# Source code for pymor.vectorarrays.interfaces

```
# -*- coding: utf-8 -*-
# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2017 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)
from numbers import Number
import sys
from packaging.version import Version
import numpy as np
from pymor.core.config import config
from pymor.core.interfaces import BasicInterface, ImmutableInterface, abstractmethod
_INDEXTYPES = (Number,) if Version(np.__version__) >= Version('1.9') else (Number, np.intp)
[docs]class VectorArrayInterface(BasicInterface):
"""Interface for vector arrays.
A vector array should be thought of as a list of (possibly high-dimensional) vectors.
While the vectors themselves will be inaccessible in general (e.g. because they are
managed by an external PDE solver code), operations on the vectors like addition can
be performed via this interface.
It is assumed that the number of vectors is small enough such that scalar data
associated to each vector can be handled on the Python side. As such, methods like
:meth:`~VectorArrayInterface.l2_norm` or :meth:`~VectorArrayInterface.gramian` will
always return |NumPy arrays|.
An implementation of the `VectorArrayInterface` via |NumPy arrays| is given by
|NumpyVectorArray|. In general, it is the implementors decision how memory is
allocated internally (e.g. continuous block of memory vs. list of pointers to the
individual vectors.) Thus, no general assumptions can be made on the costs of operations
like appending to or removing vectors from the array. As a hint for 'continuous block
of memory' implementations, :meth:`~VectorSpaceInterface.zeros` provides a `reserve`
keyword argument which allows to specify to what size the array is assumed to grow.
As with |Numpy array|, |VectorArrays| can be indexed with numbers, slices and
lists or one-dimensional |NumPy arrays|. Indexing will always return a new
|VectorArray| which acts as a view into the original data. Thus, if the indexed
array is modified via :meth:`~VectorArrayInterface.scal` or :meth:`~VectorArrayInterface.axpy`,
the vectors in the original array will be changed. Indices may be negative, in
which case the vector is selected by counting from the end of the array. Moreover
indices can be repeated, in which case the corresponding vector is selected several
times. The resulting view will be immutable, however.
.. note::
It is disallowed to append vectors to a |VectorArray| view or to remove
vectors from it. Removing vectors from an array with existing views
will lead to undefined behavior of these views. As such, it is generally
advisable to make a :meth:`~VectorArrayInterface.copy` of a view for long
term storage. Since :meth:`~VectorArrayInterface.copy` has copy-on-write
semantics, this will usually cause little overhead.
Attributes
----------
data
Implementors can provide a `data` property which returns a |NumPy array| of
shape `(len(v), v.dim)` containing the data stored in the array. Access should
be assumed to be slow and is mainly intended for debugging / visualization
purposes or to once transfer data to pyMOR and further process it using NumPy.
In the case of |NumpyVectorArray|, an actual view of the internally used
|NumPy array| is returned, so changing it, will alter the |VectorArray|.
Thus, you cannot assume to own the data returned to you, in general.
dim
The dimension of the vectors in the array.
is_view
`True` if the array is a view obtained by indexing another array.
space
The |VectorSpace| the array belongs to.
"""
is_view = False
[docs] def zeros(self, count=1, reserve=0):
"""Create a |VectorArray| of null vectors of the same |VectorSpace|.
This is a shorthand for `self.space.zeros(count, reserve)`.
Parameters
----------
count
The number of vectors.
reserve
Hint for the backend to which length the array will grow.
Returns
-------
A |VectorArray| containing `count` vectors whith each component
zero.
"""
return self.space.zeros(count, reserve=reserve)
[docs] def empty(self, reserve=0):
"""Create an empty |VectorArray| of the same |VectorSpace|.
This is a shorthand for `self.space.zeros(0, reserve)`.
Parameters
----------
reserve
Hint for the backend to which length the array will grow.
Returns
-------
An empty |VectorArray|.
"""
return self.space.zeros(0, reserve=reserve)
@property
def dim(self):
return self.space.dim
@abstractmethod
@abstractmethod
[docs] def __getitem__(self, ind):
"""Return a |VectorArray| view onto a subset of the vectors in the array."""
pass
@abstractmethod
@abstractmethod
[docs] def append(self, other, remove_from_other=False):
"""Append vectors to the array.
