pymor.operators package¶

Submodules¶

basic module¶

class pymor.operators.basic.OperatorBase[source]¶

Bases: pymor.operators.interfaces.OperatorInterface

Base class for Operators providing some default implementations.

When implementing a new operator, it is usually advisable to derive from this class.

Methods

Attributes

class pymor.operators.basic.ProjectedOperator(operator, range_basis, source_basis, product=None, solver_options=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Generic Operator representing the projection of an Operator to a subspace.

This operator is implemented as the concatenation of the linear combination with source_basis, application of the original operator and projection onto range_basis. As such, this operator can be used to obtain a reduced basis projection of any given Operator. However, no offline/online decomposition is performed, so this operator is mainly useful for testing before implementing offline/online decomposition for a specific application.

This operator is instantiated in pymor.algorithms.projection.project as a default implementation for parametric or nonlinear operators.

Parameters

operator: The Operator to project.
source_basis: See pymor.algorithms.projection.project.
range_basis: See pymor.algorithms.projection.project.
product: See pymor.algorithms.projection.project.

Methods

Attributes

block module¶

class pymor.operators.block.BlockDiagonalOperator(blocks)[source]¶

Bases: pymor.operators.block.BlockOperator

Block diagonal Operator of arbitrary Operators.

This is a specialization of BlockOperator for the block diagonal case.

Methods

Attributes

class pymor.operators.block.BlockOperator(blocks)[source]¶

Bases: pymor.operators.basic.OperatorBase

A matrix of arbitrary Operators.

This operator can be applied to a compatible BlockVectorArrays.

Parameters

blocks: Two-dimensional array-like where each entry is an Operator or None.

Methods

Attributes

classmethod hstack(operators)[source]¶

Horizontal stacking of Operators.

Parameters

operators: An iterable where each item is an Operator or None.

classmethod vstack(operators)[source]¶

Vertical stacking of Operators.

Parameters

operators: An iterable where each item is an Operator or None.

cg module¶

This module provides some operators for continuous finite element discretizations.

class pymor.operators.cg.AdvectionOperatorP1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear advection Operator for linear finite elements.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ v(x) u(x) ]

The function v is vector-valued. The current implementation works in one and two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
advection_function: The Function v(x) with shape_range = (grid.dim, ). If None, constant one is assumed.
advection_constant: The constant c. If None, c is set to one.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.
name: Name of the operator.

Methods

Attributes

class pymor.operators.cg.AdvectionOperatorQ1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear advection Operator for bilinear finite elements.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ v(x) u(x) ]

The function v has to be vector-valued. The current implementation works in two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
advection_function: The Function v(x) with shape_range = (grid.dim, ). If None, constant one is assumed.
advection_constant: The constant c. If None, c is set to one.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.
name: Name of the operator.

Methods

Attributes

pymor.operators.cg.CGVectorSpace(grid, id_='STATE')[source]¶

class pymor.operators.cg.DiffusionOperatorP1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Diffusion Operator for linear finite elements.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]

The function d can be scalar- or matrix-valued. The current implementation works in one and two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
diffusion_function: The Function d(x) with shape_range == () or shape_range = (grid.dim, grid.dim). If None, constant one is assumed.
diffusion_constant: The constant c. If None, c is set to one.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.
name: Name of the operator.

Methods

Attributes

class pymor.operators.cg.DiffusionOperatorQ1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Diffusion Operator for bilinear finite elements.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]

The function d can be scalar- or matrix-valued. The current implementation works in two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
diffusion_function: The Function d(x) with shape_range == () or shape_range = (grid.dim, grid.dim). If None, constant one is assumed.
diffusion_constant: The constant c. If None, c is set to one.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.
name: Name of the operator.

Methods

Attributes

class pymor.operators.cg.InterpolationOperator(grid, function)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Vector-like Lagrange interpolation Operator for continuous finite element spaces.

Parameters

grid: The Grid on which to interpolate.
function: The Function to interpolate.

Methods

Attributes

class pymor.operators.cg.L2ProductFunctionalP1(grid, function, boundary_info=None, dirichlet_data=None, neumann_data=None, robin_data=None, order=2, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear finite element Functional representing the inner product with an L2-Function.

Boundary treatment can be performed by providing boundary_info and dirichlet_data, in which case the DOFs corresponding to Dirichlet boundaries are set to the values provided by dirichlet_data. Neumann boundaries are handled by providing a neumann_data function, Robin boundaries by providing a robin_data tuple.

The current implementation works in one and two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: Grid for which to assemble the functional.
function: The Function with which to take the inner product.
boundary_info: BoundaryInfo determining the Dirichlet and Neumann boundaries or None. If None, no boundary treatment is performed.
dirichlet_data: Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
neumann_data: Function providing the Neumann boundary values. If None, constant-zero is assumed.
robin_data: Tuple of two Functions providing the Robin parameter and boundary values, see RobinBoundaryOperator. If None, constant-zero for both functions is assumed.
order: Order of the Gauss quadrature to use for numerical integration.
name: The name of the functional.

Methods

Attributes

class pymor.operators.cg.L2ProductFunctionalQ1(grid, function, boundary_info=None, dirichlet_data=None, neumann_data=None, robin_data=None, order=2, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Bilinear finite element Functional representing the inner product with an L2-Function.

Boundary treatment can be performed by providing boundary_info and dirichlet_data, in which case the DOFs corresponding to Dirichlet boundaries are set to the values provided by dirichlet_data. Neumann boundaries are handled by providing a neumann_data function, Robin boundaries by providing a robin_data tuple.

