pymor.operators.constructions

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

Module Contents

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

Bases: pymor.operators.interface.Operator

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.H o P_r,

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

Parameters

operator

The Operator of which the adjoint is formed.

source_product

If not None, inner product Operator for the source VectorSpace w.r.t. which to take the adjoint.

range_product

If not None, inner product Operator for the range VectorSpace 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_adjoint implementations by calling these methods on the given operator. (Is set to False in the default implementation of and apply_inverse_adjoint.)

solver_options

When with_apply_inverse is False, the solver_options to use for the apply_inverse default implementation.

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.

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

class pymor.operators.constructions.AffineOperator(operator, name=None)

Bases: ProxyOperator

Decompose an affine Operator into affine_shift and linear_part.

Methods

jacobian

Return the operator's Jacobian as a new Operator.

jacobian(U, mu=None)

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.constructions.ComponentProjectionOperator(components, source, range_id=None, name=None)

Bases: pymor.operators.interface.Operator

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

Parameters

components

List or 1D NumPy array of the indices of the vector dofs that are to be extracted by the operator.

source

Source VectorSpace of the operator.

name

Name of the operator.

Methods

apply	Apply the operator to a VectorArray.
restricted	Restrict the operator range to a given set of degrees of freedom.

linear = True

apply(U, mu=None)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

class pymor.operators.constructions.ConcatenationOperator(operators, solver_options=None, name=None)

Bases: pymor.operators.interface.Operator

Operator representing the concatenation of two Operators.

Parameters

operators

Tuple of Operators to concatenate. operators[-1] is the first applied operator, operators[0] is the last applied operator.

solver_options

The solver_options for the operator.

name

Name of the operator.

Methods

apply	Apply the operator to a VectorArray.
apply_adjoint	Apply the adjoint operator.
d_mu	Return the operator's derivative with respect to a given parameter.
jacobian	Return the operator's Jacobian as a new Operator.
restricted	Restrict the operator range to a given set of degrees of freedom.

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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.

d_mu(parameter, index=0)

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=None)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

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

Bases: pymor.operators.interface.Operator

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 VectorSpace of the operator.

name

Name of the operator.

Methods

apply	Apply the operator to a VectorArray.
apply_inverse	Apply the inverse operator.
jacobian	Return the operator's Jacobian as a new Operator.
restricted	Restrict the operator range to a given set of degrees of freedom.

linear = False

apply(U, mu=None)

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.

apply_inverse(V, mu=None, initial_guess=None, least_squares=False)

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.

jacobian(U, mu=None)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

class pymor.operators.constructions.FixedParameterOperator(operator, mu=None, name=None)

Bases: ProxyOperator

Makes an Operator Parameter-independent by setting fixed parameter values.

Parameters

operator

The Operator to wrap.

mu

The fixed parameter values that will be fed to the apply method (and related methods) of operator.

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.
jacobian	Return the operator's Jacobian as a new Operator.

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

jacobian(U, mu=None)

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.constructions.IdentityOperator(space, name=None)

Bases: pymor.operators.interface.Operator

The identity Operator.

In other words:

op.apply(U) == U

Parameters

space

The VectorSpace the operator acts on.

name

Name of the operator.

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

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

class pymor.operators.constructions.InducedNorm(product, raise_negative, tol, name)

Bases: pymor.parameters.base.ParametricObject

Instantiated by induced_norm. Do not use directly.

class pymor.operators.constructions.InverseAdjointOperator(operator, name=None)

Bases: pymor.operators.interface.Operator

Represents the inverse adjoint of a given Operator.

Parameters

operator

The Operator of which the inverse adjoint is formed.

name

If not None, name of the operator.

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.

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

class pymor.operators.constructions.InverseOperator(operator, name=None)

Bases: pymor.operators.interface.Operator

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

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.

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

class pymor.operators.constructions.LincombOperator(operators, coefficients, solver_options=None, name=None)

Bases: pymor.operators.interface.Operator

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.

solver_options

The solver_options for the operator.

name

Name of the operator.

