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Index of Notation

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Shane edited this page Feb 3, 2021 · 3 revisions

We generally denote scalars in lower case, vectors in bold lower case, matrices in upper case, and indicate low-dimensional quantities with a hat. In the code, a low-dimensional quantity ends with an underscore, so that the model classes follow some principles from the scikit-learn API.

Dimensions

Symbol Code Description
n Dimension of the full-order system (large)
r Dimension of the reduced-order system (small)
m Dimension of the input u
k Number of state snapshots, i.e., the number of training points
s Number of parameter samples for parametric training
p Dimension of the parameter space
d Number of columns of the data matrix D

Vectors

Symbol Code Size Description
x Full-order state vector
x_ Reduced-order state vector
xdot_ Reduced-order state time derivative vector
x_ROM Approximation to x produced by ROM
c_ Learned constant term
u Input vector
f(t,x,u(t)) or f(x,u) Full-order system operator
f_(t,x_,u(t)) or f(x_,u) Reduced-order system operator
np.kron(x,x) Quadratic Kronecker product of full state
np.kron(x_,x_) Quadratic Kronecker product of reduced state
utils.kron2c(x_) Compact quadratic Kronecker product of reduced state
np.kron(x,np.kron(x,x)) Cubic Kronecker product of full state
np.kron(x_,np.kron(x_,x_)) Cubic Kronecker product of reduced state
utils.kron3c(x_) Compact cubic Kronecker product of reduced state
vj jth subspace basis vector, i.e., column j of Vr

Matrices

Symbol Code Shape Description
Vr low-rank basis of rank r (usually the POD basis)
X Snapshot matrix
Xdot Snapshot time derivative matrix
U Input matrix (inputs corresonding to the snapshots)
X_ Projected snapshot matrix
Xdot_ Projected snapshot time derivative matrix
D Data matrix
O Operator matrix
R Right-hand side matrix
P Tikhonov regularization matrix
A Full-order linear state matrix
A_ Reduced-order linear state matrix
H Full-order matricized quadratic state tensor
H_ Compact reduced-order matricized quadratic state tensor
G Full-order matricized quadratic state tensor
G_ Compact reduced-order matricized quadratic state tensor
B Full-order input matrix
B_ Reduced-order input matrix

Back to Willcox-Research-Group/rom-operator-inference-Python3

Operator Inference: a brief mathematical summary of Operator Inference and some of its extensions.

Installation: getting set up with pip and/or git.

API Reference: complete rom_operator_inference documentation.

Index of Notation: list of notation used in the package and the documentation.

References: list of publications that use or build on Operator Inference.

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