# pymor.algorithms package¶

## Submodules¶

class pymor.algorithms.adaptivegreedy.AdaptiveSampleSet(parameter_space)[source]

Used by adaptive_weak_greedy.

Methods

 AdaptiveSampleSet map_vertex_to_mu, refine, visualize BasicObject

Attributes

pymor.algorithms.adaptivegreedy.adaptive_weak_greedy(surrogate, parameter_space, target_error=None, max_extensions=None, validation_mus=0, rho=1.1, gamma=0.2, theta=0.0, visualize=False, visualize_vertex_size=80, pool=None)[source]

Weak greedy basis generation algorithm with adaptively refined training set.

This method extends pyMOR’s default weak_greedy greedy basis generation algorithm by adaptive refinement of the parameter training set according to [HDO11] to prevent overfitting of the approximation basis to the training set. This is achieved by estimating the approximation error on an additional validation set of parameters. If the ratio between the estimated errors on the validation set and the validation set is larger than rho, the training set is refined using standard grid refinement techniques.

Parameters

surrogate
parameter_space

The ParameterSpace for which to compute the approximation basis.

target_error
max_extensions
validation_mus
One of the following:
• a list of Parameters to use as validation set,

• a positive number indicating the number of random parameters to use as validation set,

• a non-positive number, indicating the negative number of random parameters to use as validation set in addition to the centers of the elements of the adaptive training set.

rho

Maximum allowed ratio between maximum estimated error on validation set vs. maximum estimated error on training set. If the ratio is larger, the training set is refined.

gamma

Weight of the age penalty term in the training set refinement indicators.

theta

Ratio of training set elements to select for refinement. (One element is always refined.)

visualize

If True, visualize the refinement indicators. (Only available for 2 and 3 dimensional parameter spaces.)

visualize_vertex_size

Size of the vertices in the visualization.

pool

Returns

Dict with the following fields

extensions

Number of greedy iterations.

max_errs

Sequence of maximum errors during the greedy run.

max_err_mus

The parameters corresponding to max_errs.

max_val_errs

Sequence of maximum errors on the validation set.

max_val_err_mus

The parameters corresponding to max_val_errs.

refinements

Number of refinements made in each extension step.

training_set_sizes

The final size of the training set in each extension step.

time

Duration of the algorithm.

pymor.algorithms.adaptivegreedy.rb_adaptive_greedy(fom, reductor, parameter_space, use_error_estimator=True, error_norm=None, target_error=None, max_extensions=None, validation_mus=0, rho=1.1, gamma=0.2, theta=0.0, extension_params=None, visualize=False, visualize_vertex_size=80, pool=None)[source]

Reduced basis greedy basis generation with adaptively refined training set.

This method extends pyMOR’s default rb_greedy greedy reduced basis generation algorithm by adaptive refinement of the parameter training set [HDO11] to prevent overfitting of the reduced basis to the training set as implemented in adaptive_weak_greedy.

Parameters

fom
reductor
parameter_space

The ParameterSpace for which to compute the reduced model.

use_error_estimator
error_norm
target_error
max_extensions
validation_mus
rho
gamma
theta
extension_params
visualize
visualize_vertex_size
pool

Returns

Dict with the following fields

rom

The reduced Model obtained for the computed basis.

extensions

Number of greedy iterations.

max_errs

Sequence of maximum errors during the greedy run.

max_err_mus

The parameters corresponding to max_errs.

max_val_errs

Sequence of maximum errors on the validation set.

max_val_err_mus

The parameters corresponding to max_val_errs.

refinements

Number of refinements made in each extension step.

training_set_sizes

The final size of the training set in each extension step.

time

Duration of the algorithm.

### basic module¶

Module containing some basic but generic linear algebra algorithms.

pymor.algorithms.basic.almost_equal(U, V, product=None, sup_norm=False, rtol=1e-14, atol=1e-14)[source]

Compare U and V for almost equality.

The vectors of U and V are compared in pairs for almost equality. Two vectors u and v are considered almost equal iff

||u - v|| <= atol + ||v|| * rtol.

The norm to be used can be specified via the product or sup_norm parameters.

If the length of U resp. V is 1, the single specified vector is compared to all vectors of the other array. Otherwise, the lengths of both indexed arrays have to agree.

Parameters

U, V

VectorArrays to be compared.

product

If specified, use this inner product Operator to compute the norm. Otherwise, the Euclidean norm is used.

sup_norm

If True, use the sup norm to compute the error.

rtol

The relative tolerance.

atol

The absolute tolerance.

Defaults

rtol, atol (see pymor.core.defaults)

pymor.algorithms.basic.contains_zero_vector(vector_array, rtol=None, atol=None)[source]

returns True iff any vector in the array float_compares to 0s of the same dim

Parameters

vector_array

a VectorArray implementation

rtol

relative tolerance for float_cmp

atol

absolute tolerance for float_cmp

pymor.algorithms.basic.project_array(U, basis, product=None, orthonormal=True)[source]

Orthogonal projection of VectorArray onto subspace.

Parameters

U

The VectorArray to project.

basis

VectorArray of basis vectors for the subspace onto which to project.

product

Inner product Operator w.r.t. which to project.

orthonormal

If True, the vectors in basis are assumed to be orthonormal w.r.t. product.

Returns

The projected VectorArray.

pymor.algorithms.basic.relative_error(U, V, product=None)[source]

Compute error between U and V relative to norm of U.

### ei module¶

This module contains algorithms for the empirical interpolation of Operators.

The main work for generating the necessary interpolation data is handled by the ei_greedy method. The objects returned by this method can be used to instantiate an EmpiricalInterpolatedOperator.

As a convenience, the interpolate_operators method allows to perform the empirical interpolation of the Operators of a given model with a single function call.

pymor.algorithms.ei.deim(U, modes=None, pod=True, atol=None, rtol=None, product=None, pod_options={})[source]

Generate data for empirical interpolation using DEIM algorithm.

Given a VectorArray U, this method generates a collateral basis and interpolation DOFs for empirical interpolation of the vectors contained in U. The returned objects can be used to instantiate an EmpiricalInterpolatedOperator (with triangular=False).

The collateral basis is determined by the first pod modes of U.

Parameters

U

A VectorArray of vectors to interpolate.

modes

Dimension of the collateral basis i.e. number of POD modes of the vectors in U.

pod

If True, perform a POD of U to obtain the collateral basis. If False, U is used as collateral basis.

atol

Absolute POD tolerance.

rtol

Relative POD tolerance.

product

Inner product Operator used for the POD.

pod_options

Dictionary of additional options to pass to the pod algorithm.

Returns

interpolation_dofs

NumPy array of the DOFs at which the vectors are interpolated.

collateral_basis

VectorArray containing the generated collateral basis.

data

Dict containing the following fields:

svals

POD singular values.

pymor.algorithms.ei.ei_greedy(U, error_norm=None, atol=None, rtol=None, max_interpolation_dofs=None, copy=True, pool=DummyPool('??', '??'))[source]

Generate data for empirical interpolation using EI-Greedy algorithm.

Given a VectorArray U, this method generates a collateral basis and interpolation DOFs for empirical interpolation of the vectors contained in U. The returned objects can be used to instantiate an EmpiricalInterpolatedOperator (with triangular=True).

The interpolation data is generated by a greedy search algorithm, where in each loop iteration the worst approximated vector in U is added to the collateral basis.

Parameters

U

A VectorArray of vectors to interpolate.

error_norm

Norm w.r.t. which to calculate the interpolation error. If None, the Euclidean norm is used.

atol

Stop the greedy search if the largest approximation error is below this threshold.

rtol

Stop the greedy search if the largest relative approximation error is below this threshold.

max_interpolation_dofs

Stop the greedy search if the number of interpolation DOF (= dimension of the collateral basis) reaches this value.

copy

If False, U will be modified during executing of the algorithm.

pool

If not None, the WorkerPool to use for parallelization.

Returns

interpolation_dofs

NumPy array of the DOFs at which the vectors are evaluated.

collateral_basis

VectorArray containing the generated collateral basis.

data

Dict containing the following fields:

errors

Sequence of maximum approximation errors during greedy search.

triangularity_errors

Sequence of maximum absolute values of interoplation matrix coefficients in the upper triangle (should be near zero).

pymor.algorithms.ei.interpolate_operators(fom, operator_names, parameter_sample, error_norm=None, product=None, atol=None, rtol=None, max_interpolation_dofs=None, pod_options={}, alg='ei_greedy', pool=DummyPool('??', '??'))[source]

Empirical operator interpolation using the EI-Greedy/DEIM algorithm.

This is a convenience method to facilitate the use of ei_greedy or deim. Given a Model, names of Operators, and a sample of Parameters, first the operators are evaluated on the solution snapshots of the model for the provided parameters. These evaluations are then used as input for ei_greedy/deim. Finally the resulting interpolation data is used to create EmpiricalInterpolatedOperators and a new model with the interpolated operators is returned.

Note that this implementation creates one common collateral basis for all specified operators, which might not be what you want.

Parameters

fom

The Model whose Operators will be interpolated.

operator_names

List of keys in the operators dict of the model. The corresponding Operators will be interpolated.

parameter_sample

A list of Parameters for which solution snapshots are calculated.

error_norm

See ei_greedy. Has no effect if alg == 'deim'.

product

Inner product for POD computation in deim. Has no effect if alg == 'ei_greedy'.

atol
rtol
max_interpolation_dofs
pod_options

Further options for pod algorithm. Has no effect if alg == 'ei_greedy'.

alg

Either ei_greedy or deim.

pool

If not None, the WorkerPool to use for parallelization.

Returns

eim

Model with Operators given by operator_names replaced by EmpiricalInterpolatedOperators.

data

Dict containing the following fields:

dofs

NumPy array of the DOFs at which the Operators have to be evaluated.

basis

VectorArray containing the generated collateral basis.

In addition, data contains the fields of the data dict returned by ei_greedy/deim.