Methods

apply	Apply the operator to a VectorArray.
apply2	Treat the operator as a 2-form and apply it to V and U.
apply_adjoint	Apply the adjoint operator.
apply_inverse	Apply the inverse operator.
apply_inverse_adjoint	Apply the inverse 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.
evaluate_coefficients	Compute the linear coefficients for given parameter values.
jacobian	Return the operator's Jacobian as a new Operator.
pairwise_apply2	Treat the operator as a 2-form and apply it to V and U in pairs.

apply(U, mu=None)

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.

apply2(V, U, mu=None)

Treat the operator as a 2-form and apply it to V and U.

This method is usually implemented as V.inner(self.apply(U)). In particular, 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.

In the case of complex numbers, note that apply2 is anti-linear in the first variable by definition of inner.

Parameters

V

VectorArray of the left arguments V.

U

VectorArray of the right arguments U.

mu

The parameter values for which to evaluate the operator.

Returns

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

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

as_range_array(mu=None)

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)

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)

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)

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.

evaluate_coefficients(mu)

Compute the linear coefficients for given parameter values.

Parameters

mu

Parameter values for which to compute the linear coefficients.

Returns

List of linear coefficients.

jacobian(U, mu=None)

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.

pairwise_apply2(V, U, mu=None)

Treat the operator as a 2-form and apply it to V and U in pairs.

This method is usually implemented as V.pairwise_inner(self.apply(U)). In particular, if the operator is a linear operator given by multiplication with a matrix M, then apply2 is given as:

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

In the case of complex numbers, note that pairwise_apply2 is anti-linear in the first variable by definition of pairwise_inner.

Parameters

V

VectorArray of the left arguments V.

U

VectorArray of the right arguments U.

mu

The parameter values for which to evaluate the operator.

Returns

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

class pymor.operators.constructions.LinearInputOperator(B)

Bases: pymor.operators.interface.Operator

Operator representing the product B(mu)*input(t).

Parameters

B

The input Operator.

Methods

apply	Apply the operator to a VectorArray.
as_range_array	Return a VectorArray representation of the operator in its range space.

linear = True

apply(U, mu=None)

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.

as_range_array(mu=None)

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.

class pymor.operators.constructions.LinearOperator(operator, name=None)

Bases: ProxyOperator

Mark the wrapped Operator to be linear.

class pymor.operators.constructions.LowRankOperator(left, core, right, inverted=False, solver_options=None, name=None)

Bases: pymor.operators.interface.Operator

Non-parametric low-rank operator.

Represents an operator of the form \(L C R^H\) or \(L C^{-1} R^H\) where \(L\) and \(R\) are VectorArrays of column vectors and \(C\) a 2D NumPy array.

Parameters

left

VectorArray representing \(L\).

core

NumPy array representing \(C\).

right

VectorArray representing \(R\).

inverted

Whether \(C\) is inverted.

solver_options

The solver_options for the operator.

name

Name of the operator.

Methods

apply	Apply the operator to a VectorArray.
apply_adjoint	Apply the adjoint operator.

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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.

class pymor.operators.constructions.LowRankUpdatedOperator(operator, lr_operator, coeff, lr_coeff, solver_options=None, name=None)

Bases: LincombOperator

Operator plus LowRankOperator.

Represents a linear combination of an Operator and LowRankOperator. Uses the Sherman-Morrison-Woodbury formula in apply_inverse and apply_inverse_adjoint:

\[\begin{split}\left(\alpha A + \beta L C R^H \right)^{-1} & = \alpha^{-1} A^{-1} - \alpha^{-1} \beta A^{-1} L C \left(\alpha C + \beta C R^H A^{-1} L C \right)^{-1} C R^H A^{-1}, \\ \left(\alpha A + \beta L C^{-1} R^H \right)^{-1} & = \alpha^{-1} A^{-1} - \alpha^{-1} \beta A^{-1} L \left(\alpha C + \beta R^H A^{-1} L \right)^{-1} R^H A^{-1}.\end{split}\]

Parameters

operator

Operator.

lr_operator

LowRankOperator.

coeff

A linear coefficient for operator. Can either be a fixed number or a ParameterFunctional.

lr_coeff

A linear coefficient for lr_operator. Can either be a fixed number or a ParameterFunctional.

solver_options

The solver_options for the operator.

name

Name of the operator.

Methods

apply_inverse	Apply the inverse operator.
apply_inverse_adjoint	Apply the inverse adjoint operator.

apply_inverse(V, mu=None, initial_guess=None, least_squares=False)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

class pymor.operators.constructions.NumpyConversionOperator(space, direction='to_numpy')

Bases: pymor.operators.interface.Operator

Converts VectorArrays to NumpyVectorArrays.

