QuadraticOperator

class QuadraticOperator(entries=None)

Quadratic state operator \(\Ophat_{\ell}(\qhat,\u) = \Hhat[\qhat\otimes\qhat]\) where \(\Hhat\in\RR^{r \times r^{2}}\).

Internally, the action of the operator is computed as the product of an \(r \times r(r+1)/2\) matrix \(\tilde{\H}\) and a compressed version of the Kronecker product \(\qhat \otimes \qhat\).

Parameters:

entries(r, r^2) or (r, r(r+1)/2) or (r, r, r) ndarray or None

Operator matrix \(\Hhat\), its compressed representation \(\tilde{\H}\), or the equivalent symmetric tensor.

Examples

>>> import numpy as np
>>> H = opinf.operators.QuadraticOperator()
>>> entries = np.random.random((10, 100))   # Operator matrix.
>>> H.set_entries(entries)
>>> H.shape                                 # Compressed shape.
(10, 55)
>>> q = np.random.random(10)                # State vector.
>>> out = H.apply(q)                        # Apply the operator to q.
>>> np.allclose(out, entries @ np.kron(q, q))
True

Properties:

entries

Internal representation \(\tilde{\H}\) of the operator matrix \(\Hhat\).

shape

Shape \((r, r(r+1)/2)\) of the internal representation \(\tilde{\H}\) of the operator matrix \(\Hhat\).

state_dimension

Dimension \(r\) of the state \(\qhat\) that the operator acts on.

Methods:

apply	Apply the operator to the given state / input: \(\Ophat_{\ell}(\qhat,\u) = \Hhat[\qhat\otimes\qhat]\)
ckron	Calculate the compressed Kronecker product of a vector with itself.
ckron_indices	Construct a mask for efficiently computing the compressed Kronecker product.
compress_entries	Given \(\Hhat\in\RR^{a\times r^2}\), construct the matrix \(\tilde{\H}\in\RR^{a \times r(r+1)/2}\) such that \(\Hhat[\qhat\otimes\qhat] = \tilde{\H}[\qhat\,\hat{\otimes}\,\qhat]\) for all \(\qhat\in\RR^{r}\) where \(\hat{\otimes}\) is the compressed Kronecker product (see ckron()).
copy	Return a copy of the operator.
datablock	Return the data matrix block corresponding to the operator, the Khatri--Rao product of the state with itself: \(\Qhat\odot\Qhat\) where \(\Qhat\) is states.
expand_entries	Given \(\tilde{\H}\in\RR^{a \times r(r+1)/2}\), construct the matrix \(\Hhat\in\RR^{a\times r^2}\) such that \(\Hhat[\qhat\otimes\qhat] = \tilde{\H}[\qhat\,\hat{\otimes}\,\qhat]\) for all \(\qhat\in\RR^{r}\) where \(\hat{\otimes}\) is the compressed Kronecker product (see ckron()).
galerkin	Return the (Petrov-)Galerkin projection of the operator, \(\Hhat = (\Wr\trp\Vr)^{-1}\Wr\trp\H[\Vr\otimes\Vr]\).
jacobian	Construct the state Jacobian of the operator: \(\ddqhat\Ophat_{\ell}(\qhat,\u) = \Hhat[(\I_r\otimes\qhat) + (\qhat\otimes\I_r)]\).
load	Load an operator from an HDF5 file.
operator_dimension	Column dimension \(r(r+1)/2\) of the internal representation \(\tilde{\H}\) of the operator matrix \(\Hhat\).
save	Save the operator to an HDF5 file.
set_entries	Set the internal representation \(\tilde{\H}\) of the operator matrix \(\Hhat\).
verify	Verify consistency between dimension properties and required methods.