The current implementation works in two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: Grid for which to assemble the functional.
function: The Function with which to take the inner product.
boundary_info: BoundaryInfo determining the Dirichlet boundaries or None. If None, no boundary treatment is performed.
dirichlet_data: Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
neumann_data: Function providing the Neumann boundary values. If None, constant-zero is assumed.
robin_data: Tuple of two Functions providing the Robin parameter and boundary values, see RobinBoundaryOperator. If None, constant-zero for both functions is assumed.
order: Order of the Gauss quadrature to use for numerical integration.
name: The name of the functional.

Methods

Attributes

class pymor.operators.cg.L2ProductP1(grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Operator representing the L2-product between linear finite element functions.

The current implementation works in one and two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the product.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
dirichlet_clear_rows: If True, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if either dirichlet_clear_rows or dirichlet_clear_columns is True, the diagonal entries are set to one.
coefficient_function: Coefficient Function for product with shape_range == (). If None, constant one is assumed.
name: The name of the product.

Methods

Attributes

class pymor.operators.cg.L2ProductQ1(grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Operator representing the L2-product between bilinear finite element functions.

The current implementation works in two dimensions, but can be trivially extended to arbitrary dimensions.

Parameters

grid: The Grid for which to assemble the product.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
dirichlet_clear_rows: If True, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero.
dirichlet_clear_columns: If True, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero.
dirichlet_clear_diag: If True, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if either dirichlet_clear_rows or dirichlet_clear_columns is True, the diagonal entries are set to one.
coefficient_function: Coefficient Function for product with shape_range == (). If None, constant one is assumed.
name: The name of the product.

Methods

Attributes

class pymor.operators.cg.RobinBoundaryOperator(grid, boundary_info, robin_data=None, order=2, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Robin boundary Operator for linear finite elements.

The operator represents the contribution of Robin boundary conditions to the stiffness matrix, where the boundary condition is supposed to be given in the form

-[ d(x) ∇u(x) ] ⋅ n(x) = c(x) (u(x) - g(x))

d and n are the diffusion function (see DiffusionOperatorP1) and the unit outer normal in x, while c is the (scalar) Robin parameter function and g is the (also scalar) Robin boundary value function.

Parameters

grid: The Grid over which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
robin_data: Tuple providing two Functions that represent the Robin parameter and boundary value function. If None, the resulting operator is zero.
name: Name of the operator.

Methods

Attributes

constructions module¶

Module containing some constructions to obtain new operators from old ones.

class pymor.operators.constructions.AdjointOperator(operator, source_product=None, range_product=None, name=None, with_apply_inverse=True, solver_options=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Represents the adjoint of a given linear Operator.

For a linear Operator op the adjoint op^* of op is given by:

(op^*(v), u)_s = (v, op(u))_r,

where ( , )_s and ( , )_r denote the inner products on the source and range space of op. If two products are given by P_s and P_r, then:

op^*(v) = P_s^(-1) o op.T o P_r,

Thus, if ( , )_s and ( , )_r are the Euclidean inner products, op^*v is simply given by applycation of the :attr:transpose <pymor.operators.interface.OperatorInterface.T>` Operator.

Parameters

operator: The Operator of which the adjoint is formed.
source_product: If not None, inner product Operator for the source VectorSpaceInterface w.r.t. which to take the adjoint.
range_product: If not None, inner product Operator for the range VectorSpaceInterface w.r.t. which to take the adjoint.
name: If not None, name of the operator.
with_apply_inverse: If True, provide own apply_inverse and apply_inverse_transpose implementations by calling these methods on the given operator. (Is set to False in the default implementation of and apply_inverse_transpose.)
solver_options: When with_apply_inverse is False, the solver_options to use for the apply_inverse default implementation.

Methods

Attributes

class pymor.operators.constructions.AffineOperator(operator, name=None)[source]¶

Bases: pymor.operators.constructions.ProxyOperator

Decompose an affine Operator into affine_shift and linear_part.

Methods

Attributes

class pymor.operators.constructions.ComponentProjection(components, source, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Operator representing the projection of a VectorArray on some of its components.

Parameters

components: List or 1D NumPy array of the indices of the vector components that ar to be extracted by the operator.
source: Source VectorSpaceInterface of the operator.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.Concatenation(second, first, solver_options=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Operator representing the concatenation of two Operators.

Parameters

second: The Operator which is applied as second operator.
first: The Operator which is applied as first operator.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.ConstantOperator(value, source, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

A constant Operator always returning the same vector.

Parameters

value: A VectorArray of length 1 containing the vector which is returned by the operator.
source: Source VectorSpaceInterface of the operator.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.FixedParameterOperator(operator, mu=None, name=None)[source]¶

Bases: pymor.operators.constructions.ProxyOperator

Makes an Operator Parameter-independent by setting a fixed Parameter.

Parameters

operator: The Operator to wrap.
mu: The fixed Parameter that will be fed to the apply method of operator.

Methods

Attributes

class pymor.operators.constructions.IdentityOperator(space, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

The identity Operator.

In other words:

op.apply(U) == U

Parameters

space: The VectorSpaceInterface the operator acts on.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.InducedNorm(product, raise_negative, tol, name)[source]¶

Bases: pymor.core.interfaces.ImmutableInterface, pymor.parameters.base.Parametric

Instantiated by induced_norm. Do not use directly.

Methods

`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

Attributes

`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

class pymor.operators.constructions.InverseOperator(operator, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Represents the inverse of a given Operator.