### eigs module¶

pymor.algorithms.eigs._arnoldi(A, E, l, b)[source]

Compute an Arnoldi factorization.

pymor.algorithms.eigs._extend_arnoldi(A, E, V, H, f, p)[source]

Extend an existing Arnoldi factorization.

pymor.algorithms.eigs._qr_iteration(H, shifts)[source]

Perform the QR iteration.

pymor.algorithms.eigs.eigs(A, E=None, k=3, which='LM', b=None, l=None, maxiter=1000, tol=1e-13, imag_tol=1e-12, complex_pair_tol=1e-12, seed=0)[source]

Approximate a few eigenvalues of a linear Operator.

Computes k eigenvalues w with corresponding eigenvectors v which solve the eigenvalue problem

$A v_i = w_i v_i$

or the generalized eigenvalue problem

$A v_i = w_i E v_i$

if E is not None.

The implementation is based on Algorithm 4.2 in [RL95].

Parameters

A

The real linear Operator for which the eigenvalues are to be computed.

E

The real linear Operator which defines the generalized eigenvalue problem.

k

The number of eigenvalues and eigenvectors which are to be computed.

which

A string specifying which k eigenvalues and eigenvectors to compute:

• 'LM': select eigenvalues with largest magnitude

• 'SM': select eigenvalues with smallest magnitude

• 'LR': select eigenvalues with largest real part

• 'SR': select eigenvalues with smallest real part

• 'LI': select eigenvalues with largest imaginary part

• 'SI': select eigenvalues with smallest imaginary part

b

Initial vector for Arnoldi iteration. Default is a random vector.

l

The size of the Arnoldi factorization. Default is min(n - 1, max(2*k + 1, 20)).

maxiter

The maximum number of iterations.

tol

The relative error tolerance for the Ritz estimates.

imag_tol

Relative imaginary parts below this tolerance are set to 0.

complex_pair_tol

Tolerance for detecting pairs of complex conjugate eigenvalues.

seed

Random seed which is used for computing the initial vector for the Arnoldi iteration.

Returns

w

A 1D NumPy array which contains the computed eigenvalues.

v

A VectorArray which contains the computed eigenvectors.

### error module¶

pymor.algorithms.error.reduction_error_analysis(rom, fom, reductor, test_mus, basis_sizes=0, error_estimator=True, condition=False, error_norms=(), error_norm_names=None, error_estimator_norm_index=0, custom=(), plot=False, plot_custom_logarithmic=True, pool=DummyPool('??', '??'))[source]

Analyze the model reduction error.

The maximum model reduction error is estimated by solving the reduced Model for given random Parameters.

Parameters

rom

The reduced Model.

fom

The high-dimensional Model.

reductor

The reductor which has created rom.

test_mus

List of Parameters to compute the errors for.

basis_sizes

Either a list of reduced basis dimensions to consider, or the number of dimensions (which are then selected equidistantly, always including the maximum reduced space dimension). The dimensions are input for the dim-Parameter of reductor.reduce().

error_estimator

If True evaluate the error estimator of rom on the test Parameters.

condition

If True, compute the condition of the reduced system matrix for the given test Parameters (can only be specified if rom is an instance of StationaryModel and rom.operator is linear).

error_norms

List of norms in which to compute the model reduction error.

error_norm_names

Names of the norms given by error_norms. If None, the name attributes of the given norms are used.

error_estimator_norm_index

When error_estimator is True and error_norms are specified, this is the index of the norm in error_norms w.r.t. which to compute the effectivity of the error estimator.

custom

List of custom functions which are evaluated for each test parameter values and basis size. The functions must have the signature

def custom_value(rom, fom, reductor, mu, dim):
pass

plot

If True, generate a plot of the computed quantities w.r.t. the basis size.

plot_custom_logarithmic

If True, use a logarithmic y-axis to plot the computed custom values.

pool

If not None, the WorkerPool to use for parallelization.

Returns

Dict with the following fields

mus

The test Parameters which have been considered.

basis_sizes

The reduced basis dimensions which have been considered.

norms

NumPy array of the norms of the high-dimensional solutions w.r.t. all given test Parameters, reduced basis dimensions and norms in error_norms. (Only present when error_norms has been specified.)

max_norms

Maxima of norms over the given test Parameters.

max_norm_mus

Parameters corresponding to max_norms.

errors

NumPy array of the norms of the model reduction errors w.r.t. all given test Parameters, reduced basis dimensions and norms in error_norms. (Only present when error_norms has been specified.)

max_errors

Maxima of errors over the given test Parameters.

max_error_mus

Parameters corresponding to max_errors.

rel_errors

errors divided by norms. (Only present when error_norms has been specified.)

max_rel_errors

Maxima of rel_errors over the given test Parameters.

max_rel_error_mus

Parameters corresponding to max_rel_errors.

error_norm_names

Names of the given error_norms. (Only present when error_norms has been specified.)

error_estimates

NumPy array of the model reduction error estimates w.r.t. all given test Parameters and reduced basis dimensions. (Only present when error_estimator is True.)

max_error_estimate

Maxima of error_estimates over the given test Parameters.

max_error_estimate_mus

Parameters corresponding to max_error_estimates.

effectivities

errors divided by error_estimates. (Only present when error_estimator is True and error_norms has been specified.)

min_effectivities

Minima of effectivities over the given test Parameters.

min_effectivity_mus

Parameters corresponding to min_effectivities.

max_effectivities

Maxima of effectivities over the given test Parameters.

max_effectivity_mus

Parameters corresponding to max_effectivities.

errors

NumPy array of the reduced system matrix conditions w.r.t. all given test Parameters and reduced basis dimensions. (Only present when conditions is True.)

max_conditions

Maxima of conditions over the given test Parameters.

max_condition_mus

Parameters corresponding to max_conditions.

custom_values

NumPy array of custom function evaluations w.r.t. all given test Parameters, reduced basis dimensions and functions in custom. (Only present when custom has been specified.)

max_custom_values

Maxima of custom_values over the given test Parameters.

max_custom_values_mus

Parameters corresponding to max_custom_values.

time

Time (in seconds) needed for the error analysis.

summary

String containing a summary of all computed quantities for the largest (last) considered basis size.

figure

The figure containing the generated plots. (Only present when plot is True.)

### genericsolvers module¶

This module contains some iterative linear solvers which only use the Operator interface

pymor.algorithms.genericsolvers.apply_inverse(op, V, initial_guess=None, options=None, least_squares=False, check_finite=True, default_solver='generic_lgmres', default_least_squares_solver='generic_least_squares_lsmr')[source]

Solve linear equation system.

Applies the inverse of op to the vectors in V using a generic iterative solver.

Parameters

op

The linear, non-parametric Operator to invert.

V

VectorArray of right-hand sides for the equation system.

initial_guess

VectorArray with the same length as V containing initial guesses for the solution. Some implementations of apply_inverse may ignore this parameter. If None a solver-dependent default is used.

options

The solver_options to use (see solver_options).

least_squares

If True, return least squares solution.

check_finite

Test if solution only contains finite values.

default_solver

Default solver to use (generic_lgmres, generic_least_squares_lsmr, generic_least_squares_lsqr).

default_least_squares_solver

Default solver to use for least squares problems (generic_least_squares_lsmr, generic_least_squares_lsqr).

Returns

VectorArray of the solution vectors.

Defaults

check_finite, default_solver, default_least_squares_solver (see pymor.core.defaults)

pymor.algorithms.genericsolvers.lgmres(A, b, x0=None, tol=1e-05, maxiter=1000, M=None, callback=None, inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True)[source]

pymor.algorithms.genericsolvers.lsmr(A, b, damp=0.0, atol=1e-06, btol=1e-06, conlim=100000000.0, maxiter=None, show=False)[source]

pymor.algorithms.genericsolvers.lsqr(A, b, damp=0.0, atol=1e-08, btol=1e-08, conlim=100000000.0, iter_lim=None, show=False)[source]

pymor.algorithms.genericsolvers.solver_options(lgmres_tol=1e-05, lgmres_maxiter=1000, lgmres_inner_m=39, lgmres_outer_k=3, least_squares_lsmr_damp=0.0, least_squares_lsmr_atol=1e-06, least_squares_lsmr_btol=1e-06, least_squares_lsmr_conlim=100000000.0, least_squares_lsmr_maxiter=None, least_squares_lsmr_show=False, least_squares_lsqr_damp=0.0, least_squares_lsqr_atol=1e-06, least_squares_lsqr_btol=1e-06, least_squares_lsqr_conlim=100000000.0, least_squares_lsqr_iter_lim=None, least_squares_lsqr_show=False)[source]

Returns available solvers with default solver_options.

Parameters

lgmres_tol
lgmres_maxiter
lgmres_inner_m
lgmres_outer_k
least_squares_lsmr_damp
least_squares_lsmr_atol
least_squares_lsmr_btol
least_squares_lsmr_conlim
least_squares_lsmr_maxiter
least_squares_lsmr_show
least_squares_lsqr_damp
least_squares_lsqr_atol
least_squares_lsqr_btol
least_squares_lsqr_conlim
least_squares_lsqr_iter_lim
least_squares_lsqr_show

Returns

A dict of available solvers with default solver_options.

Defaults

lgmres_tol, lgmres_maxiter, lgmres_inner_m, lgmres_outer_k, least_squares_lsmr_damp, least_squares_lsmr_atol, least_squares_lsmr_btol, least_squares_lsmr_conlim, least_squares_lsmr_maxiter, least_squares_lsmr_show, least_squares_lsqr_atol, least_squares_lsqr_btol, least_squares_lsqr_conlim, least_squares_lsqr_iter_lim, least_squares_lsqr_show (see pymor.core.defaults)

### gram_schmidt module¶

pymor.algorithms.gram_schmidt.gram_schmidt(A, product=None, return_R=False, atol=1e-13, rtol=1e-13, offset=0, reiterate=True, reiteration_threshold=0.9, check=True, check_tol=0.001, copy=True)[source]

Orthonormalize a VectorArray using the modified Gram-Schmidt algorithm.

