pymor.reductors.h2
¶
Reductors based on H2-norm.
Module Contents¶
- class pymor.reductors.h2.GapIRKAReductor(fom, mu=None, solver_options=None)[source]¶
Bases:
GenericIRKAReductor
Gap-IRKA reductor.
Methods
Reduce using gap-IRKA.
- reduce(self, rom0_params, tol=0.0001, maxit=100, num_prev=1, conv_crit='sigma', projection='orth')[source]¶
Reduce using gap-IRKA.
See [BBG19] Algorithm 1.
Parameters
- rom0_params
Can be:
order of the reduced model (a positive integer),
initial interpolation points (a 1D
NumPy array
),dict with
'sigma'
,'b'
,'c'
as keys mapping to initial interpolation points (a 1DNumPy array
), right tangential directions (VectorArray
fromfom.input_space
), and left tangential directions (VectorArray
fromfom.output_space
), all of the same length (the order of the reduced model),initial reduced-order model (
LTIModel
).
If the order of reduced model is given, initial interpolation data is generated randomly.
- tol
Tolerance for the convergence criterion.
- maxit
Maximum number of iterations.
- num_prev
Number of previous iterations to compare the current iteration to. A larger number can avoid occasional cyclic behavior.
- conv_crit
Convergence criterion:
'sigma'
: relative change in interpolation points'htwogap'
: \(\mathcal{H}_2-gap\) distance of reduced-order models divided by \(\mathcal{L}_2\) norm of new reduced-order model'ltwo'
: relative \(\mathcal{L}_2\) distance of reduced-order models
- projection
Projection method:
'orth'
: projection matrix is orthogonalized with respect to the Euclidean inner product,'biorth'
: projection matrix is orthogonalized with respect to the E product.
Returns
- rom
Reduced
LTIModel
model.
- class pymor.reductors.h2.GenericIRKAReductor(fom, mu=None)[source]¶
Bases:
pymor.core.base.BasicObject
Generic IRKA related reductor.
Parameters
- fom
The full-order
Model
to reduce.- mu
Methods
Reconstruct high-dimensional vector from reduced vector
u
.
- class pymor.reductors.h2.IRKAReductor(fom, mu=None)[source]¶
Bases:
GenericIRKAReductor
Iterative Rational Krylov Algorithm reductor.
Parameters
- fom
The full-order
LTIModel
to reduce.- mu
Methods
Reduce using IRKA.
- reduce(self, rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, projection='orth', conv_crit='sigma', compute_errors=False)[source]¶
Reduce using IRKA.
See [GAB08] (Algorithm 4.1) and [ABG10] (Algorithm 1).
Parameters
- rom0_params
Can be:
order of the reduced model (a positive integer),
initial interpolation points (a 1D
NumPy array
),dict with
'sigma'
,'b'
,'c'
as keys mapping to initial interpolation points (a 1DNumPy array
), right tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
), and left tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
),initial reduced-order model (
LTIModel
).
If the order of reduced model is given, initial interpolation data is generated randomly.
- tol
Tolerance for the convergence criterion.
- maxit
Maximum number of iterations.
- num_prev
Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of IRKA.
- force_sigma_in_rhp
If
False
, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.- projection
Projection method:
'orth'
: projection matrices are orthogonalized with respect to the Euclidean inner product'biorth'
: projection matrices are biorthogolized with respect to the E product'arnoldi'
: projection matrices are orthogonalized using the Arnoldi process (available only for SISO systems).
- conv_crit
Convergence criterion:
'sigma'
: relative change in interpolation points'h2'
: relative \(\mathcal{H}_2\) distance of reduced-order models
- compute_errors
Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.
Warning
Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.
Returns
- rom
Reduced
LTIModel
model.
- class pymor.reductors.h2.OneSidedIRKAReductor(fom, version, mu=None)[source]¶
Bases:
GenericIRKAReductor
One-Sided Iterative Rational Krylov Algorithm reductor.
Parameters
- fom
The full-order
LTIModel
to reduce.- version
Version of the one-sided IRKA:
'V'
: Galerkin projection using the input Krylov subspace,'W'
: Galerkin projection using the output Krylov subspace.
- mu
Methods
Reduce using one-sided IRKA.
- reduce(self, rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, projection='orth', conv_crit='sigma', compute_errors=False)[source]¶
Reduce using one-sided IRKA.