Parameters

operator: The Operator of which the inverse is formed.
name: If not None, name of the operator.

Methods

Attributes

class pymor.operators.constructions.InverseTransposeOperator(operator, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Represents the inverse transpose of a given Operator.

Parameters

operator: The Operator of which the inverse transpose is formed.
name: If not None, name of the operator.

Methods

Attributes

class pymor.operators.constructions.LincombOperator(operators, coefficients, solver_options=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Linear combination of arbitrary Operators.

This Operator represents a (possibly Parameter dependent) linear combination of a given list of Operators.

Parameters

operators: List of Operators whose linear combination is formed.
coefficients: A list of linear coefficients. A linear coefficient can either be a fixed number or a ParameterFunctional.
name: Name of the operator.

Methods

Attributes

evaluate_coefficients(mu)[source]¶

Compute the linear coefficients for a given Parameter.

Parameters

mu: Parameter for which to compute the linear coefficients.

Returns

List of linear coefficients.

class pymor.operators.constructions.LinearOperator(operator, name=None)[source]¶

Bases: pymor.operators.constructions.ProxyOperator

Mark the wrapped Operator to be linear.

Methods

Attributes

class pymor.operators.constructions.ProxyOperator(operator, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Forwards all interface calls to given Operator.

Mainly useful as base class for other Operator implementations.

Parameters

operator: The Operator to wrap.
name: Name of the wrapping operator.

Methods

Attributes

class pymor.operators.constructions.SelectionOperator(operators, parameter_functional, boundaries, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

An Operator selected from a list of Operators.

operators[i] is used if parameter_functional(mu) is less or equal than boundaries[i] and greater than boundaries[i-1]:

-infty ------- boundaries[i] ---------- boundaries[i+1] ------- infty
                    |                        |
--- operators[i] ---|---- operators[i+1] ----|---- operators[i+2]
                    |                        |

Parameters

operators: List of Operators from which one Operator is selected based on the given Parameter.
parameter_functional: The ParameterFunctional used for the selection of one Operator.
boundaries: The interval boundaries as defined above.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.VectorArrayOperator(array, transposed=False, space_id=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Wraps a VectorArray as an Operator.

If transposed is False, the operator maps from NumpyVectorSpace(len(array)) to array.space by forming linear combinations of the vectors in the array with given coefficient arrays.

If transposed == True, the operator maps from array.space to NumpyVectorSpace(len(array)) by forming the inner products of the argument with the vectors in the given array.

Parameters

array: The VectorArray which is to be treated as an operator.
transposed: See description above.
name: The name of the operator.

Methods

Attributes

class pymor.operators.constructions.VectorFunctional(vector, product=None, name=None)[source]¶

Bases: pymor.operators.constructions.VectorArrayOperator

Wrap a vector as a linear Functional.

Given a vector v of dimension d, this class represents the functional

f: R^d ----> R^1
    u  |---> (u, v)

where ( , ) denotes the inner product given by product.

In particular, if product is None

VectorFunctional(vector).as_source_array() == vector.

If product is not none, we obtain

VectorFunctional(vector).as_source_array() == product.apply(vector).

Parameters

vector: VectorArray of length 1 containing the vector v.
product: Operator representing the scalar product to use.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.VectorOperator(vector, name=None)[source]¶

Bases: pymor.operators.constructions.VectorArrayOperator

Wrap a vector as a vector-like Operator.

Given a vector v of dimension d, this class represents the operator

op: R^1 ----> R^d
     x  |---> x⋅v

In particular:

VectorOperator(vector).as_range_array() == vector

Parameters

vector: VectorArray of length 1 containing the vector v.
name: Name of the operator.

Methods

Attributes

class pymor.operators.constructions.ZeroOperator(source, range, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

The Operator which maps every vector to zero.

Parameters

source: Source VectorSpaceInterface of the operator.
range: Range VectorSpaceInterface of the operator.
name: Name of the operator.

Methods

Attributes

pymor.operators.constructions.induced_norm(product, raise_negative=True, tol=1e-10, name=None)[source]¶

Obtain induced norm of an inner product.

The norm of the vectors in a VectorArray U is calculated by calling

product.pairwise_apply2(U, U, mu=mu).

In addition, negative norm squares of absolute value smaller than tol are clipped to 0. If raise_negative is True, a ValueError exception is raised if there are negative norm squares of absolute value larger than tol.

Parameters

product: The inner product Operator for which the norm is to be calculated.
raise_negative: If True, raise an exception if calculated norm is negative.
tol: See above.
name: optional, if None product’s name is used

Returns

norm: A function norm(U, mu=None) taking a VectorArray U as input together with the Parameter mu which is passed to the product.

Defaults

raise_negative, tol (see pymor.core.defaults)

ei module¶

class pymor.operators.ei.EmpiricalInterpolatedOperator(operator, interpolation_dofs, collateral_basis, triangular, solver_options=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Interpolate an Operator using Empirical Operator Interpolation.

Let L be an Operator, 0 <= c_1, ..., c_M < L.range.dim indices of interpolation DOFs and let b_1, ..., b_M in R^(L.range.dim) be collateral basis vectors. If moreover ψ_j(U) denotes the j-th component of U, the empirical interpolation L_EI of L w.r.t. the given data is given by

|                M
|   L_EI(U, μ) = ∑ b_i⋅λ_i     such that
|               i=1
|
|   ψ_(c_i)(L_EI(U, μ)) = ψ_(c_i)(L(U, μ))   for i=0,...,M

Since the original operator only has to be evaluated at the given interpolation DOFs, EmpiricalInterpolatedOperator calls restricted to obtain a restricted version of the operator which is used to quickly obtain the required evaluations. If the restricted method, is not implemented, the full operator will be evaluated (which will lead to the same result, but without any speedup).