Parameters
----------
other
A |VectorArray| containing the vectors to be appended.
remove_from_other
If `True`, the appended vectors are removed from `other`.
For list-like implementations this can be used to prevent
unnecessary copies of the involved vectors.
"""
pass
@abstractmethod
[docs] def copy(self, deep=False):
"""Returns a copy of a subarray.
All |VectorArray| implementations in pyMOR have copy-on-write semantics:
if not specified otherwise by setting `deep` to `True`, the returned
copy will hold a handle to the same array data as the original array,
and a deep copy of the data will only be performed when one of the arrays
is modified.
Note that for |NumpyVectorArray|, a deep copy is always performed when only
some vectors in the array are copied.
Parameters
----------
deep
Ensure that an actual copy of the array data is made (see above).
Returns
-------
A copy of the |VectorArray|.
"""
pass
@abstractmethod
[docs] def scal(self, alpha):
"""BLAS SCAL operation (in-place scalar multiplication).
This method calculates ::
self = alpha*self
If `alpha` is a scalar, each vector is multiplied by this scalar. Otherwise, `alpha`
has to be a one-dimensional |NumPy array| of the same length as `self`
containing the factors for each vector.
Parameters
----------
alpha
The scalar coefficient or one-dimensional |NumPy array| of coefficients
with which the vectors in `self` are multiplied.
"""
pass
@abstractmethod
[docs] def axpy(self, alpha, x):
"""BLAS AXPY operation.
This method forms the sum ::
self = alpha*x + self
If the length of `x` is 1, the same `x` vector is used for all vectors
in `self`. Otherwise, the lengths of `self` and `x` have to agree.
If `alpha` is a scalar, each `x` vector is multiplied with the same factor `alpha`.
Otherwise, `alpha` has to be a one-dimensional |NumPy array| of the same length as
`self` containing the coefficients for each `x` vector.
Parameters
----------
alpha
The scalar coefficient or one-dimensional |NumPy array| of coefficients with which
the vectors in `x` are multiplied.
x
A |VectorArray| containing the x-summands.
"""
pass
@abstractmethod
[docs] def dot(self, other):
"""Returns the inner products between |VectorArray| elements.
Parameters
----------
other
A |VectorArray| containing the second factors.
Returns
-------
A |NumPy array| `result` such that:
result[i, j] = ( self[i], other[j] ).
"""
pass
[docs] def inner(self, other, product=None):
"""Inner products w.r.t. given product |Operator|.
Equivalent to `self.dot(other)` if `product` is None,
else equivalent to `product.apply2(self, other)`.
"""
if product is None:
return self.dot(other)
else:
return product.apply2(self, other)
@abstractmethod
[docs] def pairwise_dot(self, other):
"""Returns the pairwise inner products between |VectorArray| elements.
Parameters
----------
other
A |VectorArray| containing the second factors.
Returns
-------
A |NumPy array| `result` such that:
result[i] = ( self[i], other[i] ).
"""
pass
[docs] def pairwise_inner(self, other, product=None):
"""Pairwise inner products w.r.t. given product |Operator|.
Equivalent to `self.pairwise_dot(other)` if `product` is None,
else equivalent to `product.pairwise_apply2(self, other)`.
"""
if product is None:
return self.pairwise_dot(other)
else:
return product.pairwise_apply2(self, other)
@abstractmethod
[docs] def lincomb(self, coefficients):
"""Returns linear combinations of the vectors contained in the array.
Parameters
----------
coefficients
A |NumPy array| of dimension 1 or 2 containing the linear
coefficients. `coefficients.shape[-1]` has to agree with
`len(self)`.
Returns
-------
A |VectorArray| `result` such that:
result[i] = ∑ self[j] * coefficients[i,j]
in case `coefficients` is of dimension 2, otherwise
`len(result) == 1` and
result[0] = ∑ self[j] * coefficients[j].
"""
pass
[docs] def norm(self, product=None):
"""Norm w.r.t. given inner product |Operator|.
Equivalent to `self.l2_norm()` if `product` is None,
else equivalent to `np.sqrt(product.pairwise_apply2(self, self))`.