Parameters
- rom0_params
Can be:
order of the reduced model (a positive integer),
initial interpolation points (a 1D
NumPy array
),dict with
'sigma'
,'b'
,'c'
as keys mapping to initial interpolation points (a 1DNumPy array
), right tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
), and left tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
),initial reduced-order model (
LTIModel
).
If the order of reduced model is given, initial interpolation data is generated randomly.
- tol
Tolerance for the largest change in interpolation points.
- maxit
Maximum number of iterations.
- num_prev
Number of previous iterations to compare the current iteration to. A larger number can avoid occasional cyclic behavior.
- force_sigma_in_rhp
If
False
, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.- projection
Projection method:
'orth'
: projection matrix is orthogonalized with respect to the Euclidean inner product,'Eorth'
: projection matrix is orthogonalized with respect to the E product.
- conv_crit
Convergence criterion:
'sigma'
: relative change in interpolation points,'h2'
: relative \(\mathcal{H}_2\) distance of reduced-order models.
- compute_errors
Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.
Warning
Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.
Returns
- rom
Reduced
LTIModel
model.
- class pymor.reductors.h2.TFIRKAReductor(fom, mu=None)[source]¶
Bases:
GenericIRKAReductor
Realization-independent IRKA reductor.
See [BG12].
Parameters
- fom
TransferFunction
orModel
with atransfer_function
attribute, witheval_tf
andeval_dtf
methods that should be defined at least over the open right half of the complex plane.- mu
Methods
Reconstruct high-dimensional vector from reduced vector
u
.Reduce using TF-IRKA.
- reduce(self, rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, conv_crit='sigma', compute_errors=False)[source]¶
Reduce using TF-IRKA.
Parameters
- rom0_params
Can be:
order of the reduced model (a positive integer),
initial interpolation points (a 1D
NumPy array
),dict with
'sigma'
,'b'
,'c'
as keys mapping to initial interpolation points (a 1DNumPy array
), right tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
), and left tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
),initial reduced-order model (
LTIModel
).
If the order of reduced model is given, initial interpolation data is generated randomly.
- tol
Tolerance for the convergence criterion.
- maxit
Maximum number of iterations.
- num_prev
Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of TF-IRKA.
- force_sigma_in_rhp
If
False
, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.- conv_crit
Convergence criterion:
'sigma'
: relative change in interpolation points'h2'
: relative \(\mathcal{H}_2\) distance of reduced-order models
- compute_errors
Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.
Warning
Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.
Returns
- rom
Reduced
LTIModel
model.
- class pymor.reductors.h2.TSIAReductor(fom, mu=None)[source]¶
Bases:
GenericIRKAReductor
Two-Sided Iteration Algorithm reductor.
Parameters
- fom
The full-order
LTIModel
to reduce.- mu
Methods
Reduce using TSIA.
- reduce(self, rom0_params, tol=0.0001, maxit=100, num_prev=1, projection='orth', conv_crit='sigma', compute_errors=False)[source]¶
Reduce using TSIA.
See [XZ11] (Algorithm 1) and [BKohlerS11].
In exact arithmetic, TSIA is equivalent to IRKA (under some assumptions on the poles of the reduced model). The main difference in implementation is that TSIA computes the Schur decomposition of the reduced matrices, while IRKA computes the eigenvalue decomposition. Therefore, TSIA might behave better for non-normal reduced matrices.
Parameters
- rom0_params
Can be:
order of the reduced model (a positive integer),
initial interpolation points (a 1D
NumPy array
),dict with
'sigma'
,'b'
,'c'
as keys mapping to initial interpolation points (a 1DNumPy array
), right tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
), and left tangential directions (NumPy array
of shape(len(sigma), fom.dim_input)
),initial reduced-order model (
LTIModel
).
If the order of reduced model is given, initial interpolation data is generated randomly.
- tol
Tolerance for the convergence criterion.
- maxit
Maximum number of iterations.
- num_prev
Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of TSIA.
- projection
Projection method:
'orth'
: projection matrices are orthogonalized with respect to the Euclidean inner product'biorth'
: projection matrices are biorthogolized with respect to the E product
- conv_crit
Convergence criterion:
'sigma'
: relative change in interpolation points'h2'
: relative \(\mathcal{H}_2\) distance of reduced-order models
- compute_errors
Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.
Warning
Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.
Returns
- rom
Reduced
LTIModel
.