The interpolation DOFs and the collateral basis can be generated using the algorithms provided in the pymor.algorithms.ei module.

Parameters

operator: The Operator to interpolate.
interpolation_dofs: List or 1D NumPy array of the interpolation DOFs c_1, ..., c_M.
collateral_basis: VectorArray containing the collateral basis b_1, ..., b_M.
triangular: If True, assume that ψ_(c_i)(b_j) = 0 for i < j, which means that the interpolation matrix is triangular.
name: Name of the operator.

Methods

Attributes

class pymor.operators.ei.ProjectedEmpiciralInterpolatedOperator(restricted_operator, interpolation_matrix, source_basis_dofs, projected_collateral_basis, triangular, source_id, solver_options=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

A projected EmpiricalInterpolatedOperator.

Methods

Attributes

fv module¶

This module provides some operators for finite volume discretizations.

class pymor.operators.fv.DiffusionOperator(grid, boundary_info, diffusion_function=None, diffusion_constant=None, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite Volume Diffusion Operator.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]

Parameters

grid: The Grid over which to assemble the operator.
boundary_info: BoundaryInfo for the treatment of Dirichlet boundary conditions.
diffusion_function: The scalar-valued Function d(x). If None, constant one is assumed.
diffusion_constant: The constant c. If None, c is set to one.
name: Name of the operator.

Methods

Attributes

class pymor.operators.fv.EngquistOsherFlux(flux, flux_derivative, gausspoints=5, intervals=1)[source]¶

Bases: pymor.operators.fv.NumericalConvectiveFluxInterface

Engquist-Osher numerical flux.

If f is the analytical flux, and f' its derivative, the Engquist-Osher flux is given by:

F(U_in, U_out, normal, vol) = vol * [c^+(U_in, normal)  +  c^-(U_out, normal)]

                                   U_in
c^+(U_in, normal)  = f(0)⋅normal +  ∫   max(f'(s)⋅normal, 0) ds
                                   s=0

                                  U_out
c^-(U_out, normal) =                ∫   min(f'(s)⋅normal, 0) ds
                                   s=0

Parameters

flux: Function defining the analytical flux f.
flux_derivative: Function defining the analytical flux derivative f'.
gausspoints: Number of Gauss quadrature points to be used for integration.
intervals: Number of subintervals to be used for integration.

Methods

`EngquistOsherFlux`	`evaluate_stage1`, `evaluate_stage2`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

Attributes

`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

pymor.operators.fv.FVVectorSpace(grid, id_='STATE')[source]¶

class pymor.operators.fv.L2Product(grid, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Operator representing the L2-product between finite volume functions.

Parameters

grid: The Grid for which to assemble the product.
name: The name of the product.

Methods

Attributes

class pymor.operators.fv.L2ProductFunctional(grid, function=None, boundary_info=None, dirichlet_data=None, diffusion_function=None, diffusion_constant=None, neumann_data=None, order=1, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite volume Functional representing the inner product with an L2-Function.

Additionally, boundary conditions can be enforced by providing dirichlet_data and neumann_data functions.

Parameters

grid: Grid for which to assemble the functional.
function: The Function with which to take the inner product or None.
boundary_info: BoundaryInfo determining the Dirichlet and Neumann boundaries or None. If None, no boundary treatment is performed.
dirichlet_data: Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
diffusion_function: See DiffusionOperator. Has to be specified in case dirichlet_data is given.
diffusion_constant: See DiffusionOperator. Has to be specified in case dirichlet_data is given.
neumann_data: Function providing the Neumann boundary values. If None, constant-zero is assumed.
order: Order of the Gauss quadrature to use for numerical integration.
name: The name of the functional.

Methods

Attributes

class pymor.operators.fv.LaxFriedrichsFlux(flux, lxf_lambda=1.0)[source]¶

Bases: pymor.operators.fv.NumericalConvectiveFluxInterface

Lax-Friedrichs numerical flux.

If f is the analytical flux, the Lax-Friedrichs flux F is given by:

F(U_in, U_out, normal, vol) = vol * [normal⋅(f(U_in) + f(U_out))/2 + (U_in - U_out)/(2*λ)]

Parameters

flux: Function defining the analytical flux f.
lxf_lambda: The stabilization parameter λ.

Methods

`LaxFriedrichsFlux`	`evaluate_stage1`, `evaluate_stage2`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

Attributes

`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

class pymor.operators.fv.LinearAdvectionLaxFriedrichs(grid, boundary_info, velocity_field, lxf_lambda=1.0, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear advection finite Volume Operator using Lax-Friedrichs flux.

The operator is of the form

L(u, mu)(x) = ∇ ⋅ (v(x, mu)⋅u(x))

See LaxFriedrichsFlux for the definition of the Lax-Friedrichs flux.

Parameters

grid: Grid over which to assemble the operator.
boundary_info: BoundaryInfo determining the Dirichlet and Neumann boundaries.
velocity_field: Function defining the velocity field v.
lxf_lambda: The stabilization parameter λ.
name: The name of the operator.

Methods

Attributes

class pymor.operators.fv.NonlinearAdvectionOperator(grid, boundary_info, numerical_flux, dirichlet_data=None, solver_options=None, space_id='STATE', name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Nonlinear finite volume advection Operator.

