pymor.operators.interface
¶
Module Contents¶
- class pymor.operators.interface.Operator[source]¶
Bases:
pymor.parameters.base.ParametricObject
Interface for
Parameter
dependent discrete operators.An operator in pyMOR is simply a mapping which for any given
parameter values
maps vectors from itssource
VectorSpace
to vectors in itsrange
VectorSpace
.Note that there is no special distinction between functionals and operators in pyMOR. A functional is simply an operator with
NumpyVectorSpace
(1)
as itsrange
VectorSpace
.- solver_options[source]¶
If not
None
, a dict which can contain the following keys:- ‘inverse’:
solver options used for
apply_inverse
- ‘inverse_adjoint’:
solver options used for
apply_inverse_adjoint
- ‘jacobian’:
solver options for the operators returned by
jacobian
(has no effect for linear operators)
If
solver_options
isNone
or a dict entry is missing orNone
, default options are used. The interpretation of the given solver options is up to the operator at hand. In general, values insolver_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.
- source[source]¶
The source
VectorSpace
.
- range[source]¶
The range
VectorSpace
.
- H[source]¶
The adjoint operator, i.e.
self.H.apply(V, mu) == self.apply_adjoint(V, mu)
for all V, mu.
Methods
Try to assemble a linear combination of the given operators.
Apply the operator to a
VectorArray
.Treat the operator as a 2-form and apply it to V and U.
Apply the adjoint operator.
Apply the inverse operator.
Apply the inverse adjoint operator.
Return a
VectorArray
representation of the operator in its range space.Return a
VectorArray
representation of the operator in its source space.Return a vector representation of a linear functional or vector operator.
Assemble the operator for given
parameter values
.Return the operator's derivative with respect to a given parameter.
Return the operator's Jacobian as a new
Operator
.Treat the operator as a 2-form and apply it to V and U in pairs.
Restrict the operator range to a given set of degrees of freedom.
- _assemble_lincomb(operators, coefficients, identity_shift=0.0, solver_options=None, name=None)[source]¶
Try to assemble a linear combination of the given operators.
Returns a new
Operator
which represents the sumc_1*O_1 + ... + c_N*O_N + s*I
where
O_i
areOperators
,c_i
,s
scalar coefficients andI
the identity.This method is called in the
assemble
method ofLincombOperator
on the first of its operators. 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 returnsNone
to indicate that assembly is not possible.Parameters
- operators
List of
Operators
O_i
whose linear combination is formed.- coefficients
List of the corresponding linear coefficients
c_i
.- identity_shift
The coefficient
s
.- solver_options
solver_options
for the assembled operator.- name
Name of the assembled operator.
Returns
The assembled
Operator
if assembly is possible, otherwiseNone
.
- abstract 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.
- apply2(V, U, mu=None)[source]¶
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, thenapply2
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 ofinner
.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)[source]¶
Apply the adjoint operator.
For any given linear
Operator
op
,parameter values
mu
andVectorArrays
U
,V
in thesource
resp.range
we have:op.apply_adjoint(V, mu).dot(U) == V.inner(op.apply(U, mu))
Thus, when
op
is represented by a matrixM
,apply_adjoint
is given by left-multiplication of (the complex conjugate of)M
withV
.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 asV
containing initial guesses for the solution. Some implementations ofapply_inverse
may ignore this parameter. IfNone
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)[source]¶
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 asU
containing initial guesses for the solution. Some implementations ofapply_inverse_adjoint
may ignore this parameter. IfNone
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)[source]¶
Return a
VectorArray
representation of the operator in its range space.In the case of a linear operator with
NumpyVectorSpace
assource
, this method returns for givenparameter values
mu
aVectorArray
V
in the operator’srange
, such thatV.lincomb(U.to_numpy()) == self.apply(U, mu)
for all
VectorArrays
U
.Parameters
- mu
The
parameter values
for which to return theVectorArray
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
asrange
, this method returns for givenparameter values
mu
aVectorArray
V
in the operator’ssource
, such thatself.range.make_array(V.inner(U).T) == self.apply(U, mu)
for all
VectorArrays
U
.Parameters
- mu
The
parameter values
for which to return theVectorArray
representation.
Returns
- V
The
VectorArray
defined above.
- as_vector(mu=None)[source]¶
Return a vector representation of a linear functional or vector operator.
Depending on the operator’s
source
andrange
, this method is equivalent to callingas_range_array
oras_source_array
respectively. The resultingVectorArray
is required to have length 1.Parameters
- mu
The
parameter values
for which to return the vector representation.
Returns
- V
VectorArray
of length 1 containing the vector representation.
- 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 aFixedParameterOperator
. The only assured property of the assembled operator is that it no longer depends on aParameter
.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=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 values
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 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, thenapply2
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 ofpairwise_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.
- abstract 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 arraysource_dofs
such that for anyVectorArray
U
inself.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 fewsource_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 operatorrange
to which to restrict.
Returns
- restricted_op
The restricted operator as defined above. The operator will have
NumpyVectorSpace
(len(source_dofs))
assource
andNumpyVectorSpace
(len(dofs))
asrange
.- source_dofs
One-dimensional
NumPy array
of source degrees of freedom as defined above.