and chooses the next column from the next 100 singular vectors.

If you want to use the greedy column selection in your own code, you can use the function quadmani_greedy from greedyqm/quadmani.py. The function signature is

quadmani_greedy(data_points, reduced_dimension, n_vectors_to_check, reg_magnitude, feature_map)

Here data_points is a jax.numpy array of shape (n_state, n_samples), reduced_dimension is the desired dimension of the quadratic manifold, n_vectors_to_check is the number of singular vectors to consider for the greedy column selection, and reg_magnitude is the regularization magnitude. The feature_map is a nonlinear operation on the reduced data points, which is set to the Kronecker product by default, but other nonlinear functions are possible.