The operator is of the form

L(u, mu)(x) = ∇ ⋅ f(u(x), mu)

Parameters

grid: Grid for which to evaluate the operator.
boundary_info: BoundaryInfo determining the Dirichlet and Neumann boundaries.
numerical_flux: The NumericalConvectiveFlux to use.
dirichlet_data: Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
name: The name of the operator.

Methods

Attributes

class pymor.operators.fv.NonlinearReactionOperator(grid, reaction_function, reaction_function_derivative=None, space_id='STATE', name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Methods

Attributes

class pymor.operators.fv.NumericalConvectiveFluxInterface[source]¶

Bases: pymor.core.interfaces.ImmutableInterface, pymor.parameters.base.Parametric

Interface for numerical convective fluxes for finite volume schemes.

Numerical fluxes defined by this interfaces are functions of the form F(U_inner, U_outer, unit_outer_normal, edge_volume, mu).

The flux evaluation is vectorized and happens in two stages:

evaluate_stage1 receives a NumPy array U of all values which appear as U_inner or U_outer for all edges the flux shall be evaluated at and returns a tuple of NumPy arrays each of the same length as U.
evaluate_stage2 receives the reordered stage1_data for each edge as well as the unit outer normal and the volume of the edges.

stage1_data is given as follows: If R_l is l-th entry of the tuple returned by evaluate_stage1, the l-th entry D_l of of the stage1_data tuple has the shape (num_edges, 2) + R_l.shape[1:]. If for edge k the values U_inner and U_outer are the i-th and j-th value in the U array provided to evaluate_stage1, we have
```
D_l[k, 0] == R_l[i],    D_l[k, 1] == R_l[j].
```
evaluate_stage2 returns a NumPy array of the flux evaluations for each edge.

Methods

`NumericalConvectiveFluxInterface`	`evaluate_stage1`, `evaluate_stage2`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

Attributes

`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

class pymor.operators.fv.ReactionOperator(grid, reaction_coefficient, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite Volume reaction Operator.

The operator is of the form

L(u, mu)(x) = c(x, mu)⋅u(x)

Parameters

grid: The Grid for which to assemble the operator.
reaction_coefficient: The function ‘c’
name: The name of the operator.

Methods

Attributes

class pymor.operators.fv.SimplifiedEngquistOsherFlux(flux, flux_derivative)[source]¶

Bases: pymor.operators.fv.NumericalConvectiveFluxInterface

Engquist-Osher numerical flux. Simplified Implementation for special case.

For the definition of the Engquist-Osher flux see EngquistOsherFlux. This class provides a faster and more accurate implementation for the special case that f(0) == 0 and the derivative of f only changes sign at 0.

Parameters

flux: Function defining the analytical flux f.
flux_derivative: Function defining the analytical flux derivative f'.

Methods

`SimplifiedEngquistOsherFlux`	`evaluate_stage1`, `evaluate_stage2`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

Attributes

`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

pymor.operators.fv.jacobian_options(delta=1e-07)[source]¶

pymor.operators.fv.nonlinear_advection_engquist_osher_operator(grid, boundary_info, flux, flux_derivative, gausspoints=5, intervals=1, dirichlet_data=None, solver_options=None, name=None)[source]¶: Instantiate a NonlinearAdvectionOperator using EngquistOsherFlux.

pymor.operators.fv.nonlinear_advection_lax_friedrichs_operator(grid, boundary_info, flux, lxf_lambda=1.0, dirichlet_data=None, solver_options=None, name=None)[source]¶: Instantiate a NonlinearAdvectionOperator using LaxFriedrichsFlux.

pymor.operators.fv.nonlinear_advection_simplified_engquist_osher_operator(grid, boundary_info, flux, flux_derivative, dirichlet_data=None, solver_options=None, name=None)[source]¶: Instantiate a NonlinearAdvectionOperator using SimplifiedEngquistOsherFlux.

interfaces module¶

class pymor.operators.interfaces.OperatorInterface[source]¶

Bases: pymor.core.interfaces.ImmutableInterface, pymor.parameters.base.Parametric

Interface for Parameter dependent discrete operators.

An operator in pyMOR is simply a mapping which for any given Parameter maps vectors from its source VectorSpaceInterface to vectors in its range VectorSpaceInterface.

Note that there is no special distinction between functionals and operators in pyMOR. A functional is simply an operator with NumpyVectorSpace (1) as its range VectorSpaceInterface.

Methods

Attributes

solver_options¶

If not None, a dict which can contain the following keys:

‘inverse’:	solver options used for `apply_inverse`
‘inverse_transpose’:	solver options used for `apply_inverse_transpose`
‘jacobian’:	solver options for the operators returned by `jacobian` (has no effect for linear operators)

If solver_options is None or a dict entry is missing or None, default options are used. The interpretation of the given solver options is up to the operator at hand. In general, values in solver_options should either be strings (indicating a solver type) or dicts of options, usually with an entry 'type' which specifies the solver type to use and further items which configure this solver.

linear¶: True if the operator is linear.

source¶: The source VectorSpaceInterface.

range¶: The range VectorSpaceInterface.

T¶

The transpose operator, i.e.

self.T.apply(V, mu) == self.apply_transpose(V, mu)

for all V, mu.

__add__(other)[source]¶: Sum of two operators.

__mul__(other)[source]¶: Product of operator by a scalar.

__radd__(other)[source]¶: Sum of two operators.

apply(U, mu=None)[source]¶

Apply the operator to a VectorArray.