Parameters

A

The VectorArray which is to be orthonormalized.

product

The inner product Operator w.r.t. which to orthonormalize. If None, the Euclidean product is used.

return_R

If True, the R matrix from QR decomposition is returned.

atol

Vectors of norm smaller than atol are removed from the array.

rtol

Relative tolerance used to detect linear dependent vectors (which are then removed from the array).

offset

Assume that the first offset vectors are already orthonormal and start the algorithm at the offset + 1-th vector.

reiterate

If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.

reiteration_threshold

If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.

check

If True, check if the resulting VectorArray is really orthonormal.

check_tol

Tolerance for the check.

copy

If True, create a copy of A instead of modifying A in-place.

Returns

Q

The orthonormalized VectorArray.

R

The upper-triangular/trapezoidal matrix (if compute_R is True).

Defaults

atol, rtol, reiterate, reiteration_threshold, check, check_tol (see pymor.core.defaults)

pymor.algorithms.gram_schmidt.gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshold=0.1, check=True, check_tol=0.001, copy=True)[source]

Biorthonormalize a pair of VectorArrays using the biorthonormal Gram-Schmidt process.

See Algorithm 1 in [BKS11].

Parameters

V, W

The VectorArrays which are to be biorthonormalized.

product

The inner product Operator w.r.t. which to biorthonormalize. If None, the Euclidean product is used.

reiterate

If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.

reiteration_threshold

If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.

check

If True, check if the resulting VectorArray is really orthonormal.

check_tol

Tolerance for the check.

copy

If True, create a copy of V and W instead of modifying V and W in-place.

Returns

The biorthonormalized VectorArrays.

### greedy module¶

class pymor.algorithms.greedy.RBSurrogate(fom, reductor, use_error_estimator, error_norm, extension_params, pool)[source]

Surrogate for the weak_greedy error used in rb_greedy.

Not intended to be used directly.

Methods

evaluate(mus, return_all_values=False)[source]

Evaluate the surrogate for given parameters.

Parameters

mus

List of parameters for which to estimate the approximation error. When parallelization is used, mus can be a RemoteObject.

return_all_values

See below.

Returns

If return_all_values is True, an NumPy array of the estimated errors. If return_all_values is False, the maximum estimated error as first return value and the corresponding parameter as second return value.

class pymor.algorithms.greedy.WeakGreedySurrogate[source]

Surrogate for the approximation error in weak_greedy.

Methods

abstract evaluate(mus, return_all_values=False)[source]

Evaluate the surrogate for given parameters.

Parameters

mus

List of parameters for which to estimate the approximation error. When parallelization is used, mus can be a RemoteObject.

return_all_values

See below.

Returns

If return_all_values is True, an NumPy array of the estimated errors. If return_all_values is False, the maximum estimated error as first return value and the corresponding parameter as second return value.

pymor.algorithms.greedy.rb_greedy(fom, reductor, training_set, use_error_estimator=True, error_norm=None, atol=None, rtol=None, max_extensions=None, extension_params=None, pool=None)[source]

Weak Greedy basis generation using the RB approximation error as surrogate.

This algorithm generates a reduced basis using the weak greedy algorithm [BCDDPW11], where the approximation error is estimated from computing solutions of the reduced order model for the current reduced basis and then estimating the model reduction error.

Parameters

fom

The Model to reduce.

reductor

Reductor for reducing the given Model. This has to be an object with a reduce method, such that reductor.reduce() yields the reduced model, and an exted_basis method, such that reductor.extend_basis(U, copy_U=False, **extension_params) extends the current reduced basis by the vectors contained in U. For an example see CoerciveRBReductor.

training_set

The training set of Parameters on which to perform the greedy search.

use_error_estimator

If False, exactly compute the model reduction error by also computing the solution of fom for all parameter values of the training set. This is mainly useful when no estimator for the model reduction error is available.

error_norm

If use_error_estimator is False, use this function to calculate the norm of the error. If None, the Euclidean norm is used.

atol
rtol
max_extensions
extension_params

dict of parameters passed to the reductor.extend_basis method. If None, 'gram_schmidt' basis extension will be used as a default for stationary problems (fom.solve returns VectorArrays of length 1) and 'pod' basis extension (adding a single POD mode) for instationary problems.

pool

Returns

Dict with the following fields

rom

The reduced Model obtained for the computed basis.

max_errs

Sequence of maximum errors during the greedy run.

max_err_mus

The parameters corresponding to max_errs.

extensions

Number of performed basis extensions.

time

Total runtime of the algorithm.

pymor.algorithms.greedy.weak_greedy(surrogate, training_set, atol=None, rtol=None, max_extensions=None, pool=None)[source]

Weak greedy basis generation algorithm [BCDDPW11].

This algorithm generates an approximation basis for a given set of vectors associated with a training set of parameters by iteratively evaluating a surrogate for the approximation error on the training set and adding the worst approximated vector (according to the surrogate) to the basis.

The constructed basis is extracted from the surrogate after termination of the algorithm.

Parameters

surrogate

An instance of WeakGreedySurrogate representing the surrogate for the approximation error.

training_set

The set of parameter samples on which to perform the greedy search.

atol

If not None, stop the algorithm if the maximum (estimated) error on the training set drops below this value.

rtol

If not None, stop the algorithm if the maximum (estimated) relative error on the training set drops below this value.

max_extensions

If not None, stop the algorithm after max_extensions extension steps.

pool

If not None, a WorkerPool to use for parallelization. Parallelization needs to be supported by surrogate.

Returns

Dict with the following fields

max_errs

Sequence of maximum estimated errors during the greedy run.

max_err_mus

The parameters corresponding to max_errs.

extensions

Number of performed basis extensions.

time

Total runtime of the algorithm.

### hapod module¶

class pymor.algorithms.hapod.DistHAPODTree(slices)[source]

Methods

 DistHAPODTree children Tree is_leaf BasicObject

Attributes

 Tree depth, root BasicObject

class pymor.algorithms.hapod.FakeExecutor[source]

Bases: object

Methods

 FakeExecutor submit

class pymor.algorithms.hapod.IncHAPODTree(steps)[source]

Methods

 IncHAPODTree children Tree is_leaf BasicObject

Attributes

 Tree depth, root BasicObject

class pymor.algorithms.hapod.LifoExecutor(executor, max_workers=None)[source]

Bases: object

Methods

 LifoExecutor done_callback, run_task, submit

class pymor.algorithms.hapod.Tree[source]

A rooted tree.

Methods

 Tree children, is_leaf BasicObject

Attributes

 Tree depth, root BasicObject

pymor.algorithms.hapod.default_pod_method(U, eps, is_root_node, product)[source]

pymor.algorithms.hapod.dist_hapod(num_slices, snapshots, eps, omega, product=None, executor=None, eval_snapshots_in_executor=False)[source]

Distributed Hierarchical Approximate POD.

This computes the distributed HAPOD from [HLR18].

Parameters

num_slices

The number of snapshot vector slices.

snapshots

A mapping snapshots(slice) returning for each slice number the associated snapshot vectors.

eps

Desired l2-mean approximation error.

omega

Tuning parameter (0 < omega < 1) to balance performance with approximation quality.

product

Inner product Operator w.r.t. which to compute the POD.

executor

If not None, a concurrent.futures.Executor object to use for parallelization.

eval_snapshots_in_executor

If True also parallelize the evaluation of the snapshot map.

Returns

modes

The computed POD modes.

svals

The associated singular values.

snap_count

The total number of input snapshot vectors.

pymor.algorithms.hapod.dist_vectorarray_hapod(num_slices, U, eps, omega, product=None, executor=None)[source]

Distributed Hierarchical Approximate POD.

This computes the distributed HAPOD from [HLR18] of a given VectorArray.

Parameters

num_slices

The number of snapshot vector slices.

U

The VectorArray of which to compute the HAPOD.

eps

Desired l2-mean approximation error.

omega

Tuning parameter (0 < omega < 1) to balance performance with approximation quality.

product

Inner product Operator w.r.t. which to compute the POD.

executor

If not None, a concurrent.futures.Executor object to use for parallelization.

Returns

modes

The computed POD modes.

svals

The associated singular values.

snap_count

The total number of input snapshot vectors.

pymor.algorithms.hapod.hapod(tree, snapshots, local_eps, product=None, pod_method=<function default_pod_method>, executor=None, eval_snapshots_in_executor=False)[source]

Compute the Hierarchical Approximate POD.

This is an implementation of the HAPOD algorithm from [HLR18].

Parameters

tree

A Tree defining the worker topology.

snapshots

A mapping snapshots(node) returning for each leaf node the associated snapshot vectors.

local_eps

A mapping local_eps(node, snap_count, num_vecs) assigning to each tree node node an l2 truncation error tolerance for the local pod based on the number of input vectors num_vecs and the total number of snapshot vectors below the given node snap_count.

product

Inner product Operator w.r.t. which to compute the POD.

pod_method

A function pod_method(U, eps, root_node, product) for computing the POD of the VectorArray U w.r.t. the given inner product product and the l2 error tolerance eps. root_node is set to True when the POD is computed at the root of the tree.

executor

If not None, a concurrent.futures.Executor object to use for parallelization.

eval_snapshots_in_executor

If True also parallelize the evaluation of the snapshot map.

Returns

modes

The computed POD modes.

svals

The associated singular values.

snap_count

The total number of input snapshot vectors.

pymor.algorithms.hapod.inc_hapod(steps, snapshots, eps, omega, product=None, executor=None, eval_snapshots_in_executor=False)[source]

Incremental Hierarchical Approximate POD.

This computes the incremental HAPOD from [HLR18].

Parameters

steps

The number of incremental POD updates.

snapshots

A mapping snapshots(step) returning for each incremental POD step the associated snapshot vectors.

eps

Desired l2-mean approximation error.

omega

Tuning parameter (0 < omega < 1) to balance performance with approximation quality.

product

Inner product Operator w.r.t. which to compute the POD.

executor

If not None, a concurrent.futures.Executor object to use for parallelization.

eval_snapshots_in_executor

If True also parallelize the evaluation of the snapshot map.

