# pymor.operators.block¶

## Module Contents¶

class pymor.operators.block.BlockColumnOperator(blocks, name=None)[source]

A column vector of arbitrary Operators.

blocked_range = True[source]
blocked_source = False[source]
class pymor.operators.block.BlockDiagonalOperator(blocks, name=None)[source]

Bases: BlockOperator

Block diagonal Operator of arbitrary Operators.

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

Methods

 apply Apply the operator to a VectorArray. apply_adjoint Apply the adjoint operator. apply_inverse Apply the inverse operator. apply_inverse_adjoint Apply the inverse adjoint operator. assemble Assemble the operator for given parameter values.
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 values for which to evaluate the operator.

Returns

VectorArray of the operator evaluations.

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

op.apply_adjoint(V, mu).dot(U) == V.inner(op.apply(U, mu))


Thus, when op is represented by a matrix M, apply_adjoint is given by left-multiplication of (the complex conjugate of) M with V.

Parameters

V

VectorArray of vectors to which the adjoint operator is applied.

mu

The parameter values for which to apply the adjoint operator.

Returns

VectorArray of the adjoint operator evaluations.

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

Apply the inverse operator.

Parameters

V

VectorArray of vectors to which the inverse operator is applied.

mu

The parameter values for which to evaluate the inverse operator.

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.

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.

Parameters

U

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

mu

The parameter values for which to evaluate the inverse adjoint operator.

initial_guess

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

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 adjoint operator evaluations.

Raises

InversionError

The operator could not be inverted.

assemble(mu=None)[source]

Assemble the operator for given parameter values.

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 values for which to assemble the operator.

Returns

Parameter-independent, assembled Operator.

class pymor.operators.block.BlockEmbeddingOperator(block_space, component, name=None)[source]

A column vector of arbitrary Operators.

class pymor.operators.block.BlockOperator(blocks, name=None)[source]

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.

blocked_range = True[source]
blocked_source = True[source]
class pymor.operators.block.BlockOperatorBase(blocks, name=None)[source]

Base block Operator.

Methods

 apply Apply the operator to a VectorArray. apply_adjoint Apply the adjoint operator. as_range_array Return a VectorArray representation of the operator in its range space. as_source_array Return a VectorArray representation of the operator in its source space. assemble Assemble the operator for given parameter values. d_mu Return the operator's derivative with respect to a given parameter. jacobian Return the operator's Jacobian as a new Operator.
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 values for which to evaluate the operator.

Returns

VectorArray of the operator evaluations.

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

op.apply_adjoint(V, mu).dot(U) == V.inner(op.apply(U, mu))


Thus, when op is represented by a matrix M, apply_adjoint is given by left-multiplication of (the complex conjugate of) M with V.

Parameters

V

VectorArray of vectors to which the adjoint operator is applied.

mu

The parameter values for which to apply the adjoint operator.

Returns

VectorArray of the adjoint 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 given parameter values mu a VectorArray V in the operator’s range, such that

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


for all VectorArrays U.

Parameters

mu

The parameter values 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 given parameter values mu a VectorArray V in the operator’s source, such that

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


for all VectorArrays U.

Parameters

mu

The parameter values for which to return the VectorArray representation.

Returns

V

The VectorArray defined above.

assemble(mu=None)[source]

Assemble the operator for given parameter values.

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 values for which to assemble the operator.

Returns

Parameter-independent, assembled Operator.

d_mu(parameter, index=0)[source]

Return the operator’s derivative with respect to a given parameter.

Parameters

parameter

The parameter w.r.t. which to return the derivative.

index

Index of the parameter’s component w.r.t which to return the derivative.

Returns

New Operator representing the partial derivative.

jacobian(U, mu)[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 values for which to compute the Jacobian.

Returns

Linear Operator representing the Jacobian.

class pymor.operators.block.BlockProjectionOperator(block_space, component, name=None)[source]

A row vector of arbitrary Operators.

class pymor.operators.block.BlockRowOperator(blocks, name=None)[source]

A row vector of arbitrary Operators.

blocked_range = False[source]
blocked_source = True[source]
class pymor.operators.block.SecondOrderModelOperator(alpha, beta, A, B, name=None)[source]

Bases: BlockOperator

BlockOperator appearing in SecondOrderModel.to_lti().

This represents a block operator

$\begin{split}\mathcal{A} = \begin{bmatrix} \alpha I & \beta I \\ B & A \end{bmatrix},\end{split}$

which satisfies

$\begin{split}\mathcal{A}^H &= \begin{bmatrix} \overline{\alpha} I & B^H \\ \overline{\beta} I & A^H \end{bmatrix}, \\ \mathcal{A}^{-1} &= \begin{bmatrix} (\alpha A - \beta B)^{-1} A & -\beta (\alpha A - \beta B)^{-1} \\ -(\alpha A - \beta B)^{-1} B & \alpha (\alpha A - \beta B)^{-1} \end{bmatrix}, \\ \mathcal{A}^{-H} &= \begin{bmatrix} A^H (\alpha A - \beta B)^{-H} & -B^H (\alpha A - \beta B)^{-H} \\ -\overline{\beta} (\alpha A - \beta B)^{-H} & \overline{\alpha} (\alpha A - \beta B)^{-H} \end{bmatrix}.\end{split}$

Parameters

alpha

Scalar.

beta

Scalar.

A
B

Methods

 apply Apply the operator to a VectorArray. apply_adjoint Apply the adjoint operator. apply_inverse Apply the inverse operator. apply_inverse_adjoint Apply the inverse adjoint operator. assemble Assemble the operator for given parameter values.
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 values for which to evaluate the operator.

Returns

VectorArray of the operator evaluations.

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

op.apply_adjoint(V, mu).dot(U) == V.inner(op.apply(U, mu))


Thus, when op is represented by a matrix M, apply_adjoint is given by left-multiplication of (the complex conjugate of) M with V.

Parameters

V

VectorArray of vectors to which the adjoint operator is applied.

mu

The parameter values for which to apply the adjoint operator.

Returns

VectorArray of the adjoint operator evaluations.

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

Apply the inverse operator.

Parameters

V

VectorArray of vectors to which the inverse operator is applied.

mu

The parameter values for which to evaluate the inverse operator.

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.

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.

Parameters

U

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

mu

The parameter values for which to evaluate the inverse adjoint operator.

initial_guess

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

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 adjoint operator evaluations.

Raises

InversionError

The operator could not be inverted.

assemble(mu=None)[source]

Assemble the operator for given parameter values.

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 values for which to assemble the operator.

Returns

Parameter-independent, assembled Operator.