Parameters

U: VectorArray of vectors to which the operator is applied.
mu: The Parameter for which to evaluate the operator.

Returns

VectorArray of the operator evaluations.

apply2(V, U, mu=None)[source]¶

Treat the operator as a 2-form by computing V.dot(self.apply(U)).

If the operator is a linear operator given by multiplication with a matrix M, then apply2 is given as:

op.apply2(V, U) = V^T*M*U.

Parameters

V: VectorArray of the left arguments V.
U: VectorArray of the right right arguments U.
mu: The Parameter for which to evaluate the operator.

Returns

A NumPy array with shape (len(V), len(U)) containing the 2-form evaluations.

apply_inverse(V, mu=None, least_squares=False)[source]¶

Apply the inverse operator.

Parameters

V

VectorArray of vectors to which the inverse operator is applied.

mu

The Parameter for which to evaluate the inverse operator.

least_squares

If True, solve the least squares problem:

u = argmin ||op(u) - v||_2.

Since for an invertible operator the least squares solution agrees with the result of the application of the inverse operator, setting this option should, in general, have no effect on the result for those operators. However, note that when no appropriate solver_options are set for the operator, most implementations will choose a least squares solver by default which may be undesirable.

Returns

VectorArray of the inverse operator evaluations.

Raises

InversionError: The operator could not be inverted.

apply_inverse_transpose(U, mu=None, least_squares=False)[source]¶

Apply the inverse transpose operator.

Parameters

U

VectorArray of vectors to which the inverse transpose operator is applied.

mu

The Parameter for which to evaluate the inverse transpose operator.

least_squares

If True, solve the least squares problem:

v = argmin ||op*(v) - u||_2.

Since for an invertible operator the least squares solution agrees with the result of the application of the inverse operator, setting this option should, in general, have no effect on the result for those operators. However, note that when no appropriate solver_options are set for the operator, most operator implementations will choose a least squares solver by default which may be undesirable.

Returns

VectorArray of the inverse transpose operator evaluations.

Raises

InversionError: The operator could not be inverted.

apply_transpose(V, mu=None)[source]¶

Apply the transpose operator.

For any given linear Operator op, Parameter mu and VectorArrays U, V in the source resp. range we have:

op.apply_transpose(V, mu).dot(U) == V.dot(op.apply(U, mu))

Thus, when op is represented by a matrix M, apply_transpose is given by left-multplication of M with V.

Parameters

V: VectorArray of vectors to which the transpose operator is applied.
mu: The Parameter for which to apply the transpose operator.

Returns

VectorArray of the transpose operator evaluations.

as_range_array(mu=None)[source]¶

Return a VectorArray representation of the operator in its range space.

In the case of a linear operator with NumpyVectorSpace as source, this method returns for every Parameter mu a VectorArray V in the operator’s range, such that

V.lincomb(U.data) == self.apply(U, mu)

for all VectorArrays U.

Parameters

mu: The Parameter for which to return the VectorArray representation.

Returns

V: The VectorArray defined above.

as_source_array(mu=None)[source]¶

Return a VectorArray representation of the operator in its source space.

In the case of a linear operator with NumpyVectorSpace as range, this method returns for every Parameter mu a VectorArray V in the operator’s source, such that

self.range.make_array(V.dot(U)) == self.apply(U, mu)

for all VectorArrays U.

Parameters

mu: The Parameter for which to return the VectorArray representation.

Returns

V: The VectorArray defined above.

as_vector(mu=None, *, space=None)[source]¶

Return a vector representation of a linear functional or vector operator.

Depending on the operator’s source and range, this method is equivalent to calling as_range_array or as_source_array respectively. The resulting VectorArray is required to have length 1.

Note that in case both source and range are one-dimensional NumpyVectorSpaces but with different ids, it is impossible to determine which space to choose. In this case, the desired space has to be specified via the space parameter.

Parameters

mu: The Parameter for which to return the vector representation.
space: See above.

Returns

V: VectorArray of length 1 containing the vector representation.

assemble(mu=None)[source]¶

Assemble the operator for a given parameter.

The result of the method strongly depends on the given operator. For instance, a matrix-based operator will assemble its matrix, a LincombOperator will try to form the linear combination of its operators, whereas an arbitrary operator might simply return a FixedParameterOperator. The only assured property of the assembled operator is that it no longer depends on a Parameter.

Parameters

mu: The Parameter for which to assemble the operator.

Returns

Parameter-independent, assembled Operator.

assemble_lincomb(operators, coefficients, solver_options=None, name=None)[source]¶

Try to assemble a linear combination of the given operators.

This method is called in the assemble method of LincombOperator on the first of its operator. If an assembly of the given linear combination is possible, e.g. the linear combination of the system matrices of the operators can be formed, then the assembled operator is returned. Otherwise, the method returns None to indicate that assembly is not possible.

Parameters

operators: List of Operators whose linear combination is formed.
coefficients: List of the corresponding linear coefficients.
solver_options: solver_options for the assembled operator.
name: Name of the assembled operator.

Returns

The assembled Operator if assembly is possible, otherwise None.

jacobian(U, mu=None)[source]¶

Return the operator’s Jacobian as a new Operator.

Parameters

U: Length 1 VectorArray containing the vector for which to compute the Jacobian.
mu: The Parameter for which to compute the Jacobian.

Returns

Linear Operator representing the Jacobian.

pairwise_apply2(V, U, mu=None)[source]¶

Treat the operator as a 2-form by computing V.dot(self.apply(U)).