Note that the input VectorArrays need to support to_numpy. For the adjoint, from_numpy needs to be implemented.

Parameters

space

The VectorSpace of the VectorArrays that are converted to NumpyVectorArrays.

direction

Either 'to_numpy' or 'from_numpy'. In case of 'to_numpy' apply takes a VectorArray from space and returns a NumpyVectorArray. In case of 'from_numpy', apply takes a NumpyVectorArray and returns a VectorArray from space.

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.

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

class pymor.operators.constructions.ProjectedOperator(operator, range_basis, source_basis, product=None, solver_options=None, name=None)

Bases: pymor.operators.interface.Operator

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.

range_basis

See pymor.algorithms.projection.project.

source_basis

See pymor.algorithms.projection.project.

product

See pymor.algorithms.projection.project.

solver_options

The solver_options for the projected operator.

Methods

apply	Apply the operator to a VectorArray.
apply_adjoint	Apply the adjoint operator.
assemble	Assemble the operator for given parameter values.
jacobian	Return the operator's Jacobian as a new Operator.

linear = False

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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.

assemble(mu=None)

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.

jacobian(U, mu=None)

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.constructions.ProxyOperator(operator, name=None)

Bases: pymor.operators.interface.Operator

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

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.
jacobian	Return the operator's Jacobian as a new Operator.
restricted	Restrict the operator range to a given set of degrees of freedom.

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

jacobian(U, mu=None)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

class pymor.operators.constructions.QuadraticFunctional(operator, name=None)

Bases: pymor.operators.interface.Operator

An Operator representing a quadratic functional.

Given an operator A acting on R^d, this class represents the functional:

op: R^d ----> R^1
     u  |---> (A(u), u).

Parameters

operator

The Operator defining the quadratic functional.

Methods

apply	Apply the operator to a VectorArray.
d_mu	Return the operator's derivative with respect to a given parameter.
jacobian	Return the operator's Jacobian as a new Operator.

linear = False

range

apply(U, mu=None)

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.

d_mu(parameter, index=1)

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=None)

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.constructions.QuadraticProductFunctional(left, right, product=None, name=None)

Bases: QuadraticFunctional

An Operator representing a quadratic functional by multiplying two linear functionals.

Given linear operators A, B: R^d —-> R^n, this class represents the functional:

op: R^d ----> R^1
     u  |---> (A(u), B(u)).

Parameters

left

The Operator that defines the left operator of the quadratic functional.

right

The Operator that defines the right operator of the quadratic functional.

product

The Operator that defines the inner product.

linear = False

range

class pymor.operators.constructions.SelectionOperator(operators, parameter_functional, boundaries, name=None)

Bases: pymor.operators.interface.Operator

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 values.

parameter_functional

The ParameterFunctional used for the selection of one Operator.

boundaries

The interval boundaries as defined above.

name

Name of the 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.

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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)

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)

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.constructions.VectorArrayOperator(array, adjoint=False, space_id=None, name=None)

Bases: pymor.operators.interface.Operator

Wraps a VectorArray as an Operator.

If adjoint 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 adjoint == 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.

adjoint

See description above.

space_id

Id of the source (range) VectorSpace in case adjoint is False (True).

name

The name of the operator.

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

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

as_range_array(mu=None)

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)

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

class pymor.operators.constructions.VectorFunctional(vector, product=None, name=None)

Bases: 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.

linear = True

range

class pymor.operators.constructions.VectorOperator(vector, name=None)

Bases: 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.

linear = True

source

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

Bases: pymor.operators.interface.Operator

The Operator which maps every vector to zero.

Parameters

range

Range VectorSpace of the operator.

source

Source VectorSpace of the operator.

name

Name of the operator.

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

linear = True

apply(U, mu=None)

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.

apply_adjoint(V, mu=None)

Apply the adjoint operator.

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)

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.

apply_inverse_adjoint(U, mu=None, initial_guess=None, least_squares=False)

Apply the inverse adjoint operator.

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.

restricted(dofs)

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).dofs(dofs)
    == restricted_op.apply(NumpyVectorArray(U.dofs(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.

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

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 values mu which are passed to the product.