"""
if product is None:
return self.l2_norm()
else:
return np.sqrt(product.pairwise_apply2(self, self))
[docs] def norm2(self, product=None):
"""Squared norm w.r.t. given inner product |Operator|.
Equivalent to `self.l2_norm2()` if `product` is None,
else equivalent to `product.pairwise_apply2(self, self)`.
"""
if product is None:
return self.l2_norm2()
else:
return product.pairwise_apply2(self, self)
@abstractmethod
[docs] def l1_norm(self):
"""The l1-norms of the vectors contained in the array.
Returns
-------
A |NumPy array| `result` such that `result[i]` contains the norm
of `self[i]`.
"""
pass
@abstractmethod
[docs] def l2_norm(self):
"""The l2-norms of the vectors contained in the array.
Returns
-------
A |NumPy array| `result` such that `result[i]` contains the norm
of `self[i]`.
"""
pass
@abstractmethod
[docs] def l2_norm2(self):
"""The squared l2-norms of the vectors contained in the array.
Returns
-------
A |NumPy array| `result` such that `result[i]` contains the norm
of `self[i]`.
"""
pass
[docs] def sup_norm(self):
"""The l-infinity--norms of the vectors contained in the array.
Returns
-------
A |NumPy array| `result` such that `result[i]` contains the norm
of `self[i]`.
"""
if self.dim == 0:
return np.zeros(len(self))
else:
_, max_val = self.amax()
return max_val
@abstractmethod
[docs] def components(self, component_indices):
"""Extract components of the vectors contained in the array.
Parameters
----------
component_indices
List or 1D |NumPy array| of indices of the vector components that are to
be returned.
Returns
-------
A |NumPy array| `result` such that `result[i, j]` is the `component_indices[j]`-th
component of the `i`-th vector of the array.
"""
pass
@abstractmethod
[docs] def amax(self):
"""The maximum absolute value of the vectors contained in the array.
Returns
-------
max_ind
|NumPy array| containing for each vector an index at which the maximum is
attained.
max_val
|NumPy array| containing for each vector the maximum absolute value of its
components.
"""
pass
[docs] def gramian(self, product=None):
"""Shorthand for `self.inner(self, product)`."""
return self.inner(self, product)
[docs] def __add__(self, other):
"""The pairwise sum of two |VectorArrays|."""
if isinstance(other, Number):
assert other == 0
return self.copy()
result = self.copy()
result.axpy(1, other)
return result
[docs] def __iadd__(self, other):
"""In-place pairwise addition of |VectorArrays|."""
self.axpy(1, other)
return self
__radd__ = __add__
[docs] def __sub__(self, other):
"""The pairwise difference of two |VectorArrays|."""
result = self.copy()
result.axpy(-1, other)
return result
[docs] def __isub__(self, other):
"""In-place pairwise difference of |VectorArrays|."""
self.axpy(-1, other)
return self
[docs] def __mul__(self, other):
"""Product by a scalar."""
result = self.copy()
result.scal(other)
return result
__rmul__ = __mul__
[docs] def check_ind(self, ind):
"""Check if `ind` is an admissable list of indices in the sense of the class documentation."""
l = len(self)
return (type(ind) is slice or
isinstance(ind, _INDEXTYPES) and -l <= ind < l or
isinstance(ind, (list, np.ndarray)) and all(-l <= i < l for i in ind))
[docs] def check_ind_unique(self, ind):
"""Check if `ind` is an admissable list of non-repeated indices in the sense of the class documentation."""
l = len(self)
return (type(ind) is slice or
isinstance(ind, _INDEXTYPES) and -l <= ind < l or
isinstance(ind, (list, np.ndarray)) and len(set(i if i >= 0 else l+i for i in ind if -l <= i < l)) == len(ind))
[docs] def len_ind(self, ind):
"""Return the number of given indices."""
l = len(self)
return (len(range(*ind.indices(l))) if type(ind) is slice else
1 if not hasattr(ind, '__len__') else
len(ind))
[docs] def len_ind_unique(self, ind):
"""Return the number of specified unique indices."""
l = len(self)
return (len(range(*ind.indices(l))) if type(ind) is slice else
1 if isinstance(ind, _INDEXTYPES) else
len(set(i if i >= 0 else l+i for i in ind)))
[docs] def normalize_ind(self, ind):
"""Normalize given indices such that they are independent of the array length."""
