OptDMD¶
Derived module from pydmd.dmdbase()
for the optimal closedform solution
to dmd.
Note
P. Heas & C. Herzet. Lowrank dynamic mode decomposition: optimal solution in polynomial time. arXiv:1610.02962. 2016.

class
OptDMD
(factorization='evd', svd_rank=0, tlsq_rank=0, opt=False)[source] Bases:
pydmd.dmdbase.DMDBase
Dynamic Mode Decomposition
This class implements the closedform solution to the DMD minimization problem. It relies on the optimal solution given by [HeasHerzet16].
 HeasHerzet16
P. Heas & C. Herzet. Lowrank dynamic mode decomposition: optimal solution in polynomial time. arXiv:1610.02962. 2016.
 Parameters
factorization (str) – compute either the eigenvalue decomposition of the unknown highdimensional DMD operator (factorization=”evd”) or its singular value decomposition (factorization=”svd”). Default is “evd”.
svd_rank (int or float) – the rank for the truncation; If 0, the method computes the optimal rank and uses it for truncation; if positive interger, the method uses the argument for the truncation; if float between 0 and 1, the rank is the number of the biggest singular values that are needed to reach the ‘energy’ specified by svd_rank; if 1, the method does not compute truncation.
tlsq_rank (int) – rank truncation computing Total Least Square. Default is 0, that means TLSQ is not applied.
opt (bool or int) – argument to control the computation of DMD modes amplitudes. See
DMDBase
. Default is False.

_compute_amplitudes
(modes, snapshots, eigs, opt)[source] Compute the amplitude coefficients. If self.opt is False the amplitudes are computed by minimizing the error between the modes and the first snapshot; if self.opt is True the amplitudes are computed by minimizing the error between the modes and all the snapshots, at the expense of bigger computational cost.
This method uses the class variables self._snapshots (for the snapshots), self.modes and self.eigs.
 Returns
the amplitudes array
 Return type
References for optimal amplitudes: Jovanovic et al. 2014, Sparsitypromoting dynamic mode decomposition, https://halpolytechnique.archivesouvertes.fr/hal00995141/document

property
amplitudes
Get the coefficients that minimize the error between the original system and the reconstructed one. For futher information, see dmdbase._compute_amplitudes.
 Returns
the array that contains the amplitudes coefficient.
 Return type

property
dynamics
Get the time evolution of each mode.
\mathbf{x}(t) \approx \sum_{k=1}^{r} \boldsymbol{\phi}_{k} \exp \left( \omega_{k} t \right) b_{k} = \sum_{k=1}^{r} \boldsymbol{\phi}_{k} \left( \lambda_{k} \right)^{\left( t / \Delta t \right)} b_{k}
 Returns
the matrix that contains all the time evolution, stored by row.
 Return type

property
eigs
Get the eigenvalues of A tilde.
 Returns
the eigenvalues from the eigendecomposition of atilde.
 Return type

property
factorization

fit
(X, Y=None)[source] Compute the Dynamic Modes Decomposition to the input data.
 Parameters
X (numpy.ndarray or iterable) – the input snapshots.
Y (numpy.ndarray or iterable) – the input snapshots at sequential timestep, if passed. Default is None.

property
fitted
Check whether this DMD instance has been fitted.
 Returns
True is the instance has been fitted, False otherwise.
 Return type

property
modes
Get the matrix containing the DMD modes, stored by column.
 Returns
the matrix containing the DMD modes.
 Return type

property
modes_activation_bitmask
Get the bitmask which controls which DMD modes are enabled at the moment in this DMD instance.
The DMD instance must be fitted before this property becomes valid. After
fit()
is called, the defalt value of modes_activation_bitmask is an array of True values of the same shape ofamplitudes()
.The array returned is readonly (this allow us to react appropriately to changes in the bitmask). In order to modify the bitmask you need to set the field to a brandnew value (see example below).
Example:
>>> # this is an error >>> dmd.modes_activation_bitmask[[1,2]] = False ValueError: assignment destination is readonly >>> tmp = np.array(dmd.modes_activation_bitmask) >>> tmp[[1,2]] = False >>> dmd.modes_activation_bitmask = tmp
 Returns
The DMD modes activation bitmask.
 Return type

predict
(X)[source] Predict the output Y given the input X using the fitted DMD model.
 Parameters
X (numpy.ndarray) – the input vector.
 Returns
one timestep ahead predicted output.
 Return type