Returns

modes

The computed POD modes.

svals

The associated singular values.

snap_count

The total number of input snapshot vectors.

pymor.algorithms.hapod.inc_vectorarray_hapod(steps, U, eps, omega, product=None, executor=None)[source]

Incremental Hierarchical Approximate POD.

This computes the incremental HAPOD from [HLR18] for a given VectorArray.

Parameters

steps

The number of incremental POD updates.

U

The VectorArray of which to compute the HAPOD.

eps

Desired l2-mean approximation error.

omega

Tuning parameter (0 < omega < 1) to balance performance with approximation quality.

product

Inner product Operator w.r.t. which to compute the POD.

executor

If not None, a concurrent.futures.Executor object to use for parallelization.

eval_snapshots_in_executor

If True also parallelize the evaluation of the snapshot map.

Returns

modes

The computed POD modes.

svals

The associated singular values.

snap_count

The total number of input snapshot vectors.

pymor.algorithms.hapod.std_local_eps(tree, eps, omega, pod_on_leafs=True)[source]

### image module¶

class pymor.algorithms.image.CollectOperatorRangeRules(source, image, extends)[source]

RuleTable for the estimate_image algorithm.

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 CollectOperatorRangeRules action_apply_operator, action_ConcatenationOperator, action_EmpiricalInterpolatedOperator, action_recurse, rules BasicObject

class pymor.algorithms.image.CollectVectorRangeRules(image)[source]

RuleTable for the estimate_image algorithm.

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 CollectVectorRangeRules action_as_range_array, action_ConcatenationOperator, action_recurse, action_VectorArray, rules BasicObject

pymor.algorithms.image.estimate_image(operators=(), vectors=(), domain=None, extends=False, orthonormalize=True, product=None, riesz_representatives=False)[source]

Estimate the image of given Operators for all mu.

Let operators be a list of Operators with common source and range, and let vectors be a list of VectorArrays or vector-like Operators in the range of these operators. Given a VectorArray domain of vectors in the source of the operators, this algorithms determines a VectorArray image of range vectors such that the linear span of image contains:

The algorithm will try to choose image as small as possible. However, no optimality is guaranteed. The image estimation algorithm is specified by CollectOperatorRangeRules and CollectVectorRangeRules.

Parameters

operators

See above.

vectors

See above.

domain

See above. If None, an empty domain VectorArray is assumed.

extends

For some operators, e.g. EmpiricalInterpolatedOperator, as well as for all elements of vectors, image is estimated independently from the choice of domain. If extends is True, such operators are ignored. (This is useful in case these vectors have already been obtained by earlier calls to this function.)

orthonormalize

Compute an orthonormal basis for the linear span of image using the gram_schmidt algorithm.

product

Inner product Operator w.r.t. which to orthonormalize.

riesz_representatives

If True, compute Riesz representatives of the vectors in image before orthonormalizing (useful for dual norm computation when the range of the operators is a dual space).

Returns

The VectorArray image.

Raises

ImageCollectionError

Is raised when for a given Operator no image estimate is possible.

pymor.algorithms.image.estimate_image_hierarchical(operators=(), vectors=(), domain=None, extends=None, orthonormalize=True, product=None, riesz_representatives=False)[source]

Estimate the image of given Operators for all mu.

This is an extended version of estimate_image, which calls estimate_image individually for each vector of domain.

As a result, the vectors in the returned image VectorArray will be ordered by the domain vector they correspond to (starting with vectors which correspond to the elements of vectors and to Operators for which the image is estimated independently from domain).

This function also returns an image_dims list, such that the first image_dims[i+1] vectors of image correspond to the first i vectors of domain (the first image_dims[0] vectors correspond to vectors and to Operators with fixed image estimate).

Parameters

operators
vectors
domain
extends

When additional vectors have been appended to the domain VectorArray after estimate_image_hierarchical has been called, and estimate_image_hierarchical shall be called again for the extended domain array, extends can be set to (image, image_dims), where image, image_dims are the return values of the last estimate_image_hierarchical call. The old domain vectors will then be skipped during computation and image, image_dims will be modified in-place.

orthonormalize
product
riesz_representatives

Returns

image

See above.

image_dims

See above.

Raises

ImageCollectionError

Is raised when for a given Operator no image estimate is possible.

### krylov module¶

Module for computing (rational) Krylov subspaces’ bases.

pymor.algorithms.krylov.rational_arnoldi(A, E, b, sigma, trans=False)[source]

Rational Arnoldi algorithm.

If trans == False, using Arnoldi process, computes a real orthonormal basis for the rational Krylov subspace

$\mathrm{span}\{ (\sigma_1 E - A)^{-1} b, (\sigma_2 E - A)^{-1} b, \ldots, (\sigma_r E - A)^{-1} b \},$

otherwise, computes the same for

$\mathrm{span}\{ (\sigma_1 E - A)^{-T} b^T, (\sigma_2 E - A)^{-T} b^T, \ldots, (\sigma_r E - A)^{-T} b^T \}.$

Interpolation points in sigma are allowed to repeat (in any order). Then, in the above expression,

$\underbrace{ (\sigma_i E - A)^{-1} b, \ldots, (\sigma_i E - A)^{-1} b }_{m \text{ times}}$

is replaced by

$(\sigma_i E - A)^{-1} b, (\sigma_i E - A)^{-1} E (\sigma_i E - A)^{-1} b, \ldots, \left((\sigma_i E - A)^{-1} E\right)^{m - 1} (\sigma_i E - A)^{-1} b.$

Analogously for the trans == True case.

Parameters

A

Real Operator A.

E

Real Operator E.

b

Real vector-like operator (if trans is False) or functional (if trans is True).

sigma

Sequence of interpolation points (closed under conjugation).

trans

Boolean, see above.

Returns

V

Orthonormal basis for the Krylov subspace VectorArray.

pymor.algorithms.krylov.tangential_rational_krylov(A, E, B, b, sigma, trans=False, orth=True)[source]

Tangential Rational Krylov subspace.

If trans == False, computes a real basis for the rational Krylov subspace

$\mathrm{span}\{ (\sigma_1 E - A)^{-1} B b_1, (\sigma_2 E - A)^{-1} B b_2, \ldots, (\sigma_r E - A)^{-1} B b_r \},$

otherwise, computes the same for

$\mathrm{span}\{ (\sigma_1 E - A)^{-T} B^T b_1, (\sigma_2 E - A)^{-T} B^T b_2, \ldots, (\sigma_r E - A)^{-T} B^T b_r \}.$

Interpolation points in sigma are assumed to be pairwise distinct.

Parameters

A

Real Operator A.

E

Real Operator E.

B

Real Operator B.

b
VectorArray from B.source, if trans == False, or

B.range, if trans == True.

sigma

Sequence of interpolation points (closed under conjugation), of the same length as b.

trans

Boolean, see above.

orth

If True, orthonormalizes the basis using pymor.algorithms.gram_schmidt.gram_schmidt.

Returns

V

Optionally orthonormal basis for the Krylov subspace VectorArray.

### lincomb module¶

class pymor.algorithms.lincomb.AssembleLincombRules(coefficients, solver_options, name)[source]

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 AssembleLincombRules action_BlockDiagonalOperator, action_BlockOperatorBase, action_BlockSpaceIdentityOperator, action_call_assemble_lincomb_method, action_failed, action_IdentityAndSecondOrderModelOperator, action_IdentityOperator, action_merge_into_low_rank_updated_operator, action_merge_low_rank_operators, action_only_one_operator, action_VectorArrayOperator, action_zero_coeff, action_ZeroOperator, rules BasicObject

pymor.algorithms.lincomb.assemble_lincomb(operators, coefficients, solver_options=None, name=None)[source]

Try to assemble a linear combination of the given operators.

Returns a new Operator which represents the sum

c_1*O_1 + ... + c_N*O_N


where O_i are Operators and c_i scalar coefficients.

This function is called in the assemble method of LincombOperator and is not intended to be used directly. The assembled Operator is expected to no longer be a LincombOperator nor should it contain any LincombOperators. If an assembly of the given linear combination is not possible, None is returned. The special case of a LincombOperator with a single operator (i.e. a scaled Operator) is allowed as assemble_lincomb implements apply_inverse for this special case.

To form the linear combination of backend Operators (containing actual matrix data), _assemble_lincomb will be called on the first Operator in the linear combination.

Parameters

operators

List of Operators O_i whose linear combination is formed.

coefficients

List of the corresponding linear coefficients c_i.

solver_options

solver_options for the assembled operator.

name

Name of the assembled operator.

Returns

The assembled Operator if assembly is possible, otherwise None.

pymor.algorithms.line_search.armijo(f, starting_point, direction, grad=None, initial_value=None, alpha_init=1.0, tau=0.5, beta=0.0001, maxiter=10)[source]

Armijo line search algorithm.

This method computes a step size such that the Armijo condition (see [NW06], p. 33) is fulfilled.

Parameters

f

Real-valued function that can be evaluated for its value.

starting_point

A VectorArray of length 1 containing the starting point of the line search.

direction

Descent direction along which the line search is performed.

Gradient of f in the point starting_point.

initial_value

Value of f in the point starting_point.

alpha_init

Initial step size that is gradually reduced.

tau

The fraction by which the step size is reduced in each iteration.

beta

Control parameter to adjust the required decrease of the function value of f.

maxiter

Use alpha_init as default if the iteration count reaches this value without finding a point fulfilling the Armijo condition.

Returns

alpha

Step size computed according to the Armijo condition.

Defaults

alpha_init, tau, beta, maxiter (see pymor.core.defaults)

pymor.algorithms.lradi.lyap_lrcf_solver_options(lradi_tol=1e-10, lradi_maxiter=500, lradi_shifts='projection_shifts', projection_shifts_init_maxiter=20, projection_shifts_init_seed=None)[source]

Return available Lyapunov solvers with default options.