Same as OperatorInterface.apply2, except that vectors from V and U are applied in pairs.

Parameters

V: VectorArray of the left arguments V.
U: VectorArray of the right right arguments U.
mu: The Parameter for which to evaluate the operator.

Returns

A NumPy array with shape (len(V),) == (len(U),) containing the 2-form evaluations.

restricted(dofs)[source]¶

Restrict the operator range to a given set of degrees of freedom.

This method returns a restricted version restricted_op of the operator along with an array source_dofs such that for any VectorArray U in self.source the following is true:

self.apply(U, mu).components(dofs)
    == restricted_op.apply(NumpyVectorArray(U.components(source_dofs)), mu))

Such an operator is mainly useful for empirical interpolation where the evaluation of the original operator only needs to be known for few selected degrees of freedom. If the operator has a small stencil, only few source_dofs will be needed to evaluate the restricted operator which can make its evaluation very fast compared to evaluating the original operator.

Parameters

dofs: One-dimensional NumPy array of degrees of freedom in the operator range to which to restrict.

Returns

restricted_op: The restricted operator as defined above. The operator will have NumpyVectorSpace (len(source_dofs)) as source and NumpyVectorSpace (len(dofs)) as range.
source_dofs: One-dimensional NumPy array of source degrees of freedom as defined above.

mpi module¶

class pymor.operators.mpi.MPIOperator(obj_id, with_apply2=False, pickle_local_spaces=True, space_type=<class 'pymor.vectorarrays.mpi.MPIVectorSpace'>)[source]¶

Bases: pymor.operators.basic.OperatorBase

MPI distributed Operator.

Given a single-rank implementation of an Operator, this wrapper class uses the event loop from pymor.tools.mpi to allow an MPI distributed usage of the Operator.

Instances of MPIOperator can be used on rank 0 like any other (non-distributed) Operator.

Note, however, that the underlying Operator implementation needs to be MPI aware. For instance, the operator’s apply method has to perform the necessary MPI communication to obtain all DOFs hosted on other MPI ranks which are required for the local operator evaluation.

Instead of instantiating MPIOperator directly, it is usually preferable to use mpi_wrap_operator instead.

Parameters

obj_id: ObjectId of the local Operators on each rank.
with_apply2: Set to True if the operator implementation has its own MPI aware implementation of apply2 and pairwise_apply2. Otherwise, the default implementations using apply and dot will be used.
pickle_local_spaces: If pickle_local_spaces is False, a unique identifier is computed for each local source/range VectorSpaceInterface, which is then transferred to rank 0 instead of the true VectorSpaceInterface. This allows the useage of MPIVectorArray even when the local VectorSpaces are not picklable.
space_type: This class will be used to wrap the local VectorArrays returned by the local operators into an MPI distributed VectorArray managed from rank 0. By default, MPIVectorSpace will be used, other options are MPIVectorSpaceAutoComm and MPIVectorSpaceNoComm.

Methods

Attributes

pymor.operators.mpi.mpi_wrap_operator(obj_id, with_apply2=False, pickle_local_spaces=True, space_type=<class 'pymor.vectorarrays.mpi.MPIVectorSpace'>)[source]¶

Wrap MPI distributed local Operators to a global Operator on rank 0.

Given MPI distributed local Operators referred to by the ObjectId obj_id, return a new Operator which manages these distributed operators from rank 0. This is done by instantiating MPIOperator. Additionally, the structure of the wrapped operators is preserved. E.g. LincombOperators will be wrapped as a LincombOperator of MPIOperators.

Parameters

See : class:MPIOperator.

Returns

The wrapped Operator.

numpy module¶

This module provides the following NumPy based Operators:

NumpyMatrixOperator wraps a 2D NumPy array as an Operator.
NumpyMatrixBasedOperator should be used as base class for all Operators which assemble into a NumpyMatrixOperator.
NumpyGenericOperator wraps an arbitrary Python function between NumPy arrays as an Operator.

class pymor.operators.numpy.NumpyGenericOperator(mapping, transpose_mapping=None, dim_source=1, dim_range=1, linear=False, parameter_type=None, source_id=None, range_id=None, solver_options=None, name=None)[source]¶

Bases: pymor.operators.basic.OperatorBase

Wraps an arbitrary Python function between NumPy arrays as a an Operator.

Parameters

mapping

The function to wrap. If parameter_type is None, the function is of the form mapping(U) and is expected to be vectorized. In particular:

mapping(U).shape == U.shape[:-1] + (dim_range,).

If parameter_type is not None, the function has to have the signature mapping(U, mu).

transpose_mapping

The transpsoe function to wrap. If parameter_type is None, the function is of the form transpose_mapping(U) and is expected to be vectorized. In particular:

transpose_mapping(U).shape == U.shape[:-1] + (dim_source,).

If parameter_type is not None, the function has to have the signature transpose_mapping(U, mu).

dim_source

Dimension of the operator’s source.

dim_range

Dimension of the operator’s range.

linear

Set to True if the provided mapping and transpose_mapping are linear.

parameter_type

The ParameterType of the Parameters the mapping accepts.

name

Name of the operator.

Methods

Attributes

class pymor.operators.numpy.NumpyMatrixBasedOperator[source]¶

Bases: pymor.operators.basic.OperatorBase

Base class for operators which assemble into a NumpyMatrixOperator.

Methods

Attributes

sparse¶: True if the operator assembles into a sparse matrix, False if the operator assembles into a dense matrix, None if unknown.

assemble(mu=None)[source]¶

Assembles the operator for a given Parameter.

