# SpDMD¶

Derived module from dmdbase.py for sparsity-promoting DMD.

 SpDMD.atilde Get the reduced Koopman operator A, called A tilde. SpDMD.dmd_timesteps Get the timesteps of the reconstructed states. SpDMD.dynamics Get the time evolution of each mode. SpDMD.eigs Get the eigenvalues of A tilde. SpDMD.fit Compute the Dynamic Modes Decomposition of the input data. SpDMD.modes Get the matrix containing the DMD modes, stored by column. SpDMD.original_timesteps Get the timesteps of the original snapshot. SpDMD.plot_eigs Plot the eigenvalues. SpDMD.plot_modes_2D Plot the DMD Modes. SpDMD.plot_snapshots_2D Plot the snapshots. SpDMD.reconstructed_data Get the reconstructed data. SpDMD.snapshots Get the original input data.
class SpDMD(svd_rank=0, tlsq_rank=0, exact=True, opt=False, rescale_mode=None, forward_backward=False, sorted_eigs=False, abs_tolerance=1e-06, rel_tolerance=0.0001, max_iterations=10000, rho=1, gamma=10, verbose=True, enforce_zero=True, release_memory=True, zero_absolute_tolerance=1e-12)[source]

Bases: pydmd.dmd.DMD

Sparsity-Promoting Dynamic Mode Decomposition. Promotes solutions having an high number of amplitudes set to zero (i.e. sparse solutions). Reference: 10.1063/1.4863670

Parameters
• 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.

• exact (bool) – flag to compute either exact DMD or projected DMD. Default is True.

• opt (bool or int) – argument to control the computation of DMD modes amplitudes. See DMDBase. Default is False.

• rescale_mode ({'auto'} or None or numpy.ndarray) – Scale Atilde as shown in 10.1016/j.jneumeth.2015.10.010 (section 2.4) before computing its eigendecomposition. None means no rescaling, ‘auto’ means automatic rescaling using singular values, otherwise the scaling factors.

• forward_backward (bool) – If True, the low-rank operator is computed like in fbDMD (reference: https://arxiv.org/abs/1507.02264). Default is False.

• sorted_eigs ({'real', 'abs'} or False) – Sort eigenvalues (and modes/dynamics accordingly) by magnitude if sorted_eigs=’abs’, by real part (and then by imaginary part to break ties) if sorted_eigs=’real’. Default: False.

• abs_tolerance (float) – Controls the convergence of ADMM. See _loop_condition() for more details.

• rel_tolerance (float) – Controls the convergence of ADMM. See _loop_condition() for more details.

• max_iterations (int) – The maximum number of iterations performed by ADMM, after that the algorithm is stopped.

• rho (float) – Controls the convergence of ADMM. For a reference on the optimal value for rho see 10.1109/TAC.2014.2354892 or 10.3182/20120914-2-US-4030.00038.

• gamma (float) – Controls the level of “promotion” assigned to sparse solution. Increasing gamma will result in an higher number of zero-amplitudes.

• verbose (bool) – If False, the information provided by SpDMD (like the number of iterations performed by ADMM) are not shown.

• enforce_zero (bool) – If True the DMD amplitudes which should be set to zero according to the solution of ADMM are manually set to 0 (since we solve a sparse linear system to find the optimal vector of DMD amplitudes very small terms may survive in some cases).

• release_memory – If True the intermediate matrices computed by the algorithm are deleted after the termination of a call to fit().

_find_zero_amplitudes()[source]

Use ADMM to find which amplitudes (i.e. their position in the DMD amplitudes array) which can be set to zero according to the given parameters. Note that this method does not compute amplitudes, but only which amplitudes are to be set to 0. Optimal amplitudes should be computed separately afterwards (see _find_sparsity_promoting_amplitudes()). :return np.ndarray: A boolean vector whose True items correspond to

amplitudes which should be set to 0.

_loop_condition(alpha, beta, lmbd, old_beta)[source]

Check whether ADMM can stop now, or should perform another iteration. :param np.ndarray alpha: Current value of \alpha_k (vector

of DMD amplitudes).

Parameters
• beta (np.ndarray) – Current value of \beta_k (vector of non-zero amplitudes).

• lmbd (np.ndarray) – Current value of \lambda_k (vector of Lagrange multipliers).

• old_beta (np.ndarray) – Old value of \beta_{k-1} (vector of non-zero amplitudes).

Return bool

True if ADMM can stop now, False otherwise.

_optimal_amplitudes(zero_amplitudes)[source]

Find the optimal DMD amplitudes with the constraint that the given indexes should be set to 0. :param np.ndarray zero_amplitudes: Boolean vector. :return np.ndarray: Vector of optimal DMD amplitudes. Amplitudes at

indexes corresponding to True indexes in zero_amplitudes are set to 0.

_update(beta, lmbd)[source]

Operate an entire step of ADMM. :param np.ndarray beta: Current value of \beta_k (vector of

non-zero amplitudes).

Parameters

lmbd (np.ndarray) – Current value of \lambda_k (vector of Lagrande multipliers).

Returns

A tuple containing the updated values \alpha_{k+1},\beta_{k+1},\lambda_{k+1} (in this order).

Return type

tuple

_update_alpha(beta, lmbd)[source]

Update the vector \alpha_k of DMD amplitudes. :param np.ndarray beta: Current value of \beta_k (vector of

non-zero amplitudes).

Parameters

lmbd (np.ndarray) – Current value of \lambda_k (vector of Lagrande multipliers).

Returns

The updated value \alpha_{k+1}.

Return type

np.ndarray

_update_beta(alpha, lmbd)[source]

Update the vector \beta of non-zero amplitudes. :param np.ndarray alpha: Updated value of \alpha_{k+1} (vector

of DMD amplitudes).

Parameters

lmbd (np.ndarray) – Current value of \lambda_k (vector of Lagrange multipliers).

Returns

The updated value \beta_{k+1}.

Return type

np.ndarray

_update_lagrangian(alpha, beta, lmbd)[source]

Update the vector \lambda of Lagrange multipliers. :param np.ndarray alpha: Updated value of \alpha_{k+1} (vector

of DMD amplitudes).

Parameters
• beta (np.ndarray) – Updated value of \beta_{k+1} (vector of non-zero amplitudes).

• lmbd (np.ndarray) – Current value of \lambda_k (vector of Lagrange multipliers).

Returns

The updated value \lambda_{k+1}.

Return type

np.ndarray

fit(X)[source]

Compute the Dynamic Modes Decomposition of the input data. :param X: the input snapshots. :type X: numpy.ndarray or iterable