Parameters

projection_shifts_init_maxiter
projection_shifts_init_seed

Returns

A dict of available solvers with default solver options.

Defaults

lradi_tol, lradi_maxiter, lradi_shifts, projection_shifts_init_maxiter, projection_shifts_init_seed (see pymor.core.defaults)

pymor.algorithms.lradi.projection_shifts(A, E, V, prev_shifts)[source]

Find further shift parameters for low-rank ADI iteration using Galerkin projection on spaces spanned by LR-ADI iterates.

See [PK16], pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

V

A VectorArray representing the currently computed iterate.

prev_shifts

A NumPy array containing the set of all previously used shift parameters.

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.projection_shifts_init(A, E, B, shift_options)[source]

Find starting shift parameters for low-rank ADI iteration using Galerkin projection on spaces spanned by LR-ADI iterates.

See [PK16], pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

B

The VectorArray B from the corresponding Lyapunov equation.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.solve_lyap_lrcf(A, E, B, trans=False, options=None)[source]

Compute an approximate low-rank solution of a Lyapunov equation.

See pymor.algorithms.lyapunov.solve_lyap_lrcf for a general description.

This function uses the low-rank ADI iteration as described in Algorithm 4.3 in [PK16].

Parameters

A

The non-parametric Operator A.

E

The non-parametric Operator E or None.

B

The operator B as a VectorArray from A.source.

trans

Whether the first Operator in the Lyapunov equation is transposed.

options

The solver options to use (see lyap_lrcf_solver_options).

Returns

Z

Low-rank Cholesky factor of the Lyapunov equation solution, VectorArray from A.source.

pymor.algorithms.lrradi.hamiltonian_shifts(A, E, B, R, K, Z, shift_options)[source]

Compute further shift parameters for low-rank RADI iteration.

Compute Galerkin projection of Hamiltonian matrix on space spanned by last few columns of $$Z$$ and return the eigenvalue of the projected Hamiltonian with the most impact on convergence as the next shift parameter.

See [BBKS18], pp. 318-321.

Parameters

A

The Operator A from the corresponding Riccati equation.

E

The Operator E from the corresponding Riccati equation.

B

The VectorArray B from the corresponding Riccati equation.

R

A VectorArray representing the currently computed residual factor.

K

A VectorArray representing the currently computed iterate.

Z

A VectorArray representing the currently computed solution factor.

shift_options

The shift options to use (see ricc_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lrradi.hamiltonian_shifts_init(A, E, B, C, shift_options)[source]

Compute initial shift parameters for low-rank RADI iteration.

Compute Galerkin projection of Hamiltonian matrix on space spanned by $$C$$ and return the eigenvalue of the projected Hamiltonian with the most impact on convergence as the next shift parameter.

See [BBKS18], pp. 318-321.

Parameters

A

The Operator A from the corresponding Riccati equation.

E

The Operator E from the corresponding Riccati equation.

B

The VectorArray B from the corresponding Riccati equation.

C

The VectorArray C from the corresponding Riccati equation.

shift_options

The shift options to use (see ricc_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lrradi.ricc_lrcf_solver_options(lrradi_tol=1e-10, lrradi_maxiter=500, lrradi_shifts='hamiltonian_shifts', hamiltonian_shifts_init_maxiter=20, hamiltonian_shifts_init_seed=None, hamiltonian_shifts_subspace_columns=6)[source]

Returns available Riccati equation solvers with default solver options.

Parameters

hamiltonian_shifts_init_maxiter
hamiltonian_shifts_init_seed
hamiltonian_shifts_subspace_columns

Returns

A dict of available solvers with default solver options.

Defaults

lrradi_tol, lrradi_maxiter, lrradi_shifts, hamiltonian_shifts_init_maxiter, hamiltonian_shifts_init_seed, hamiltonian_shifts_subspace_columns (see pymor.core.defaults)

pymor.algorithms.lrradi.solve_ricc_lrcf(A, E, B, C, R=None, trans=False, options=None)[source]

Compute an approximate low-rank solution of a Riccati equation.

See pymor.algorithms.riccati.solve_ricc_lrcf for a general description.

This function is an implementation of Algorithm 2 in [BBKS18].

Parameters

A

The Operator A.

E

The Operator E or None.

B

The operator B as a VectorArray from A.source.

C

The operator C as a VectorArray from A.source.

R

The operator R as a 2D NumPy array or None.

trans

Whether the first Operator in the Riccati equation is transposed.

options

The solver options to use. (see ricc_lrcf_solver_options)

Returns

Z

Low-rank Cholesky factor of the Riccati equation solution, VectorArray from A.source.

### lyapunov module¶

pymor.algorithms.lyapunov._chol(A)[source]

Cholesky decomposition.

This implementation uses SVD to compute the Cholesky factor (can be used for singular matrices).

Parameters

A

Symmetric positive semidefinite matrix as a NumPy array.

Returns

L

Cholesky factor of A (in the sense that L * L^T approximates A).

pymor.algorithms.lyapunov.mat_eqn_sparse_min_size(value=1000)[source]

Returns minimal size for which a sparse solver will be used by default.

Defaults

value (see pymor.core.defaults)

pymor.algorithms.lyapunov.solve_lyap_dense(A, E, B, trans=False, options=None, default_solver_backend='pymess')[source]

Compute the solution of a Lyapunov equation.

Returns the solution $$X$$ of a (generalized) continuous-time algebraic Lyapunov equation:

• if trans is False and E is None:

$A X + X A^T + B B^T = 0,$
• if trans is False and E is an Operator:

$A X E^T + E X A^T + B B^T = 0,$
• if trans is True and E is None:

$A^T X + X A + B^T B = 0,$
• if trans is True and E is an Operator:

$A^T X E + E^T X A + B^T B = 0.$

We assume A and E are real NumPy arrays, E is invertible, and that no two eigenvalues of (A, E) sum to zero (i.e., there exists a unique solution X).

If the solver is not specified using the options argument, a solver backend is chosen based on availability in the following order:

1. pymess (see pymor.bindings.pymess.solve_lyap_dense)

2. slycot (see pymor.bindings.slycot.solve_lyap_dense)

3. scipy (see pymor.bindings.scipy.solve_lyap_dense)

Parameters

A

The operator A as a 2D NumPy array.

E

The operator E as a 2D NumPy array or None.

B

The operator B as a 2D NumPy array.

trans

Whether the first operator in the Lyapunov equation is transposed.

options

The solver options to use. See:

default_solver_backend

Default solver backend to use (pymess, slycot, scipy).

Returns

X

Lyapunov equation solution as a NumPy array.

Defaults

default_solver_backend (see pymor.core.defaults)

pymor.algorithms.lyapunov.solve_lyap_lrcf(A, E, B, trans=False, options=None, default_sparse_solver_backend='pymess', default_dense_solver_backend='pymess')[source]

Compute an approximate low-rank solution of a Lyapunov equation.

Returns a low-rank Cholesky factor $$Z$$ such that $$Z Z^T$$ approximates the solution $$X$$ of a (generalized) continuous-time algebraic Lyapunov equation:

• if trans is False and E is None:

$A X + X A^T + B B^T = 0,$
• if trans is False and E is an Operator:

$A X E^T + E X A^T + B B^T = 0,$
• if trans is True and E is None:

$A^T X + X A + B^T B = 0,$
• if trans is True and E is an Operator:

$A^T X E + E^T X A + B^T B = 0.$

We assume A and E are real Operators, E is invertible, and all the eigenvalues of (A, E) all lie in the open left half-plane. Operator B needs to be given as a VectorArray from A.source, and for large-scale problems, we assume len(B) is small.

If the solver is not specified using the options argument, a solver backend is chosen based on availability in the following order:

Parameters

A

The non-parametric Operator A.

E

The non-parametric Operator E or None.

B

The operator B as a VectorArray from A.source.

trans

Whether the first Operator in the Lyapunov equation is transposed.

options

The solver options to use. See:

default_sparse_solver_backend

Default sparse solver backend to use (pymess, lradi).

default_dense_solver_backend

Default dense solver backend to use (pymess, slycot, scipy).

Returns

Z

Low-rank Cholesky factor of the Lyapunov equation solution, VectorArray from A.source.

Defaults

default_sparse_solver_backend, default_dense_solver_backend (see pymor.core.defaults)

### newton module¶

pymor.algorithms.newton.newton(operator, rhs, initial_guess=None, mu=None, range_product=None, source_product=None, least_squares=False, miniter=0, maxiter=100, atol=0.0, rtol=1e-07, relax='armijo', line_search_params=None, stagnation_window=3, stagnation_threshold=inf, error_measure='update', return_stages=False, return_residuals=False)[source]

Newton algorithm.

This method solves the nonlinear equation

A(U, mu) = V


for U using the Newton method.

Parameters

operator

The Operator A. A must implement the jacobian interface method.

rhs

VectorArray of length 1 containing the vector V.

initial_guess

If not None, a VectorArray of length 1 containing an initial guess for the solution U.

mu

The parameter values for which to solve the equation.

range_product

The inner product Operator on operator.range with which the norm of the resiudal is computed. If None, the Euclidean inner product is used.

source_product

The inner product Operator on operator.source with which the norm of the solution and update vectors is computed. If None, the Euclidean inner product is used.

least_squares

If True, use a least squares linear solver (e.g. for residual minimization).

miniter

Minimum amount of iterations to perform.

maxiter

Fail if the iteration count reaches this value without converging.

atol

Finish when the error measure is below this threshold.

rtol

Finish when the error measure has been reduced by this factor relative to the norm of the initial residual resp. the norm of the current solution.

relax

If real valued, relaxation factor for Newton updates; otherwise 'armijo' to indicate that the armijo line search algorithm shall be used.

line_search_params

Dictionary of additional parameters passed to the line search method.

stagnation_window

Finish when the error measure has not been reduced by a factor of stagnation_threshold during the last stagnation_window iterations.

stagnation_threshold

See stagnation_window.

error_measure

If 'residual', convergence depends on the norm of the residual. If 'update', convergence depends on the norm of the update vector.

return_stages

If True, return a VectorArray of the intermediate approximations of U after each iteration.

return_residuals

If True, return a VectorArray of all residual vectors which have been computed during the Newton iterations.