Parameters

mu: The Parameter for which to assemble the operator.

Returns

The assembled parameter independent Operator.

export_matrix(filename, matrix_name=None, output_format='matlab', mu=None)[source]¶

Save the matrix of the operator to a file.

Parameters

filename: Name of output file.
matrix_name: The name, the output matrix is given. (Comment field is used in case of Matrix Market output_format.) If None, the Operator‘s name is used.
output_format: Output file format. Either matlab or matrixmarket.
mu: The Parameter to assemble the to be exported matrix for.

class pymor.operators.numpy.NumpyMatrixOperator(matrix, source_id=None, range_id=None, solver_options=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Wraps a 2D NumPy array as an Operator.

Parameters

matrix: The NumPy array which is to be wrapped.
name: Name of the operator.

Methods

Attributes

apply_inverse(V, mu=None, least_squares=False, check_finite=True, default_sparse_solver_backend='scipy')[source]¶

Apply the inverse operator.

Parameters

V

VectorArray of vectors to which the inverse operator is applied.

mu

The Parameter for which to evaluate the inverse operator.

least_squares

If True, solve the least squares problem:

u = argmin ||op(u) - v||_2.

Since for an invertible operator the least squares solution agrees with the result of the application of the inverse operator, setting this option should, in general, have no effect on the result for those operators. However, note that when no appropriate solver_options are set for the operator, most implementations will choose a least squares solver by default which may be undesirable.

check_finite

Test if solution only containes finite values.

default_solver

Default sparse solver backend to use (scipy, pyamg, generic).

Returns

VectorArray of the inverse operator evaluations.

Raises

InversionError: The operator could not be inverted.

Defaults

check_finite, default_sparse_solver_backend (see pymor.core.defaults)

pymor.operators package¶

Submodules¶

basic module¶

block module¶

cg module¶

constructions module¶

ei module¶

fv module¶

interfaces module¶

mpi module¶

numpy module¶

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`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`apply`, `as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`OperatorInterface`	`linear`, `range`, `solver_options`, `source`, `T`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`ProjectedOperator`	`apply`, `assemble`, `jacobian`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `as_range_array`, `as_source_array`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`ProjectedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`BlockDiagonalOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `assemble`, `assemble_lincomb`
`BlockOperator`	`hstack`, `vstack`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`BlockOperator`	`apply`, `apply_transpose`, `assemble`, `assemble_lincomb`, `hstack`, `vstack`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `as_range_array`, `as_source_array`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`NumpyMatrixBasedOperator`	`apply`, `apply_inverse`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`, `export_matrix`
`OperatorBase`	`apply2`, `apply_inverse_transpose`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`AdvectionOperatorP1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`AdvectionOperatorQ1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`DiffusionOperatorP1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`DiffusionOperatorQ1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`InterpolationOperator`	`linear`, `source`
`NumpyMatrixBasedOperator`	`sparse`, `T`
`OperatorInterface`	`range`, `solver_options`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`L2ProductFunctionalP1`	`range`, `sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`L2ProductFunctionalQ1`	`range`, `sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`L2ProductP1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`L2ProductQ1`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`RobinBoundaryOperator`	`sparse`
`NumpyMatrixBasedOperator`	`linear`, `T`
`OperatorInterface`	`range`, `solver_options`, `source`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`AdjointOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`AffineOperator`	`jacobian`
`ProxyOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `restricted`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `assemble`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`ComponentProjection`	`apply`, `restricted`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`ComponentProjection`	`linear`
`OperatorInterface`	`range`, `solver_options`, `source`, `T`
`ImmutableInterface`	`add_with_arguments`, `sid`, `sid_ignore`, `with_arguments`
`BasicInterface`	`logger`, `logging_disabled`, `name`, `uid`
`Parametric`	`parameter_space`, `parameter_type`, `parametric`

`Concatenation`	`apply`, `apply_transpose`, `jacobian`, `restricted`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `as_range_array`, `as_source_array`, `assemble`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`ConstantOperator`	`apply`, `jacobian`, `restricted`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`FixedParameterOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `jacobian`
`ProxyOperator`	`restricted`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `assemble`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`IdentityOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`, `assemble`, `assemble_lincomb`, `restricted`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`InverseOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`InverseTransposeOperator`	`apply`, `apply_inverse`, `apply_inverse_transpose`, `apply_transpose`
`OperatorBase`	`apply2`, `as_range_array`, `as_source_array`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`LincombOperator`	`apply`, `apply2`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`, `evaluate_coefficients`, `jacobian`, `pairwise_apply2`
`OperatorBase`	`apply_inverse`, `apply_inverse_transpose`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`SelectionOperator`	`apply`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble`
`OperatorBase`	`apply2`, `apply_inverse`, `apply_inverse_transpose`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`, `assemble_lincomb`, `restricted`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`

`VectorArrayOperator`	`apply`, `apply_inverse_transpose`, `apply_transpose`, `as_range_array`, `as_source_array`, `assemble_lincomb`, `restricted`
`OperatorBase`	`apply2`, `apply_inverse`, `assemble`, `jacobian`, `pairwise_apply2`
`OperatorInterface`	`as_vector`
`ImmutableInterface`	`generate_sid`, `with_`, `__setattr__`
`BasicInterface`	`disable_logging`, `enable_logging`, `has_interface_name`, `implementor_names`, `implementors`
`Parametric`	`build_parameter_type`, `parse_parameter`, `strip_parameter`