if type(ind) is slice:
return slice(*ind.indices(len(self)))
elif not hasattr(ind, '__len__'):
ind = ind if 0 <= ind else len(self)+ind
return slice(ind, ind+1)
else:
l = len(self)
return [i if 0 <= i else l+i for i in ind]
[docs] def sub_index(self, ind, ind_ind):
"""Return indices corresponding to the view `self[ind][ind_ind]`"""
if type(ind) is slice:
ind = range(*ind.indices(len(self)))
if type(ind_ind) is slice:
if config.PY2: # make ind a list since is xrange is not sliceable
return list(ind)[ind_ind]
else:
result = ind[ind_ind]
return slice(result.start, result.stop, result.step)
elif hasattr(ind_ind, '__len__'):
return [ind[i] for i in ind_ind]
else:
return [ind[ind_ind]]
else:
if not hasattr(ind, '__len__'):
ind = [ind]
if type(ind_ind) is slice:
return ind[ind_ind]
elif hasattr(ind_ind, '__len__'):
return [ind[i] for i in ind_ind]
else:
return [ind[ind_ind]]
[docs]class VectorSpaceInterface(ImmutableInterface):
"""Class describing a vector space.
Vector spaces act as factories for |VectorArrays| of vectors
contained in them. As such, they hold all data necessary to
create |VectorArrays| of a given type (e.g. the dimension of
the vectors, or a socket for communication with an external
PDE solver).
New |VectorArrays| of null vectors are created via
:meth:`~VectorSpaceInterface.zeros`. The
:meth:`~VectorSpaceInterface.make_array` method builds a new
|VectorArray| from given raw data of the underlying linear algebra
backend (e.g. a |Numpy array| in the case of |NumpyVectorSpace|).
Some vector spaces can create new |VectorArrays| from a given
|Numpy array| via the :meth:`~VectorSpaceInterface.from_data`
method.
Each vector space has a string :attr:`~VectorSpaceInterface.id`
to distinguish mathematically different spaces appearing
in the formulation of a given problem.
Vector spaces can be compared for equality via the `==` and `!=`
operators. To test if a given |VectorArray| is an element of
the space, the `in` operator can be used.
Attributes
----------
id
None, or a string describing the mathematical identity
of the vector space (for instance to distinguish different
components in an equation system).
dim
The dimension (number of degrees of freedom) of the
vectors contained in the space.
is_scalar
Equivalent to `isinstance(space, NumpyVectorSpace) and space.dim == 1`.
"""
id = None
dim = None
is_scalar = False
@abstractmethod
[docs] def make_array(*args, **kwargs):
"""Create a |VectorArray| from raw data.
This method is used in the implementation of |Operators|
and |Discretizations| to create new |VectorArrays| from
raw data of the underlying solver backends. The ownership
of the data is transferred to the newly created array.
The exact signature of this method depends on the wrapped
solver backend.
"""
pass
@abstractmethod
[docs] def zeros(self, count=1, reserve=0):
"""Create a |VectorArray| of null vectors
Parameters
----------
count
The number of vectors.
reserve
Hint for the backend to which length the array will grow.
Returns
-------
A |VectorArray| containing `count` vectors with each component zero.
"""
pass
[docs] def empty(self, reserve=0):
"""Create an empty |VectorArray|
This is a shorthand for `self.zeros(0, reserve)`.
Parameters
----------
reserve
Hint for the backend to which length the array will grow.
Returns
-------
An empty |VectorArray|.
"""
return self.zeros(0, reserve=reserve)
[docs] def from_data(self, data):
"""Create a |VectorArray| from a |NumPy array|
Note that this method will not be supported by all vector
space implementations.
Parameters
----------
data
|NumPy| array.
Returns
-------
A |VectorArray| with `data` as data.
"""
raise NotImplementedError
def __eq__(self, other):
return other is self
def __ne__(self, other):
return not (self == other)
def __contains__(self, other):
return self == getattr(other, 'space', None)
def __hash__(self):
return hash(self.id)
def __repr__(self):
return '{}({})'.format(self.__class__.__name__, self.id)
```