Returns

U

VectorArray of length 1 containing the computed solution

data

Dict containing the following fields:

solution_norms

NumPy array of the solution norms after each iteration.

update_norms

NumPy array of the norms of the update vectors for each iteration.

residual_norms

NumPy array of the residual norms after each iteration.

stages

See return_stages.

residuals

See return_residuals.

Raises

NewtonError

Raised if the Newton algorithm failed to converge.

Defaults

miniter, maxiter, rtol, atol, relax, stagnation_window, stagnation_threshold (see pymor.core.defaults)

### pod module¶

pymor.algorithms.pod.pod(A, product=None, modes=None, rtol=1e-07, atol=0.0, l2_err=0.0, method='method_of_snapshots', orth_tol=1e-10)[source]

Proper orthogonal decomposition of A.

Viewing the VectorArray A as a A.dim x len(A) matrix, the return values of this method are the VectorArray of left singular vectors and a NumPy array of singular values of the singular value decomposition of A, where the inner product on R^(dim(A)) is given by product and the inner product on R^(len(A)) is the Euclidean inner product.

Parameters

A

The VectorArray for which the POD is to be computed.

product

Inner product Operator w.r.t. which the POD is computed.

modes

If not None, at most the first modes POD modes (singular vectors) are returned.

rtol

Singular values smaller than this value multiplied by the largest singular value are ignored.

atol

Singular values smaller than this value are ignored.

l2_err

Do not return more modes than needed to bound the l2-approximation error by this value. I.e. the number of returned modes is at most

argmin_N { sum_{n=N+1}^{infty} s_n^2 <= l2_err^2 }


where s_n denotes the n-th singular value.

method

Which SVD method from svd_va to use ('method_of_snapshots' or 'qr_svd').

orth_tol

POD modes are reorthogonalized if the orthogonality error is above this value.

Returns

POD

VectorArray of POD modes.

SVALS

One-dimensional NumPy array of singular values.

Defaults

rtol, atol, l2_err, method, orth_tol (see pymor.core.defaults)

### preassemble module¶

class pymor.algorithms.preassemble.PreAssembleRules[source]

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 PreAssembleRules action_AdjointOperator, action_assemble, action_identity, action_recurse, action_recurse_and_assemble, rules BasicObject

pymor.algorithms.preassemble.preassemble(obj)[source]

Preassemble non-parametric operators.

If obj is a non-parametric Operator, return obj.assemble() otherwise return obj. Recursively replaces children of obj.

### projection module¶

class pymor.algorithms.projection.ProjectRules(range_basis, source_basis)[source]

RuleTable for the project algorithm.

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 ProjectRules action_AdjointOperator, action_AffineOperator, action_apply_basis, action_BlockOperatorBase, action_ConcatenationOperator, action_ConstantOperator, action_EmpiricalInterpolatedOperator, action_LincombOperator, action_no_bases, action_SelectionOperator, action_ZeroOperator, rules BasicObject

class pymor.algorithms.projection.ProjectToSubbasisRules(dim_range, dim_source)[source]

RuleTable for the project_to_subbasis algorithm.

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 ProjectToSubbasisRules action_BlockColumnOperator, action_BlockRowOperator, action_ConstantOperator, action_IdentityOperator, action_NumpyMatrixOperator, action_ProjectedEmpiciralInterpolatedOperator, action_ProjectedOperator, action_recurse, action_VectorArrayOperator, action_ZeroOperator, rules BasicObject

pymor.algorithms.projection.project(op, range_basis, source_basis, product=None)[source]

Petrov-Galerkin projection of a given Operator.

Given an inner product ( ⋅, ⋅), source vectors b_1, ..., b_N and range vectors c_1, ..., c_M, the projection op_proj of op is defined by

[ op_proj(e_j) ]_i = ( c_i, op(b_j) )


for all i,j, where e_j denotes the j-th canonical basis vector of R^N.

In particular, if the c_i are orthonormal w.r.t. the given product, then op_proj is the coordinate representation w.r.t. the b_i/c_i bases of the restriction of op to span(b_i) concatenated with the orthogonal projection onto span(c_i).

From another point of view, if op is viewed as a bilinear form (see apply2) and ( ⋅, ⋅ ) is the Euclidean inner product, then op_proj represents the matrix of the bilinear form restricted to span(b_i) / span(c_i) (w.r.t. the b_i/c_i bases).

How the projection is realized will depend on the given Operator. While a projected NumpyMatrixOperator will again be a NumpyMatrixOperator, only a generic ProjectedOperator can be returned in general. The exact algorithm is specified in ProjectRules.

Parameters

range_basis

The vectors c_1, ..., c_M as a VectorArray. If None, no projection in the range space is performed.

source_basis

The vectors b_1, ..., b_N as a VectorArray or None. If None, no restriction of the source space is performed.

product

An Operator representing the inner product. If None, the Euclidean inner product is chosen.

Returns

The projected Operator op_proj.

pymor.algorithms.projection.project_to_subbasis(op, dim_range=None, dim_source=None)[source]

Project already projected Operator to a subbasis.

The purpose of this method is to further project an operator that has been obtained through project to subbases of the original projection bases, i.e.

project_to_subbasis(project(op, r_basis, s_basis, prod), dim_range, dim_source)


should be the same as

project(op, r_basis[:dim_range], s_basis[:dim_source], prod)


For a NumpyMatrixOperator this amounts to extracting the upper-left (dim_range, dim_source) corner of its matrix.

The subbasis projection algorithm is specified in ProjectToSubbasisRules.

Parameters

dim_range

Dimension of the range subbasis.

dim_source

Dimension of the source subbasis.

Returns

The projected Operator.

### randrangefinder module¶

pymor.algorithms.randrangefinder.adaptive_rrf(A, source_product=None, range_product=None, tol=0.0001, failure_tolerance=1e-15, num_testvecs=20, lambda_min=None, iscomplex=False)[source]

Adaptive randomized range approximation of A.

This is an implementation of Algorithm 1 in [BS18].

Given the Operator A, the return value of this method is the VectorArray B with the property

$\Vert A - P_{span(B)} A \Vert \leq tol$

with a failure probability smaller than failure_tolerance, where the norm denotes the operator norm. The inner product of the range of A is given by range_product and the inner product of the source of A is given by source_product.

Parameters

A

The Operator A.

source_product

Inner product Operator of the source of A.

range_product

Inner product Operator of the range of A.

tol

Error tolerance for the algorithm.

failure_tolerance

Maximum failure probability.

num_testvecs

Number of test vectors.

lambda_min

The smallest eigenvalue of source_product. If None, the smallest eigenvalue is computed using scipy.

iscomplex

If True, the random vectors are chosen complex.

Returns

B

VectorArray which contains the basis, whose span approximates the range of A.

Defaults

tol, failure_tolerance, num_testvecs (see pymor.core.defaults)

pymor.algorithms.randrangefinder.rrf(A, source_product=None, range_product=None, q=2, l=8, iscomplex=False)[source]

Randomized range approximation of A.

This is an implementation of Algorithm 4.4 in [HMT11].

Given the Operator A, the return value of this method is the VectorArray Q whose vectors form an approximate orthonomal basis for the range of A.

Parameters

A

The Operator A.

source_product

Inner product Operator of the source of A.

range_product

Inner product Operator of the range of A.

q

The number of power iterations.

l

The block size of the normalized power iterations.

iscomplex

If True, the random vectors are chosen complex.

Returns

Q

VectorArray which contains the basis, whose span approximates the range of A.

Defaults

q, l (see pymor.core.defaults)

### riccati module¶

pymor.algorithms.riccati.solve_pos_ricc_lrcf(A, E, B, C, R=None, trans=False, options=None, default_dense_solver_backend='pymess')[source]

Compute an approximate low-rank solution of a positive Riccati equation.

Returns a low-rank Cholesky factor $$Z$$ such that $$Z Z^T$$ approximates the solution $$X$$ of a (generalized) positive continuous-time algebraic Riccati equation:

• if trans is False

$A X E^T + E X A^T + E X C^T R^{-1} C X E^T + B B^T = 0.$
• if trans is True

$A^T X E + E^T X A + E^T X B R^{-1} B^T X E + C^T C = 0.$

If E is None, it is taken to be identity, and similarly for R.

If the solver is not specified using the options argument, a solver backend is chosen based on availability in the following order:

1. pymess (see pymor.bindings.pymess.solve_pos_ricc_lrcf),

2. slycot (see pymor.bindings.slycot.solve_pos_ricc_lrcf),

3. scipy (see pymor.bindings.scipy.solve_pos_ricc_lrcf).

Parameters

A

The non-parametric Operator A.

E

The non-parametric Operator E or None.

B

The operator B as a VectorArray from A.source.

C

The operator C as a VectorArray from A.source.

R

The operator R as a 2D NumPy array or None.

trans

Whether the first Operator in the positive Riccati equation is transposed.

options

The solver options to use. See:

default_dense_solver_backend

Default dense solver backend to use (pymess, slycot, scipy).

Returns

Z

Low-rank Cholesky factor of the positive Riccati equation solution, VectorArray from A.source.

Defaults

default_dense_solver_backend (see pymor.core.defaults)

pymor.algorithms.riccati.solve_ricc_lrcf(A, E, B, C, R=None, trans=False, options=None, default_sparse_solver_backend='pymess', default_dense_solver_backend='pymess')[source]

Compute an approximate low-rank solution of a Riccati equation.

Returns a low-rank Cholesky factor $$Z$$ such that $$Z Z^T$$ approximates the solution $$X$$ of a (generalized) continuous-time algebraic Riccati equation:

• if trans is False

$A X E^T + E X A^T - E X C^T R^{-1} C X E^T + B B^T = 0.$
• if trans is True

$A^T X E + E^T X A - E^T X B R^{-1} B^T X E + C^T C = 0.$

If E is None, it is taken to be identity, and similarly for R.

We assume:

For large-scale problems, we additionally assume that len(B) and len(C) are small.

If the solver is not specified using the options argument, a solver backend is chosen based on availability in the following order:

Parameters

A

The non-parametric Operator A.

E

The non-parametric Operator E or None.

B

The operator B as a VectorArray from A.source.

C

The operator C as a VectorArray from A.source.

R

The operator R as a 2D NumPy array or None.

trans

Whether the first Operator in the Riccati equation is transposed.

options

The solver options to use. See:

default_sparse_solver_backend

Default sparse solver backend to use (pymess, lrradi).

default_dense_solver_backend

Default dense solver backend to use (pymess, slycot, scipy).

Returns

Z

Low-rank Cholesky factor of the Riccati equation solution, VectorArray from A.source.

Defaults

default_sparse_solver_backend, default_dense_solver_backend (see pymor.core.defaults)

### rules module¶

class pymor.algorithms.rules.RuleTable(use_caching=False)[source]

Define algorithm by a table of match conditions and corresponding actions.

RuleTable manages a table of rules, stored in the rules attributes, which can be applied to given objects.

A new table is created by subclassing RuleTable and defining new methods which are decorated with match_class, match_generic or another rule subclass. The order of the method definitions determines the order in which the defined rules are applied.

Parameters

use_caching

If True, cache results of apply.

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject
rules

list of all defined rules.

__repr__()[source]

Return repr(self).

apply(obj)[source]

Sequentially apply rules to given object.

This method iterates over all rules of the given RuleTable. For each rule, it is checked if it matches the given object. If False, the next rule in the table is considered. If True the corresponding action is executed with obj as parameter. If execution of action raises RuleNotMatchingError, the rule is considered as not matching, and execution continues with evaluation of the next rule. Otherwise, execution is stopped and the return value of rule.action is returned to the caller.

If no rule matches, a NoMatchingRuleError is raised.

Parameters

obj

The object to apply the RuleTable to.

Returns

Return value of the action of the first matching rule in the table.

Raises

NoMatchingRuleError

No rule could be applied to the given object.

apply_children(obj, children=None)[source]

Apply rules to all children of the given object.

This method calls apply to each child of the given object. The children of the object are either provided by the children parameter or automatically inferred by the get_children method.

Parameters

obj

The object to apply the RuleTable to.

children

None or a list of attribute names defining the children to consider.

Returns

Result of : meth:apply for all given children.

classmethod breakpoint_for_name(name)[source]

Add a conditional breakpoint for objects of given name.

Break execution in apply, when being applied to an object with a certain name.

Parameters

name

name of the object for which to add the conditional breakpoint.

classmethod breakpoint_for_obj(obj)[source]

Add a conditional breakpoint for given object.

Break execution in apply, when being applied to a certain object.

Parameters

obj

Object for which to add the conditional breakpoint.

classmethod get_children(obj)[source]

Determine children of given object.

This method returns a list of the names of all attributes a, for which one of the folling is true:

1. a is an Operator.

2. a is a mapping and each of its values is either an Operator or None.

3. a is an iterable and each of its elements is either an Operator or None.

replace_children(obj, children=None)[source]

Replace children of object according to rule table.

Same as apply_children, but additionally calls obj.with_ to replace the children of obj with the result of the corresponding apply call.

class pymor.algorithms.rules.RuleTableMeta(name, parents, dct)[source]

Meta class for RuleTable.

Methods

 ABCMeta register, __instancecheck__, __subclasscheck__ type mro, __dir__, __prepare__, __sizeof__, __subclasses__
static __new__(cls, name, parents, dct)[source]

I copy docstrings from base class methods to deriving classes.

Copying of docstrings is disabled when the PYMOR_WITH_SPHINX environment variable is set to 1.

__repr__()[source]

Return repr(self).

__str__()

Return repr(self).

pymor.algorithms.rules.format_rules(rules)[source]

class pymor.algorithms.rules.match_always(action)[source]

rule that always matches.

Methods

Attributes

 match_always condition_type rule action, action_description, condition_description, source
_matches(obj)[source]

Returns True if given object matches the condition.

class pymor.algorithms.rules.match_class(*classes)[source]

rule that matches when obj is instance of one of the given classes.

Methods

Attributes

 match_class condition_type rule action, action_description, condition_description, source
_matches(obj)[source]

Returns True if given object matches the condition.

class pymor.algorithms.rules.match_class_all(*classes)[source]

rule that matches when each item of obj is instance of one of the given classes.

Methods

Attributes

 match_class_all condition_type rule action, action_description, condition_description, source
_matches(obj)[source]

Returns True if given object matches the condition.

class pymor.algorithms.rules.match_class_any(*classes)[source]

rule that matches when any item of obj is instance of one of the given classes.

Methods

Attributes

 match_class_any condition_type rule action, action_description, condition_description, source
_matches(obj)[source]

Returns True if given object matches the condition.

class pymor.algorithms.rules.match_class_base(*classes)[source]

Methods

Attributes

 rule action, action_description, condition_description, condition_type, source

class pymor.algorithms.rules.match_generic(condition, condition_description=None)[source]

rule with matching condition given by an arbitrary function.

Parameters

condition

Function of one argument which checks if given object matches condition.

condition_description

Optional string describing the condition implemented by condition.

Methods

Attributes

 match_generic condition_type rule action, action_description, condition_description, source
_matches(obj)[source]

Returns True if given object matches the condition.

pymor.algorithms.rules.print_children(obj)[source]

pymor.algorithms.rules.print_rules(rules)[source]

class pymor.algorithms.rules.rule[source]

Bases: object

Decorator to make a method a rule in a given RuleTable.

The decorated function will become the action to perform in case the rule matches. Matching conditions are specified by subclassing and overriding the matches method.

If an action is decorated by multiple rules, all these rules must match for the action to apply.

Methods

Attributes

 rule action, action_description, condition_description, condition_type, source
action

Method to call in case the rule matches.

__call__(action)[source]

Call self as a function.

__repr__()[source]

Return repr(self).

abstract _matches(obj)[source]

Returns True if given object matches the condition.

matches(obj)[source]

Returns True if given object matches the condition.

### samdp module¶

pymor.algorithms.samdp._select_max_eig(H, G, X, V, B, C, which)[source]

Compute poles sorted from largest to smallest residual.

Parameters

H

The NumPy array H from the SAMDP algorithm.

G

The NumPy array G from the SAMDP algorithm.

X

A VectorArray describing the orthogonal search space used in the SAMDP algorithm.

V

A VectorArray describing the orthogonal search space used in the SAMDP algorithm.

B

The VectorArray B from the corresponding LTI system modified by deflation.

C

The VectorArray C from the corresponding LTI system modified by deflation.

which

A string that indicates which poles to select. See samdp.

Returns

poles

A NumPy array containing poles sorted according to the chosen strategy.

rightevs

A NumPy array containing the right eigenvectors of the computed poles.

leftevs

A NumPy array containing the left eigenvectors of the computed poles.

residue

A 1D NumPy array containing the norms of the residues.

pymor.algorithms.samdp._twosided_rqi(A, E, x, y, theta, init_res, imagtol, rqitol, maxiter)[source]

Refine an initial guess for an eigentriplet of the matrix pair (A, E).

Parameters

A

The Operator A from the LTI system.

E

The Operator E from the LTI system.

x

Initial guess for right eigenvector of matrix pair (A, E).

y

Initial guess for left eigenvector of matrix pair (A, E).

theta

Initial guess for eigenvalue of matrix pair (A, E).

init_res

Residual of initial guess.

imagtol

Relative tolerance for imaginary parts of pairs of complex conjugate eigenvalues.

rqitol

Convergence tolerance for the relative change of the pole.

maxiter

Maximum number of iteration.

Returns

x

Refined right eigenvector of matrix pair (A, E).

y

Refined left eigenvector of matrix pair (A, E).

theta

Refined eigenvalue of matrix pair (A, E).

residual

Residual of the computed triplet.

pymor.algorithms.samdp.samdp(A, E, B, C, nwanted, init_shifts=None, which='LR', tol=1e-10, imagtol=1e-06, conjtol=1e-08, dorqitol=0.0001, rqitol=1e-10, maxrestart=100, krestart=20, rqi_maxiter=10, seed=0)[source]

Compute the dominant pole triplets and residues of the transfer function of an LTI system.

This function uses the subspace accelerated dominant pole (SAMDP) algorithm as described in [RM06] in Algorithm 2 in order to compute dominant pole triplets and residues of the transfer function

$H(s) = C (s E - A)^{-1} B$

of an LTI system. It is possible to take advantage of prior knowledge about the poles by specifying shift parameters, which are injected after a new pole has been found.

Parameters

A

The Operator A.

E

The Operator E or None.

B

The operator B as a VectorArray from A.source.

C

The operator C as a VectorArray from A.source.

nwanted

The number of dominant poles that should be computed.

init_shifts

A NumPy array containing shifts which are injected after a new pole has been found.

which

A string specifying the strategy by which the dominant poles and residues are selected. Possible values are:

• 'LR': select poles with largest norm(residual) / abs(Re(pole))

• 'LS': select poles with largest norm(residual) / abs(pole)

• 'LM': select poles with largest norm(residual)

tol

Tolerance for the residual of the poles.

imagtol

Relative tolerance for imaginary parts of pairs of complex conjugate eigenvalues.

conjtol

Tolerance for the residual of the complex conjugate of a pole.

dorqitol

If the residual is smaller than dorqitol the two-sided Rayleigh quotient iteration is executed.

rqitol

Tolerance for the relative change of a pole in the two-sided Rayleigh quotient iteration.

maxrestart

The maximum number of restarts.

krestart

Maximum dimension of search space before performing a restart.

rqi_maxiter

Maximum number of iterations for the two-sided Rayleigh quotient iteration.

seed

Random seed which is used for computing the initial shift and random restarts.

Returns

poles

A 1D NumPy array containing the computed dominant poles.

residues

A 3D NumPy array of shape (len(poles), len(C), len(B)) containing the computed residues.

rightev

A VectorArray containing the right eigenvectors of the computed poles.

leftev

A VectorArray containing the left eigenvectors of the computed poles.

Defaults

which, tol, imagtol, conjtol, dorqitol, rqitol, maxrestart, krestart, init_shifts, rqi_maxiter, seed (see pymor.core.defaults)

### simplify module¶

class pymor.algorithms.simplify.ExpandRules[source]

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 ExpandRules action_ConcatenationOperator, action_LincombOperator, action_recurse, rules BasicObject

pymor.algorithms.simplify.expand(obj)[source]

Expand concatenations of LincombOperators.

To any given Operator or Model, the following transformations are applied recursively:

Parameters

obj

Either a Model or an Operator to which the expansion rules are applied recursively for all children.

Returns

The transformed object.

### svd_va module¶

Module for SVD method of operators represented by VectorArrays.

pymor.algorithms.svd_va.method_of_snapshots(A, product=None, modes=None, rtol=1e-07, atol=0.0, l2_err=0.0)[source]

SVD of a VectorArray using the method of snapshots.

Viewing the VectorArray A as a A.dim x len(A) matrix, the return value of this method is the singular value decomposition of A, where the inner product on R^(dim(A)) is given by product and the inner product on R^(len(A)) is the Euclidean inner product.

Warning

The left singular vectors may not be numerically orthonormal for ill-conditioned A.

Parameters

A

The VectorArray for which the SVD is to be computed.

product

Inner product Operator w.r.t. which the SVD is computed.

modes

If not None, at most the first modes singular values and vectors are returned.

rtol

Singular values smaller than this value multiplied by the largest singular value are ignored.

atol

Singular values smaller than this value are ignored.

l2_err

Do not return more modes than needed to bound the l2-approximation error by this value. I.e. the number of returned modes is at most

argmin_N { sum_{n=N+1}^{infty} s_n^2 <= l2_err^2 }


where s_n denotes the n-th singular value.

Returns

U

VectorArray of left singular vectors.

s

One-dimensional NumPy array of singular values.

Vh

NumPy array of right singular vectors.

Defaults

rtol, atol, l2_err (see pymor.core.defaults)

pymor.algorithms.svd_va.qr_svd(A, product=None, modes=None, rtol=4e-08, atol=0.0, l2_err=0.0)[source]

SVD of a VectorArray using Gram-Schmidt orthogonalization.

Viewing the VectorArray A as a A.dim x len(A) matrix, the return value of this method is the singular value decomposition of A, where the inner product on R^(dim(A)) is given by product and the inner product on R^(len(A)) is the Euclidean inner product.

Parameters

A

The VectorArray for which the SVD is to be computed.

product

Inner product Operator w.r.t. which the left singular vectors are computed.

modes

If not None, at most the first modes singular values and vectors are returned.

rtol

Singular values smaller than this value multiplied by the largest singular value are ignored.

atol

Singular values smaller than this value are ignored.

l2_err

Do not return more modes than needed to bound the l2-approximation error by this value. I.e. the number of returned modes is at most

argmin_N { sum_{n=N+1}^{infty} s_n^2 <= l2_err^2 }


where s_n denotes the n-th singular value.

Returns

U

VectorArray of left singular vectors.

s

One-dimensional NumPy array of singular values.

Vh

NumPy array of right singular vectors.

Defaults

rtol, atol, l2_err (see pymor.core.defaults)

### sylvester module¶

pymor.algorithms.sylvester.solve_sylv_schur(A, Ar, E=None, Er=None, B=None, Br=None, C=None, Cr=None)[source]

Solve Sylvester equation by Schur decomposition.

Solves Sylvester equation

$A V E_r^T + E V A_r^T + B B_r^T = 0$

or

$A^T W E_r + E^T W A_r + C^T C_r = 0$

or both using (generalized) Schur decomposition (Algorithms 3 and 4 in [BKS11]), if the necessary parameters are given.

Parameters

A

Real Operator.

Ar

Real Operator. It is converted into a NumPy array using to_matrix.

E

Real Operator or None (then assumed to be the identity).

Er

Real Operator or None (then assumed to be the identity). It is converted into a NumPy array using to_matrix.

B

Real Operator or None.

Br

Real Operator or None. It is assumed that Br.range.from_numpy is implemented.

C

Real Operator or None.

Cr

Real Operator or None. It is assumed that Cr.source.from_numpy is implemented.

Returns

V

Returned if B and Br are given, VectorArray from A.source.

W

Returned if C and Cr are given, VectorArray from A.source.

Raises

ValueError

If V and W cannot be returned.

### timestepping module¶

This module provides generic time-stepping algorithms for the solution of instationary problems.

The algorithms are generic in the sense that each algorithms operates exclusively on Operators and VectorArrays. In particular, the algorithms can also be used to turn an arbitrary stationary Model provided by an external library into an instationary Model.

Currently, implementations of explicit_euler and implicit_euler time-stepping are provided. The TimeStepper defines a common interface that has to be fulfilled by the time-steppers used by InstationaryModel. The classes ExplicitEulerTimeStepper and ImplicitEulerTimeStepper encapsulate explicit_euler and implicit_euler to provide this interface.

class pymor.algorithms.timestepping.ExplicitEulerTimeStepper(*args, **kwargs)[source]

Explicit Euler time-stepper.

Solves equations of the form

M * d_t u + A(u, mu, t) = F(mu, t).


Parameters

nt

The number of time-steps the time-stepper will perform.

solve(initial_time, end_time, initial_data, operator, rhs=None, mass=None, mu=None, num_values=None)[source]

Apply time-stepper to the equation

M * d_t u + A(u, mu, t) = F(mu, t).


Parameters

initial_time

The time at which to begin time-stepping.

end_time

The time until which to perform time-stepping.

initial_data

The solution vector at initial_time.

operator

The Operator A.

rhs

The right-hand side F (either VectorArray of length 1 or Operator with source.dim == 1). If None, zero right-hand side is assumed.

mass

The Operator M. If None, the identity operator is assumed.

mu

Parameter values for which operator and rhs are evaluated. The current time is added to mu with key t.

num_values

The number of returned vectors of the solution trajectory. If None, each intermediate vector that is calculated is returned.

Returns

VectorArray containing the solution trajectory.

class pymor.algorithms.timestepping.ImplicitEulerTimeStepper(*args, **kwargs)[source]

Implict Euler time-stepper.

Solves equations of the form

M * d_t u + A(u, mu, t) = F(mu, t).


Parameters

nt

The number of time-steps the time-stepper will perform.

solver_options

The solver_options used to invert M + dt*A. The special values 'mass' and 'operator' are recognized, in which case the solver_options of M (resp. A) are used.

solve(initial_time, end_time, initial_data, operator, rhs=None, mass=None, mu=None, num_values=None)[source]

Apply time-stepper to the equation

M * d_t u + A(u, mu, t) = F(mu, t).


Parameters

initial_time

The time at which to begin time-stepping.

end_time

The time until which to perform time-stepping.

initial_data

The solution vector at initial_time.

operator

The Operator A.

rhs

The right-hand side F (either VectorArray of length 1 or Operator with source.dim == 1). If None, zero right-hand side is assumed.

mass

The Operator M. If None, the identity operator is assumed.

mu

Parameter values for which operator and rhs are evaluated. The current time is added to mu with key t.

num_values

The number of returned vectors of the solution trajectory. If None, each intermediate vector that is calculated is returned.

Returns

VectorArray containing the solution trajectory.

class pymor.algorithms.timestepping.TimeStepper(*args, **kwargs)[source]

Interface for time-stepping algorithms.

Algorithms implementing this interface solve time-dependent problems of the form

M * d_t u + A(u, mu, t) = F(mu, t).


Time-steppers used by InstationaryModel have to fulfill this interface.

abstract solve(initial_time, end_time, initial_data, operator, rhs=None, mass=None, mu=None, num_values=None)[source]

Apply time-stepper to the equation

M * d_t u + A(u, mu, t) = F(mu, t).


Parameters

initial_time

The time at which to begin time-stepping.

end_time

The time until which to perform time-stepping.

initial_data

The solution vector at initial_time.

operator

The Operator A.

rhs

The right-hand side F (either VectorArray of length 1 or Operator with source.dim == 1). If None, zero right-hand side is assumed.

mass

The Operator M. If None, the identity operator is assumed.

mu

Parameter values for which operator and rhs are evaluated. The current time is added to mu with key t.

num_values

The number of returned vectors of the solution trajectory. If None, each intermediate vector that is calculated is returned.

Returns

VectorArray containing the solution trajectory.

pymor.algorithms.timestepping.explicit_euler(A, F, U0, t0, t1, nt, mu=None, num_values=None)[source]

pymor.algorithms.timestepping.implicit_euler(A, F, M, U0, t0, t1, nt, mu=None, num_values=None, solver_options='operator')[source]

### to_matrix module¶

class pymor.algorithms.to_matrix.ToMatrixRules(format, mu)[source]

Methods

 RuleTable append_rule, apply, apply_children, breakpoint_for_name, breakpoint_for_obj, get_children, insert_rule, replace_children BasicObject

Attributes

 ToMatrixRules action_AdjointOperator, action_BlockOperator, action_ComponentProjectionOperator, action_ConcatenationOperator, action_IdentityOperator, action_LincombOperator, action_LowRankOperator, action_LowRankUpdatedOperator, action_NumpyMatrixOperator, action_VectorArrayOperator, action_ZeroOperator, rules BasicObject

pymor.algorithms.to_matrix.to_matrix(op, format=None, mu=None)[source]

Convert a linear Operator to a matrix.

Parameters

op

The Operator to convert.

format

Format of the resulting matrix: NumPy array if ‘dense’, otherwise the appropriate SciPy spmatrix. If None, a choice between dense and sparse format is automatically made.

mu

The parameter values for which to convert op.

Returns

res

The matrix equivalent to op.